Infinite Dimensional Lie Algebras
Pavel Etingof
scribed/augmented by Darij Grinberg/Raeez Lorgat
URL: http://math.raeez.com
Abstract. 18.747: (unfinished) notes on infinite-dimensional lie algebras held
in the spring semesters of 2012 and 2016 at MIT
Contents
Nomenclature 5
1. The main examples 7
2. Representation theory: generalities 28
3. Representation theory: concrete examples 140
4. Affine Lie algebras 427
5. [unfinished] ... 566
3
4 CONTENTS
Nomenclature
k (Σ) Function field of a curve Σ defined over k
CommIndSch
k
The category of commutative indschemes over a ring k
AJ The Abel Jacobi map taking a divisor p supported on a space X to AJ : p 7→
O
X
(p)I
I
p
BunG The moduli space of G bundles.
B(G) everyone’s favourite quotient category
C The complex number field
D Cartier duality on a category
C
k
The class formation group associated to k.
d = d
a
The formal disc defined over k with the distinguished pt = a labeling the
origin.
p, I
p
A divisor on some ambient space along with its ideal sheaf
DMod(X) The category DMod of D-modules defined on X
F
p
A finite field of p elements; for example the quotient ring Z/hpi
FinIndProSch The category of finite Ind Pro-Schemes
FinIndProSch The category of finite Ind-Schemes
Fields The Category of Fields
G
a
The additive group as commutative reductive algebraic group
G
P
(I, Q) The generalized loop grassmannian of G
G
m
The multiplicative group as commutative reductive algebraic group.
Gr The affine grassmannian of G.
k The ground field, often the complex numbers C, and almost always embedded
in the total field K.
Z [G] The group ring of G
Hil
X
The hilbert scheme of X representing the moduli of finite length subschemes
of X.
Ind N Stacks The Category of Ind-N-Stacks
IndProSch The category of Ind-Pro-Schemes
IndSch
k
The category of IndSchemes defined over k
IndStacks The Category of Ind-Stacks
K The abstract total field
K (( z)) The field of laurent series valued in K; equivalently, the fraction field of series
valued in k; equivalently, the ring of integers of the completed local field.
A LMod The category of left modules over a ring A.
G The classical loop grassmannian of G.
Mat
n×n
k n × n matrices with coefficients in k
A Mod The category of modules over a ring A.
NoethIntSch The category of Integral Noetherian Schemes
NoethSch The category of Noetherian Schemes
5
6 Nomenclature
R The ring of R-valued polynomials in the formal variables z
1
, . . . , z
n
valued in the
ring R.
P
1
The projective line as algebro-geometric object, for example, as represented in
the functor-of-points yoga by two copies of Z[x] glued along a common Z [x, x
1
]
pt/G everyone’s favourite quotient category
Q The rational number field
R The real number field
RH
Right derived functor of homology
O The local ring k [[ z]] ring of formal power series.
A RMod The category of right modules over a ring A.
Sch The category of Schemes
Sch
k
The category of Schemes defined over k
Set The (small) category of Sets
SL
2
The reductive affine algebraic group scheme associated to the Special Linear
Group of invertible determinant 1 matrices i.e. the fiber det
1
(1) in Mat
2×2
.
Equivalently characterized via a functor of points formalism e.g. k Algebras
Set represented in Z Algebras by Z [SL
2
] '
Spec(A), Specf(A [[ z]]) The respective Prime and formal spectra of the commutative
rings A and A [[ z]]
J
B
The tate category associated to a group H, defined as the categorical quotient
A
G
N
G
(A)
Vec
k
Category of Vector Spaces over k
Z The ring of integers.
Z
p
The inverse limit of the inverse system of rings Z/p
n
Z, also called the formal
completion at the point p in the curve Spec(Z)
G A reductive group.
I, I For an ideal I in a commutative ring A, I denotes the associated ideal sheaf on
Spec(A)
X [n] The nth Tate Shift
M
The dual module Hom A(M, A) associated to an object M A Mod for some
ring A
0.1. Notes on the notes. Chapters 1 and 2 of these notes are the most complete,
due to Darij Grinberg. Chapters 3 and 4 are a mess, and are to an even greater extent
potentially incorrect and almost certainly unusable;
Raeez Lorgat has a prelimanry rewrite on chapters 3 and 4; should you wish be
willing to proofread these, email him at rootatraeezdotcom.
It is suggested that the reader at hand a copy of Pasha Etingof’s hand-written
notes
http://www-math.mit.edu/
~
etingof/ The following index into Pasha’s notes may
help (or harm):
(1) Course introduction, Virasoro Algebra.
(2) Dixmier’s Lemma, Fock Spaces, Triangular Decomposition and the first re-
marks on duality.
(3) Genericity, Simple Quotient, Characters, Grading, Definition of Highest-Weight
modules.
(4) Fundamental concepts for the examples of sl]
2
, Vir
¡++¿
1. THE MAIN EXAMPLES 7
0.2. References. The standard text on infinite-dimensional Lie algebras (although
we will not really follow it) is:
V. G. Kac, A. K. Raina, (Bombay Lectures on) Highest Weight Representations
of Infinite Dimensional Lie Algebras, World Scientific 1987.
Further recommended sources are:
Victor G. Kac, Infinite dimensional Lie algebras, Third Edition, CUP 1995.
B. L. Feigin, A. Zelevinsky, Representations of contragredient Lie algebras and
the Kac-Macdonald identities, a paper in: Representations of Lie groups and
Lie algebras (Budapest, 1971), pp. 25-77, Akad. Kiad´o, Budapest, 1985.
0.3. General conventions. We will almost always work over C in this course.
All algebras are over C unless specified otherwise. Characteristic p is too complicated
for us, although very interesting. Sometimes we will work over R, and occasionally
even over rings (as auxiliary constructions require this).
Some remarks on notation:
In the following, N will always denote the set {0, 1, 2, ...} (and not {1, 2, 3, ...}).
All rings are required to have a unity (but not necessarily be commutative).
If R is a ring, then all R-algebras are required to have a unity and satisfy
(λa) b = a (λb) = λ (ab) for all λ R and all a and b in the algebra. (Some
people call such R-algebras central R-algebras, but for us this is part of the
notion of an R-algebra.)
When a Lie algebra g acts on a vector space M, we will denote the image
of an element m M under the action of an element a g by any of the
three notations am, a · m and a * m. (One day, I will probably come to an
agreement with myself and decide which of these notations to use, but for now
expect to see all of them used synonymously in this text. Some authors also
use the notation a m for the image of m under the action of a, but we won’t
use this notation.)
If V is a vector space, then the tensor algebra of V will be denoted by T (V );
the symmetric algebra of V will be denoted by S (V ); the exterior algebra of
V will be denoted by V .
For every n N, we let S
n
denote the n-th symmetric group (that is, the
group of all permutations of the set {1, 2, . . . , n}). On occasion, the notation
S
n
will denote some other things as well; we hope that context will suffice to
keep these meanings apart.
1. The main examples
1.1. The Heisenberg algebra. We start with the definition of the Heisenberg
algebra. Before we formulate it, let us introduce polynomial differential forms on C
×
(in the algebraic sense):
Definition 1.1.1. Recall that C [t, t
1
] denotes the C-algebra of Laurent poly-
nomials in the variable t over C.
Consider the free C [t, t
1
]-module on the basis (dt) (where dt is just a symbol).
The elements of this module are called polynomial differential forms on C
×
. Thus,
polynomial differential forms on C
×
are just formal expressions of the form fdt where
f C [t, t
1
].
8 Nomenclature
Whenever g C [t, t
1
] is a Laurent polynomial, we define a polynomial differ-
ential form dg by dg = g
0
dt. This notation dg does not conflict with the previously
defined notation dt (which was a symbol), because the polynomial t satisfies t
0
= 1.
Definition 1.1.2. For every polynomial differential form fdt on C
×
(with f
C [t, t
1
]), we define a complex number Res
t=0
(fdt) to be the coefficient of the
Laurent polynomial f before t
1
. In other words, we define Res
t=0
(fdt) to be a
1
,
where f is written as
P
iZ
a
i
t
i
(with a
i
C for all i Z).
This number Res
t=0
(fdt) is called the residue of the form fdt at 0.
(The same definition could have been done for Laurent series instead of Laurent
polynomials, but this would require us to consider a slightly different notion of differ-
ential forms, and we do not want to do this here.)
Remark 1.1.3. (a) Every Laurent polynomial f C [t, t
1
] satisfies
Res
t=0
(df) = 0.
(b) Every Laurent polynomial f C [t, t
1
] satisfies Res
t=0
(fdf) = 0.
Proof of Remark 1.1.3. (a) Write f in the form
P
iZ
b
i
t
i
(with b
i
C for all i Z).
Then, f
0
=
P
iZ
ib
i
t
i1
=
P
iZ
(i + 1) b
i+1
t
i
. Now, df = f
0
dt, so that
Res
t=0
(df) = Res
t=0
(f
0
dt) =
the coefficient of the Laurent polynomial f
0
before t
1
= (1 + 1)
| {z }
=0
b
1+1
since f
0
=
X
iZ
(i + 1) b
i+1
t
i
!
= 0,
proving Remark 1.1.3 (a).
(b) First proof of Remark 1.1.3 (b): By the Leibniz identity, (f
2
)
0
= f f
0
+ f
0
f =
2ff
0
, so that ff
0
=
1
2
(f
2
)
0
and thus f df
|{z}
=f
0
dt
= ff
0
|{z}
=
1
2
(f
2
)
0
dt =
1
2
f
2
0
dt
| {z }
=d(f
2
)
=
1
2
d (f
2
). Thus,
Res
t=0
(fdf) = Res
t=0
1
2
d
f
2
=
1
2
Res
t=0
d
f
2

| {z }
=0 (by Remark 1.1.3 (a),
applied to f
2
instead of f)
= 0,
and Remark 1.1.3 (b) is proven.
Second proof of Remark 1.1.3 (b): Write f in the form
P
iZ
b
i
t
i
(with b
i
C for all
i Z). Then, f
0
=
P
iZ
ib
i
t
i1
=
P
iZ
(i + 1) b
i+1
t
i
. Now,
ff
0
=
X
iZ
b
i
t
i
!
X
iZ
(i + 1) b
i+1
t
i
!
=
X
nZ
X
(i,j)Z
2
;
i+j=n
b
i
· (j + 1) b
j+1
t
n
1. THE MAIN EXAMPLES 9
(by the definition of the product of Laurent polynomials). Also, df = f
0
dt, so that
Res
t=0
(fdf) = Res
t=0
(ff
0
dt) =
the coefficient of the Laurent polynomial f f
0
before t
1
=
X
(i,j)Z
2
;
i+j=1
b
i
· (j + 1) b
j+1
since ff
0
=
X
nZ
X
(i,j)Z
2
;
i+j=n
b
i
· (j + 1) b
j+1
t
n
=
X
(i,j)Z
2
;
i+j=0
b
i
· jb
j
(here, we substituted (i, j) for (i, j + 1) in the sum)
=
X
jZ
b
j
· jb
j
=
X
jZ;
j<0
b
j
· jb
j
| {z }
=
P
jZ;
j>0
b
(j)
·(j)b
j
(here, we substituted j for j in the sum)
+ b
0
· 0b
0
| {z }
=0
+
X
jZ;
j>0
b
j
· jb
j
=
X
jZ;
j>0
b
(j)
· (j) b
j
| {z }
=b
j
(j)b
j
=b
j
·jb
j
+
X
jZ;
j>0
b
j
· jb
j
=
X
jZ;
j>0
(b
j
· jb
j
) +
X
jZ;
j>0
b
j
· jb
j
= 0.
This proves Remark 1.1.3 (b).
Note that the first proof of Remark 1.1.3 (b) made use of the fact that 2 is invertible
in C, whereas the second proof works over any commutative ring instead of C.
Now, finally, we define the Heisenberg algebra:
Definition 1.1.4. The oscillator algebra A is the vector space C [t, t
1
] C
endowed with the Lie bracket
[(f, α) , (g, β)] = (0, Res
t=0
(gdf)) .
Since this Lie bracket satisfies the Jacobi identity (because the definition quickly
yields that [[x, y] , z] = 0 for all x, y, z A) and is skew-symmetric (due to Remark
1.1.3 (b)), this A is a Lie algebra.
This oscillator algebra A is also known as the Heisenberg algebra.
Thus, A has a basis
{a
n
| n Z} {K},
where a
n
= (t
n
, 0) and K = (0, 1). The bracket is given by
[a
n
, K] = 0 (thus, K is central) ;
[a
n
, a
m
] =
n,m
K
(in fact, [a
n
, a
n
] = Res
t=0
(t
n
dt
n
) K = Res
t=0
(nt
1
dt) K = nK). Thus, A is a 1-
dimensional central extension of the abelian Lie algebra C [t, t
1
]; this means that we
have a short exact sequence
0
//
CK
//
A
//
C [t, t
1
]
//
0 ,
where CK is contained in the center of A and where C [t, t
1
] is an abelian Lie algebra.
Note that A is a 2-nilpotent Lie algebra. Also note that the center of A is spanned
by a
0
and K.
10 Nomenclature
1.2. The Witt algebra. The next introductory example will be the Lie algebra
of vector fields:
Definition 1.2.1. Consider the free C [t, t
1
]-module on the basis () (where
is just a symbol). This module, regarded as a C-vector space, will be denoted by W .
Thus, the elements of W are formal expressions of the form f where f C [t, t
1
].
(Thus, W
=
C [t, t
1
].)
Define a Lie bracket on the C-vector space W by
[f, g] = (fg
0
gf
0
) for all f C
t, t
1
and g C
t, t
1
.
This Lie bracket is easily seen to be skew-symmetric and satisfy the Jacobi identity.
Thus, it makes W into a Lie algebra. This Lie algebra is called the Witt algebra.
The elements of W are called polynomial vector fields on C
×
.
The symbol is often denoted by
d
dt
.
Remark 1.2.2. It is not by chance that is also known as
d
dt
. In fact, this
notation allows us to view the elements of W as actual polynomial vector fields on
C
×
in the sense of algebraic geometry over C. The Lie bracket of the Witt algebra
W is then exactly the usual Lie bracket of vector fields (because if f C [t, t
1
] and
g C [t, t
1
] are two Laurent polynomials, then a simple application of the Leibniz
rule shows that the commutator of the differential operators f
d
dt
and g
d
dt
is indeed
the differential operator (fg
0
gf
0
)
d
dt
).
A basis of the Witt algebra W is {L
n
| n Z}, where L
n
means t
n+1
d
dt
=
t
n+1
. (Note that some other references like to define L
n
as t
n+1
instead, thus
getting a different sign in many formulas.) It is easy to see that the Lie bracket of the
Witt algebra is given on this basis by
[L
n
, L
m
] = (n m) L
n+m
for every n Z and m Z.
1.3. A digression: Lie groups (and the absence thereof). Let us make some
remarks about the relationship between Lie algebras and Lie groups. In analysis and
geometry, linearizations (tangent spaces etc.) usually only give a crude approximation
of non-linear things (manifolds etc.). This is what makes the theory of Lie groups
special: The linearization of a finite-dimensional Lie group (i. e., its corresponding
Lie algebra) carries very much information about the Lie group. The relation between
finite-dimensional Lie groups and finite-dimensional Lie algebras is almost a one-to-
one correspondence (at least if we restrict ourselves to simply connected Lie groups).
This correspondence breaks down in the infinite-dimensional case. There are lots of
important infinite-dimensional Lie groups, but their relation to Lie algebras is not as
close as in the finite-dimensional case anymore. One example for this is that there is
no Lie group corresponding to the Witt algebra W . There are a few things that come
close to such a Lie group:
We can consider the real subalgebra W
R
of W , consisting of the vector fields in W
which are tangent to S
1
(the unit circle in C). This is a real Lie algebra satisfying
W
R
R
C
=
W (thus, W
R
is what is called a real form of W ). And we can say that
d
W
R
= Lie (Diff S
1
) (where Diff S
1
denotes the group of all diffeomorphisms S
1
S
1
)
1. THE MAIN EXAMPLES 11
for some kind of completion
d
W
R
of W
R
(although W
R
itself is not the Lie algebra of
any Lie group).
1
Now if we take two one-parameter families
g
s
Diff S
1
, g
s
|
s=0
= id, g
0
s
|
s=0
= ϕ;
h
u
Diff S
1
, h
u
|
u=0
= id, h
0
u
|
u=0
= ψ,
then
g
s
(θ) = θ + (θ) + O
s
2
;
h
u
(θ) = θ + (θ) + O
u
2
;
g
s
h
u
g
1
s
h
1
u
(θ) = θ + su (ϕψ
0
ψϕ
0
) (θ) + (cubic terms in s and u and higher) .
So we get something resembling the standard Lie-group-Lie-algebra correspondence,
but only for the completion of the real part. For the complex one, some people have
done some work yielding something like Lie semigroups (the so-called “semigroup of
annuli” of G. Segal), but no Lie groups.
Anyway, this was a digression, just to show that we don’t have Lie groups corre-
sponding to our Lie algebras. Still, this should not keep us from heuristically thinking
of Lie algebras as linearizations of Lie groups. We can even formalize this heuristic, by
using the purely algebraic notion of formal groups.
1.4. The Witt algebra acts on the Heisenberg algebra by derivations.
Let’s return to topic. The following proposition is a variation on a well-known theme:
Proposition 1.4.1. Let n be a Lie algebra. Let f : n n and g : n n be
two derivations of n. Then, [f, g] is a derivation of n. (Here, the Lie bracket is to be
understood as the Lie bracket on End n, so that we have [f, g] = f g g f.)
1
Here is how this completion
d
W
R
is defined exactly: Notice that
W
R
=
ϕ (θ)
d
|
ϕ is a trigonometric polynomial, i. e.,
ϕ (θ) = a
0
+
P
n>0
a
n
cos +
P
n>0
b
n
sin
where both sums are finite
,
where θ =
1
i
ln t and
d
= it
d
dt
. Now, define the completion
d
W
R
by
d
W
R
=
ϕ (θ)
d
|
ϕ (θ) = a
0
+
P
n>0
a
n
cos +
P
n>0
b
n
sin
where both sums are infinite sums with rapidly
decreasing coefficients
.
12 Nomenclature
Proof of Proposition 1.4.1. Let a n and b n. Since f is a derivation, we have
f ([a, b]) = [f (a) , b] + [a, f (b)]. Thus,
(g f) ([a, b]) = g
f ([a, b])
| {z }
=[f(a),b]+[a,f(b)]
= g ([f (a) , b] + [a, f (b)])
= g ([f (a) , b])
| {z }
=[g(f(a)),b]+[f(a),g(b)]
(since g is a derivation)
+ g ([a, f (b)])
| {z }
=[g(a),f(b)]+[a,g(f (b))]
(since g is a derivation)
=
g (f (a))
| {z }
=(gf)(a)
, b
+ [f (a) , g (b)] + [g (a) , f (b)] +
a, g (f (b))
| {z }
=(gf)(b)
= [(g f) (a) , b] + [f (a) , g (b)] + [g (a) , f (b)] + [a, (g f) (b)] .
The same argument, with f and g replaced by g and f, shows that
(f g) ([a, b]) = [(f g) (a) , b] + [g (a) , f (b)] + [f (a) , g (b)] + [a, (f g) (b)] .
Thus,
[f, g]
|{z}
=fggf
([a, b]) = (f g g f) ([a, b])
= (f g) ([a, b])
| {z }
=[(fg)(a),b]+[g(a),f (b)]+[f (a),g(b)]+[a,(fg)(b)]
(g f) ([a, b])
| {z }
=[(gf)(a),b]+[f(a),g(b)]+[g(a),f (b)]+[a,(gf)(b)]
= ([(f g) (a) , b] + [g (a) , f (b)] + [f (a) , g (b)] + [a, (f g) (b)])
([(g f) (a) , b] + [f (a) , g (b)] + [g (a) , f (b)] + [a, (g f) (b)])
= [(f g) (a) , b] [(g f) (a) , b]
| {z }
=[(fg)(a)(gf )(a),b]
+ [a, (f g) (b)] [a, (g f) (b)]
| {z }
=[a,(fg)(b)(gf )(b)]
=
(f g) (a) (g f) (a)
| {z }
=(fggf )(a)
, b
+
a, (f g) (b) (g f) (b)
| {z }
=(fggf )(b)
=
(f g g f)
| {z }
=[f,g]
(a) , b
+
a, (f g g f)
| {z }
=[f,g]
(b)
= [[f, g] (a) , b] + [a, [f, g] (b)] .
We have thus proven that any a n and b n satisfy [f, g] ([a, b]) = [[f, g] (a) , b] +
[a, [f, g] (b)]. In other words, [f, g] is a derivation. This proves Proposition 1.4.1.
Definition 1.4.2. For every Lie algebra g, we will denote by Der g the Lie
subalgebra {f End g | f is a derivation} of End g. (This is well-defined because
Proposition 1.4.1 shows that {f End g | f is a derivation} is a Lie subalgebra of
End g.) We call Der g the Lie algebra of derivations of g.
Lemma 1.4.3. There is a natural homomorphism η : W Der A of Lie algebras
given by
(η (f)) (g, α) = (fg
0
, 0) for all f C
t, t
1
, g C
t, t
1
and α C.
1. THE MAIN EXAMPLES 13
First proof of Lemma 1.4.3. Lemma 1.4.3 can be proven by direct calculation:
For every f W , the map
A A, (g, α) 7→ (f g
0
, 0)
is a derivation of A
2
, thus lies in Der A. Hence, we can define a map η : W Der A
by
η (f) = (A A, (g, α) 7→ (fg
0
, 0)) for all f C
t, t
1
.
In other words, we can define a map η : W Der A by
(η (f)) (g, α) = (fg
0
, 0) for all f C
t, t
1
, g C
t, t
1
and α C.
Now, it remains to show that this map η is a homomorphism of Lie algebras.
In fact, any f
1
C [t, t
1
] and f
2
C [t, t
1
] and any g C [t, t
1
] and α C satisfy
η
[f
1
, f
2
]
| {z }
=
(
f
1
f
0
2
f
2
f
0
1
)
(g, α) = (η ((f
1
f
0
2
f
2
f
0
1
) )) (g, α) = ((f
1
f
0
2
f
2
f
0
1
) g
0
, 0)
2
Proof. Let f be an element of W . (In other words, let f be an element of C
t, t
1
.) Let τ
denote the map
A A, (g, α) 7→ (fg
0
, 0) .
Then, we must prove that τ is a derivation of A.
In fact, first it is clear that τ is C-linear. Moreover, any (u, β) A and (v, γ) A satisfy
τ
[(u, β) , (v, γ)]
| {z }
=(0,Res
t=0
(vdu))
= τ (0, Res
t=0
(vdu)) = (f0, 0) (by the definition of τ)
= (0, 0)
and
τ (u, β)
| {z }
=(fu
0
,0)
, (v, γ)
+
(u, β) , τ (v, γ)
| {z }
=(fv
0
,0)
= [(fu
0
, 0) , (v, γ)]
| {z }
=(0,Res
t=0
(vd(fu
0
)))
+ [(u, β) , (fv
0
, 0)]
| {z }
=(0,Res
t=0
(fv
0
du))
= (0, Res
t=0
(vd (fu
0
))) + (0, Res
t=0
(fv
0
du))
= (0, Res
t=0
(vd (fu
0
) + fv
0
du)) = (0, Res
t=0
(d (vfu
0
)))
since v d (fu
0
)
| {z }
=(fu
0
)
0
dt
+fv
0
du
|{z}
=u
0
dt
= v (fu
0
)
0
dt + fv
0
u
0
dt
=
v (fu
0
)
0
+ f v
0
u
0
dt =
v (fu
0
)
0
+ v
0
(fu
0
)
| {z }
=(vfu
0
)
0
dt = (vfu
0
)
0
dt = d (vf u
0
)
= (0, 0) (since Remark 1.1.3 (a) (applied to vfu
0
instead of f) yields Res
t=0
(d (vfu
0
)) = 0) ,
so that τ ([(u, β) , (v, γ)]) = [τ (u, β) , (v, γ)] + [(u, β) , τ (v, γ)]. Thus, τ is a derivation of A, qed.
14 Nomenclature
and
[η (f
1
) , η (f
2
)] (g, α)
= (η (f
1
)) ((η (f
2
)) (g, α))
| {z }
=(f
2
g
0
,0)
(η (f
2
)) ((η (f
1
)) (g, α))
| {z }
=(f
1
g
0
,0)
= (η (f
1
)) (f
2
g
0
, 0)
| {z }
=
(
f
1
(f
2
g
0
)
0
,0
)
(η (f
2
)) (f
1
g
0
, 0)
| {z }
=
(
f
2
(f
1
g
0
)
0
,0
)
=
f
1
(f
2
g
0
)
0
, 0
f
2
(f
1
g
0
)
0
, 0
=
f
1
(f
2
g
0
)
0
f
2
(f
1
g
0
)
0
, 0
= ((f
1
f
0
2
f
2
f
0
1
) g
0
, 0)
since f
1
(f
2
g
0
)
0
|{z}
=f
0
2
g
0
+f
2
g
00
f
2
(f
1
g
0
)
0
|{z}
=f
0
1
g
0
+f
1
g
00
= f
1
(f
0
2
g
0
+ f
2
g
00
) f
2
(f
0
1
g
0
+ f
1
g
00
)
= f
1
f
0
2
g
0
+ f
1
f
2
g
00
f
2
f
0
1
g
0
f
1
f
2
g
00
= f
1
f
0
2
g
0
f
2
f
0
1
g
0
= (f
1
f
0
2
f
2
f
0
1
) g
0
,
so that
(η ([f
1
, f
2
])) (g, α) = ((f
1
f
0
2
f
2
f
0
1
) g
0
, 0) = [η (f
1
) , η (f
2
)] (g, α) .
Thus, any f
1
C [t, t
1
] and f
2
C [t, t
1
] satisfy η ([f
1
, f
2
])) = [η (f
1
) , η (f
2
)].
This proves that η is a Lie algebra homomorphism, and thus Lemma 1.4.3 is proven.
Second proof of Lemma 1.4.3 (sketched). The following proof I don’t understand,
so don’t expect my version of it to make any sense. See Akhil Matthew’s blog post
http://amathew.wordpress.com/2012/03/01/the-heisenberg-and-witt-algebras/
for a much better writeup.
The following proof is a bit of an overkill; however, it is supposed to provide some
motivation for Lemma 1.4.3. We won’t be working completely formally, so the reader
should expect some imprecision.
Let us really interpret the elements of W as vector fields on C
×
. The bracket [·, ·]
of the Lie algebra A was defined in an invariant way:
[f, g] = Res
t=0
(gdf) =
1
2πi
I
|z|=1
gdf (by Cauchy’s residue theorem)
is an integral of a 1-form, thus invariant under diffeomorphisms, thus invariant under
“infinitesimal diffeomorphisms” such as the ones given by elements of W . Thus, Lemma
1.4.3 becomes obvious. [This proof needs revision.]
The first of these two proofs is obviously the more straightforward one (and gener-
alizes better to fields other than C), but it does not offer any explanation why Lemma
1.4.3 is more than a mere coincidence. Meanwhile, the second proof gives Lemma 1.4.3
a philosophical reason to be true.
1.5. The Virasoro algebra. In representation theory, one often doesn’t encounter
representations of W directly, but instead one finds representations of a 1-dimensional
central extension of W called the Virasoro algebra. I will now construct this extension
and show that it is the only one (up to isomorphism of extensions).
Let us recollect the theory of central extensions of Lie algebras (more precisely, the
1-dimensional ones):
Definition 1.5.1. If L is a Lie algebra, then a 1-dimensional central extension
of L is a Lie algebra
b
L along with an exact sequence
(1) 0 C
b
L L 0,
1. THE MAIN EXAMPLES 15
where C is central in
b
L. Since all exact sequences of vector spaces split, we can pick
a splitting of this exact sequence on the level of vector spaces, and thus identify
b
L
with L C as a vector space (not as a Lie algebra). Upon this identification, the
Lie bracket of
b
L can be written as
(2) [(a, α) , (b, β)] = ([a, b] , ω (a, b)) for a L, α C, b L, β C,
for some skew-symmetric bilinear form ω : L × L C. (We can also write this
skew-symmetric bilinear form ω : L × L C as a linear form
2
L C.) But ω
cannot be a completely arbitrary skew-symmetric bilinear form. It needs to satisfy
the so-called 2-cocycle condition
(3) ω ([a, b] , c) + ω ([b, c] , a) + ω ([c, a] , b) = 0 for all a, b, c L.
This condition comes from the requirement that the bracket in
b
L have to satisfy the
Jacobi identity.
In the following, a 2-cocycle on L will mean a skew-symmetric bilinear form
ω : L × L C (not necessarily obtained from a central extension!) which satisfies
the equation (3). (The name “2-cocycle” comes from Lie algebra cohomology, where
2-cocycles are indeed the cocycles in the 2-nd degree.) Thus, we have assigned a 2-
cocycle on L to every 1-dimensional central extension of L (although the assignment
depended on the splitting).
Conversely, if ω is any 2-cocycle on L, then we can define a 1-dimensional central
extension
b
L
ω
of L such that the 2-cocycle corresponding to this extension is ω. In
fact, we can construct such a central extension
b
L
ω
by setting
b
L
ω
= L C as a vector
space, and defining the Lie bracket on this vector space by (2). (The maps C
b
L
ω
and
b
L
ω
L are the canonical ones coming from the direct sum decomposition
b
L
ω
= L C.) Thus, every 2-cocycle on L canonically determines a 1-dimensional
central extension of L.
However, our assignment of the 2-cocycle ω to the central extension
b
L was not
canonical, but depended on the splitting of the exact sequence (1). If we change
the splitting by some ξ L
, then ω is changed by (this means that ω is being
replaced by ω + ), where is the 2-cocycle on L defined by
(a, b) = ξ ([a, b]) for all a, b L.
The 2-cocycle is called a 2-coboundary. As a conclusion, 1-dimensional central
extensions of L are parametrized up to isomorphism by the vector space
(2-cocycles) (2-coboundaries) = H
2
(L) .
(Note that “up to isomorphism” means “up to isomorphism of extensions” here, not
“up to isomorphism of Lie algebras”.) The vector space H
2
(L) is called the 2-nd
cohomology space (or just the 2-nd cohomology) of the Lie algebra L.
Theorem 1.5.2. The vector space H
2
(W ) is 1-dimensional and is spanned by
the residue class of the 2-cocycle ω given by
ω (L
n
, L
m
) =
n
3
n
6
δ
n,m
for all n, m Z.
Note that in this theorem, we could have replaced the factor
n
3
n
6
by n
3
n (since
the vector space spanned by a vector obviously doesn’t change if we rescale the vector
16 Nomenclature
by a nonzero scalar factor), or even by n
3
(since the 2-cocycle (L
n
, L
m
) 7→
n,m
is
a coboundary, and two 2-cocycles which differ by a coboundary give the same residue
class in H
2
(W )). But we prefer
n
3
n
6
since this is closer to how this class appears in
representation theory (and, also, comes up in the proof below).
Proof of Theorem 1.5.2. First of all, it is easy to prove by computation that the
bilinear form ω : W × W C given by
ω (L
n
, L
m
) =
n
3
n
6
δ
n,m
for all n, m Z
is indeed a 2-cocycle. Now, let us prove that every 2-cocycle on W is congruent to a
multiple of ω modulo the 2-coboundaries.
Let β be a 2-cocycle on W . We must prove that β is congruent to a multiple of ω
modulo the 2-coboundaries.
Pick ξ W
such that ξ (L
n
) =
1
n
β (L
n
, L
0
) for all n 6= 0 (such a ξ clearly exists,
but is not unique since we have complete freedom in choosing ξ (L
0
)). Let
e
β be the
2-cocycle β . Then,
e
β (L
n
, L
0
) = β (L
n
, L
0
)
| {z }
=(L
n
)
(since ξ(L
n
)=
1
n
β(L
n
,L
0
))
ξ
[L
n
, L
0
]
| {z }
=nL
n
= (L
n
) ξ (nL
n
) = 0
for every n 6= 0. Thus, by replacing β by
e
β, we can WLOG assume that β (L
n
, L
0
) = 0
for every n 6= 0. This clearly also holds for n = 0 since β is skew-symmetric. Hence,
β (X, L
0
) = 0 for every X W . Now, by the 2-cocycle condition, we have
β ([L
0
, L
m
] , L
n
) + β ([L
n
, L
0
] , L
m
) + β ([L
m
, L
n
] , L
0
) = 0
for all n Z and m Z. Thus,
0 = β
[L
0
, L
m
]
| {z }
=mL
m
, L
n
+ β
[L
n
, L
0
]
| {z }
=nL
n
, L
m
+ β ([L
m
, L
n
] , L
0
)
| {z }
=0 (since β(X,L
0
)=0 for every XW )
= m β (L
m
, L
n
)
| {z }
=β(L
n
,L
m
)
(since β is skew-symmetric)
+ (L
n
, L
m
) = (L
n
, L
m
) + (L
n
, L
m
)
= (n + m) β (L
n
, L
m
)
for all n Z and m Z. Hence, for all n Z and m Z with n + m 6= 0, we have
β (L
n
, L
m
) = 0. In other words, there exists some sequence (b
n
)
nZ
C
Z
such that
(4) β (L
n
, L
m
) = b
n
δ
n,m
for all n Z and m Z.
This sequence satisfies
(5) b
n
= b
n
for every n Z
(since β is skew-symmetric and thus β (L
n
, L
n
) = β (L
n
, L
n
)) and thus, in partic-
ular, b
0
= 0. We will now try to get a recursive equation for this sequence.
Let m, n and p be three integers satisfying m + n + p = 0. Then, the 2-cocycle
condition yields
β ([L
p
, L
n
] , L
m
) + β ([L
m
, L
p
] , L
n
) + β ([L
n
, L
m
] , L
p
) = 0.
1. THE MAIN EXAMPLES 17
Due to
β
[L
p
, L
n
]
| {z }
=(pn)L
p+n
, L
m
= (p n) β (L
p+n
, L
m
)
| {z }
=β(L
m
,L
p+n
)
(since β is skew-symmetric)
= (p n) β (L
m
, L
p+n
)
| {z }
=b
m
δ
m,(p+n)
(by (4))
= (p n) b
m
δ
m,(p+n)
| {z }
=1
(since m+n+p=0)
= (p n) b
m
and the two cyclic permutations of this equality, this rewrites as
((p n) b
m
) + ((m p) b
n
) + ((n m) b
p
) = 0.
In other words,
(6) (n m) b
p
+ (m p) b
n
+ (p n) b
m
= 0.
Now define a form ξ
0
W
by ξ
0
(L
0
) = 1 and ξ
0
(L
i
) = 0 for all i 6= 0.
By replacing β with β
b
1
2
0
, we can assume WLOG that b
1
= 0.
Now let n Z be arbitrary. Setting m = 1 and p = (n + 1) in (6) (this is allowed
since 1 + n + ((n + 1)) = 0), we get
(n 1) b
(n+1)
+ (1 ((n + 1))) b
n
+ (n 1) b
1
= 0.
Thus,
0 = (n 1) b
(n+1)
| {z }
=b
n+1
(by (5))
+ (1 ((n + 1)))
| {z }
=n+2
b
n
+ (n 1) b
1
|{z}
=0
= (n 1) b
n+1
+ (n + 2) b
n
,
so that (n 1) b
n+1
= (n + 2) b
n
. This recurrence equation rewrites as b
n+1
=
n + 2
n 1
b
n
for n 2. Thus, by induction we see that every n 2 satisfies
b
n
=
n + 1
n 2
·
n
n 3
·
n 1
n 4
·...·
4
1
b
2
=
(n + 1) ·n ·... · 4
(n 2) ·(n 3) · ... · 1
b
2
=
(n + 1) (n 1) n
6
b
2
=
n
3
n
6
b
2
.
But b
n
=
n
3
n
6
b
2
also holds for n = 1 (since b
1
= 0 and
1
3
1
6
= 0) and for n = 0
(since b
0
= 0 and
0
3
0
6
= 0). Hence, b
n
=
n
3
n
6
b
2
holds for every n 0. By (5),
we conclude that b
n
=
n
3
n
6
b
2
holds also for every n 0. Thus, every n Z satisfies
b
n
=
n
3
n
6
b
2
. From (4), we thus see that β is a scalar multiple of ω.
We thus have proven that every 2-cocycle β on W is congruent to a multiple of ω
modulo the 2-coboundaries. This yields that the space H
2
(W ) is at most 1-dimensional
and is spanned by the residue class of the 2-cocycle ω. In order to complete the proof
of Theorem 1.5.2, we have yet to prove that H
2
(W ) is indeed 1-dimensional (and not
18 Nomenclature
0-dimensional), i. e., that the 2-cocycle ω is not a 2-coboundary. But this is easy
3
.
The proof of Theorem 1.5.2 is thus complete.
The 2-cocycle
1
2
ω (where ω is the 2-cocycle introduced in Theorem 1.5.2) gives a
central extension of the Witt algebra W : the so-called Virasoro algebra. Let us recast
the definition of this algebra in elementary terms:
Definition 1.5.3. The Virasoro algebra Vir is defined as the vector space W C
with Lie bracket defined by
[L
n
, L
m
] = (n m) L
n+m
+
n
3
n
12
δ
n,m
C;
[L
n
, C] = 0,
where L
n
denotes (L
n
, 0) for every n Z, and where C denotes (0, 1). Note that
{L
n
| n Z} {C} is a basis of Vir.
If we change the denominator 12 to any other nonzero complex number, we get a
Lie algebra isomorphic to Vir (it is just a rescaling of C). It is easy to show that the
Virasoro algebra is not isomorphic to the Lie-algebraic direct sum W C. Thus, Vir is
the unique (up to Lie algebra isomorphism) nontrivial 1-dimensional central extension
of W .
1.6. Recollection on g-invariant forms. Before we show the next important
family of infinite-dimensional Lie algebras, let us define some standard notions. First,
let us define the notion of a g-invariant form, in full generality (that is, for any two
g-modules):
Definition 1.6.1. Let g be a Lie algebra over a field k. Let M and N be two
g-modules. Let β : M × N k be a k-bilinear form. Then, this form β is said to
be g-invariant if and only if every x g, a M and b N satisfy
β (x * a, b) + β (a, x * b) = 0.
Instead of g-invariant”, one often says “invariant”.
The following remark gives an alternative characterization of g-invariant bilinear
forms (which is occasionally used as an alternative definition thereof):
3
Proof. Assume the contrary. Then, the 2-cocycle ω is a 2-coboundary. This means that there
exists a linear map η : W C such that ω = . Pick such a η. Then,
ω (L
2
, L
2
) = () (L
2
, L
2
) = η
[L
2
, L
2
]
| {z }
=4L
0
= 4η (L
0
)
and
ω (L
1
, L
1
) = () (L
1
, L
1
) = η
[L
1
, L
1
]
| {z }
=2L
0
= 2η (L
0
) .
Hence,
2 ω (L
1
, L
1
)
| {z }
=2η(L
0
)
= 4η (L
0
) = ω (L
2
, L
2
) .
But this contradicts with the equalities ω (L
1
, L
1
) = 0 and ω (L
2
, L
2
) = 1 (which easily follow from
the definition of ω). This contradiction shows that our assumption was wrong, and thus the 2-cocycle
ω is not a 2-coboundary, qed.
1. THE MAIN EXAMPLES 19
Remark 1.6.2. Let g be a Lie algebra over a field k. Let M and N be two
g-modules. Consider the tensor product M N of the two g-modules M and N;
this is known to be a g-module again. Consider also k as a g-module (with the trivial
g-module structure).
Let β : M × N k be a k-bilinear form. Let B be the linear map M N k
induced by the k-bilinear map β : M × N k using the universal property of the
tensor product.
Then, β is g-invariant if and only if B is a g-module homomorphism.
Proof of Remark 1.6.2. We know that B is the linear map M N k induced by
the k-bilinear map β : M ×N k using the universal property of the tensor product.
Hence, any a M and b N satisfy
(7) B (a b) = β (a, b) .
We are going to prove the following two assertions:
Assertion 1.6.2.1: If β is g-invariant, then B is a g-module homomorphism.
Assertion 1.6.2.2: If B is a g-module homomorphism, then β is g-invariant.
Proof of Assertion 1.6.2.1: Assume that β is g-invariant. Therefore, every x g,
a M and b N satisfy
(8) β (x * a, b) + β (a, x * b) = 0
(because Definition 1.6.1 states that β is g-invariant if and only if every x g, a M
and b N satisfy (8)).
Now, let x g and u M N. Since u is a tensor in M N, we can write u in the
form u =
n
P
i=1
λ
i
a
i
b
i
for some n N, some elements λ
1
, λ
2
, ..., λ
n
of k, some elements
a
1
, a
2
, ..., a
n
of M and some elements b
1
, b
2
, ..., b
n
of N . Consider this n, these λ
1
, λ
2
,
..., λ
n
, these a
1
, a
2
, ..., a
n
, and these b
1
, b
2
, ..., b
n
.
Since u =
n
P
i=1
λ
i
a
i
b
i
, we have
x * u = x *
n
X
i=1
λ
i
a
i
b
i
!
=
n
X
i=1
λ
i
x * (a
i
b
i
)
| {z }
=(x*a
i
)b
i
+a
i
(x*b
i
)
(by the definition of the g-module MN)
=
n
X
i=1
λ
i
((x * a
i
) b
i
+ a
i
(x * b
i
)) .
20 Nomenclature
Hence,
B (x * u) = B
n
X
i=1
λ
i
((x * a
i
) b
i
+ a
i
(x * b
i
))
!
=
n
X
i=1
λ
i
B ((x * a
i
) b
i
+ a
i
(x * b
i
))
| {z }
=B((x*a
i
)b
i
)+B(a
i
(x*b
i
))
(since B is k-linear)
(since B is k-linear)
=
n
X
i=1
λ
i
B ((x * a
i
) b
i
)
| {z }
=β(x*a
i
,b
i
)
(by (7), applied
to x*a
i
and b
i
instead of a and b)
+ B (a
i
(x * b
i
))
| {z }
=β(a
i
,x*b
i
)
(by (7), applied
to a
i
and x*b
i
instead of a and b)
=
n
X
i=1
λ
i
(β (x * a
i
, b
i
) + β (a
i
, x * b
i
))
| {z }
=0
(by (8), applied to
a=a
i
and b=b
i
)
=
n
X
i=1
λ
i
0 = 0.
Comparing this with x * (B (u)) = 0 (because the g-module structure on k is trivial),
this yields B (x * u) = x * (B (u)).
Now, forget that we fixed x and u. We thus have shown that B (x * u) = x *
(B (u)) for all x g and u M N. In other words, the map B is a g-module
homomorphism. This proves Assertion 1.6.2.1.
Proof of Assertion 1.6.2.2: Assume that B is a g-module homomorphism. Now, let
x g, a M and b N. By the definition of the g-module M N, we have
x * (a b) = (x * a) b + a (x * b) ,
so that
B (x * (a b)) = B ((x * a) b + a (x * b))
= B ((x * a) b)
| {z }
=β(x*a,b)
(by (7), applied
to x*a instead of a)
+ B (a (x * b))
| {z }
=β(a,x*b)
(by (7), applied
to x*b instead of b)
(since B is k-linear)
= β (x * a, b) + β (a, x * b) .
Comparing this with
B (x * (a b)) = x * (B (a b)) (since B is a g-module homomorphism)
= 0 (since the g-module structure on k is trivial) ,
this yields β (x * a, b) + β (a, x * b) = 0.
Now, forget that we fixed x, a and b. We thus have shown that every x g, a M
and b N satisfy
(9) β (x * a, b) + β (a, x * b) = 0.
In other words, β is g-invariant (because Definition 1.6.1 states that β is g-invariant if
and only if every x g, a M and b N satisfy (9)). This proves Assertion 1.6.2.2.
1. THE MAIN EXAMPLES 21
Now, both Assertion 1.6.2.1 and Assertion 1.6.2.2 are proven. Combining these two
assertions, we conclude that β is g-invariant if and only if B is a g-module homomor-
phism. This proves Remark 1.6.2.
Very often, the notion of a g-invariant” bilinear form (as defined in Definition
1.6.1) is applied to forms on g itself. In this case, it has to be interpreted as follows:
Convention 1.6.3. Let g be a Lie algebra over a field k. Let β : g ×g k be
a bilinear form. When we say that β is g-invariant without specifying the g-module
structure on g, we always tacitly understand that the g-module structure on g is the
adjoint one (i. e., the one defined by x * a = [x, a] for all x g and a g).
The following remark provides two equivalent criteria for a bilinear form on the Lie
algebra g itself to be g-invariant; they will often be used tacitly:
Remark 1.6.4. Let g be a Lie algebra over a field k. Let β : g × g k be a
k-bilinear form.
(a) The form β is g-invariant if and only if every elements a, b and c of g satisfy
β ([a, b] , c) + β (b, [a, c]) = 0.
(b) The form β is g-invariant if and only if every elements a, b and c of g satisfy
β ([a, b] , c) = β (a, [b, c]).
Proof of Remark 1.6.4. Consider g as a g-module using the adjoint action. Then,
(10) x * a = [x, a] for any x g and a g.
(a) By Definition 1.6.1 (applied to M = g and N = g), we know that the form β is g-
invariant if and only if every x g, a g and b g satisfy β (x * a, b)+β (a, x * b) =
0. Thus, we have the following equivalence of assertions:
(the form β is g-invariant)
every x g, a g and b g satisfy β
x * a
|{z}
=[x,a]
(by (10))
, b
+ β
a, x * b
|{z}
=[x,b]
(by (10), applied
to b instead of a)
= 0
(every x g, a g and b g satisfy β ([x, a] , b) + β (a, [x, b]) = 0)
(11)
(every a g, b g and c g satisfy β ([a, b] , c) + β (b, [a, c]) = 0)
(here, we renamed the indices x, a and b as a, b and c)
(every elements a, b and c of g satisfy β ([a, b] , c) + β (b, [a, c]) = 0) .
In other words, Remark 1.6.4 (a) is proven.
22 Nomenclature
(b) We have the following equivalence of assertions:
(the form β is g-invariant)
(every x g, a g and b g satisfy β ([x, a] , b) + β (a, [x, b]) = 0)
(by (11))
(every b g, a g and c g satisfy β ([b, a] , c) + β (a, [b, c]) = 0)
(here, we renamed the indices x and b as b and c)
(every elements a, b and c of g satisfy β ([b, a] , c) + β (a, [b, c]) = 0)
(every elements a, b and c of g satisfy β ([a, b] , c) + β (a, [b, c]) = 0)
since every elements a, b and c of g satisfy
β
[b, a]
|{z}
=[a,b]
, c
= β ([a, b] , c) = β ([a, b] , c)
(since β is k-bilinear)
(every elements a, b and c of g satisfy β (a, [b, c]) = β ([a, b] , c))
(every elements a, b and c of g satisfy β ([a, b] , c) = β (a, [b, c])) .
In other words, Remark 1.6.4 (b) is proven.
An example of a g-invariant bilinear form on g itself for g finite-dimensional is given
by the so-called Killing form:
Proposition 1.6.5. Let g be a finite-dimensional Lie algebra over a field k.
Then, the form
g ×g k,
(x, y) 7→ Tr
g
((ad x) (ad y))
is a symmetric g-invariant bilinear form. This form is called the Killing form of the
Lie algebra g.
Proposition 1.6.6. Let g be a finite-dimensional semisimple Lie algebra over
C.
(a) The Killing form of g is nondegenerate.
(b) Any g-invariant bilinear form on g is a scalar multiple of the Killing form
of g. (Hence, if g 6= 0, then the vector space of g-invariant bilinear forms on g is
1-dimensional and spanned by the Killing form.)
1.7. Affine Lie algebras. Now let us introduce the so-called affine Lie algebras;
this is a very general construction from which a lot of infinite-dimensional Lie algebras
emerge (including the Heisenberg algebra defined above).
Definition 1.7.1. Let g be a Lie algebra.
(a) The C-Lie algebra g induces (by extension of scalars) a C [t, t
1
]-Lie algebra
C
t, t
1
g =
(
X
iZ
a
i
t
i
| a
i
g; all but finitely many i Z satisfy a
i
= 0
)
.
This Lie algebra C [t, t
1
] g, considered as a C-Lie algebra, will be called the loop
algebra of g, and denoted by g [t, t
1
].
1. THE MAIN EXAMPLES 23
(b) Let (·, ·) be a symmetric bilinear form on g (that is, a symmetric bilinear
map g × g C) which is g-invariant (this means that ([a, b] , c) + (b, [a, c]) = 0 for
all a, b, c g).
Then, we can define a 2-cocycle ω on the loop algebra g [t, t
1
] by
(12) ω (f, g) =
X
iZ
i (f
i
, g
i
) for every f g
t, t
1
and g g
t, t
1
(where we write f in the form f =
P
iZ
f
i
t
i
with f
i
g, and where we write g in the
form g =
P
iZ
g
i
t
i
with g
i
g).
Proving that ω is a 2-cocycle is an exercise. So we can define a 1-dimensional
central extension g [t, t
1
]
ω
= g [t, t
1
] C with bracket defined by ω.
We are going to abbreviate g [t, t
1
]
ω
by
b
g
ω
, or, more radically, by
b
g.
Remark 1.7.2. The equation (12) can be rewritten in the (laconical but sugges-
tive) form ω (f, g) = Res
t=0
(df, g). Here, (df, g) is to be understood as follows: Ex-
tend the bilinear form (·, ·) : g×g C to a bilinear form (·, ·) : g [t, t
1
]×g [t, t
1
]
C [t, t
1
] by setting
at
i
, bt
j
= (a, b) t
i+j
for all a g, b g, i Z and j Z.
Also, for every f g [t, t
1
], define the “derivative” f
0
of f to be the element
P
iZ
if
i
t
i1
of g [t, t
1
] (where we write f in the form f =
P
iZ
f
i
t
i
with f
i
g). In
analogy to the notation dg = g
0
dt which we introduced in Definition 1.1.1, set
(df, g) to mean the polynomial differential form (f
0
, g) dt for any f g [t, t
1
] and
g g [t, t
1
]. Then, it is very easy to see that Res
t=0
(df, g) =
P
iZ
i (f
i
, g
i
) (where
we write f in the form f =
P
iZ
f
i
t
i
with f
i
g, and where we write g in the form
g =
P
iZ
g
i
t
i
with g
i
g), so that we can rewrite (12) as ω (f, g) = Res
t=0
(df, g).
We already know one example of the construction in Definition 1.7.1:
Remark 1.7.3. If g is the abelian Lie algebra C, and (·, ·) is the bilinear form
C ×C C, (x, y) 7→ xy, then the 2-cocycle ω on the loop algebra C [t, t
1
] is given
by
ω (f, g) = Res
t=0
(gdf) =
X
iZ
if
i
g
i
for every f, g C
t, t
1
(where we write f in the form f =
P
iZ
f
i
t
i
with f
i
C, and where we write g
in the form g =
P
iZ
g
i
t
i
with g
i
C). Hence, in this case, the central extension
g [t, t
1
]
ω
=
b
g
ω
is precisely the Heisenberg algebra A as introduced in Definition
1.1.4.
The main example that we will care about is when g is a simple finite-dimensional
Lie algebra and (·, ·) is the unique (up to scalar) invariant symmetric bilinear form (i.
e., a multiple of the Killing form). In this case, the Lie algebra
b
g =
b
g
ω
is called an
affine Lie algebra.
24 Nomenclature
Theorem 1.7.4. If g is a simple finite-dimensional Lie algebra, then H
2
(g [t, t
1
])
is 1-dimensional and spanned by the cocycle ω corresponding to (·, ·).
Corollary 1.7.5. If g is a simple finite-dimensional Lie algebra, then the Lie
algebra g [t, t
1
] has a unique (up to isomorphism of Lie algebras, not up to isomor-
phism of extensions) nontrivial 1-dimensional central extension
b
g
ω
.
Definition 1.7.6. The Lie algebra
b
g
ω
defined in Corollary 1.7.5 (for (·, ·) being
the Killing form of g) is called the affine Kac-Moody algebra corresponding to g.
(Or, more precisely, the untwisted affine Kac-Moody algebra corresponding to g.)
In order to prepare for the proof of Theorem 1.7.4, we recollect some facts from the
cohomology of Lie algebras:
Definition 1.7.7. Let g be a Lie algebra. Let M be a g-module. We define the
semidirect product g n M to be the Lie algebra which, as a vector space, is g M,
but whose Lie bracket is defined by
[(a, α) , (b, β)] = ([a, b] , a * β b * α)
for all a g, α M, b g and β M .
(The symbol * means action here; i. e., a term like c * m (with c g and m M)
means the action of c on m.) Thus, the canonical injection g g n M, a 7→ (a, 0)
is a Lie algebra homomorphism, and so is the canonical projection g n M g,
(a, α) 7→ a. Also, M is embedded into g n M by the injection M g n M,
α 7→ (0, α); this makes M an abelian Lie subalgebra of g n M.
All statements made in Definition 1.7.7 (including the tacit statement that the Lie
bracket on g n M defined in Definition 1.7.7 satisfies antisymmetry and the Jacobi
identity) are easy to verify by computation. The semidirect product that we have just
defined is not the most general notion of a semidirect product. We will later (Definition
3.2.1) define a more general one, where M itself may have a Lie algebra structure and
this structure has an effect on that of g n M. But for now, Definition 1.7.7 suffices for
us.
Definition 1.7.8. Let g be a Lie algebra. Let M be a g-module.
(a) A 1-cocycle of g with coefficients in M is a linear map η : g M such that
η ([a, b]) = a * η (b) b * η (a) for all a g and b g.
(The symbol * means action here; i. e., a term like c * m (with c g and m M)
means the action of c on m.)
It is easy to see (and known) that 1-cocycles of g with coefficients in M are in
bijection with Lie algebra homomorphisms g g n M. This bijection sends every
1-cocycle η to the map g g n M, a 7→ (a, η (a)).
Notice that 1-cocycles of g with coefficients in the g-module g are exactly the
same as derivations of g.
(b) A 1-coboundary of g with coefficients in M means a linear map η : g M
which has the form a 7→ a * m for some m M. Every 1-coboundary of g with
coefficients in M is a 1-cocycle.
(c) The space of 1-cocycles of g with coefficients in M is denoted by Z
1
(g, M).
The space of 1-coboundaries of g with coefficients in M is denoted by B
1
(g, M). We
1. THE MAIN EXAMPLES 25
have B
1
(g, M) Z
1
(g, M). The quotient space Z
1
(g, M) B
1
(g, M) is denoted
by H
1
(g, M) is called the 1-st cohomology space of g with coefficients in M.
Of course, these spaces Z
1
(g, M), B
1
(g, M) and H
1
(g, M) are but par-
ticular cases of more general constructions Z
i
(g, M), B
i
(g, M) and H
i
(g, M)
which are defined for every i N. (In particular, H
0
(g, M) is the subspace
{m M | a * m = 0 for all a g} of M , and often denoted by M
g
.) The spaces
H
i
(g, M) (or, more precisely, the functors assigning these spaces to every g-module
M) can be understood as the so-called derived functors of the functor M 7→ M
g
.
However, we won’t use H
i
(g, M) for any i other than 1 here.
We record a relation between H
1
(g, M) and the Ext bifunctor:
H
1
(g, M) = Ext
1
g
(C, M) .
More generally, Ext
1
g
(N, M) = H
1
(g, Hom
C
(N, M)) for any two g-modules N and
M.
Theorem 1.7.9 (Whitehead). If g is a simple finite-dimensional Lie algebra, and
M is a finite-dimensional g-module, then H
1
(g, M) = 0.
Proof of Theorem 1.7.9. Since g is a simple Lie algebra, Weyl’s theorem says that
finite-dimensional g-modules are completely reducible. Hence, if N and M are finite-
dimensional g-modules, we have Ext
1
g
(N, M) = 0. In particular, Ext
1
g
(C, M) = 0.
Since H
1
(g, M) = Ext
1
g
(C, M), this yields H
1
(g, M) = 0. Theorem 1.7.9 is thus
proven.
Lemma 1.7.10. Let ω be a 2-cocycle on a Lie algebra g. Let g
0
g be a Lie
subalgebra, and M g be a g
0
-submodule. Then, ω |
g
0
×M
, when considered as a
map g
0
M
, belongs to Z
1
(g
0
, M
).
The proof of Lemma 1.7.10 is a straightforward manipulation of formulas:
Proof of Lemma 1.7.10. Let η denote the 2-cocycle ω |
g
0
×M
, considered as a map
g
0
M
. Thus, η is defined by
η (x) = (M C, y 7→ ω (x, y)) for all x g
0
.
Hence,
(13) (η (x)) (y) = ω (x, y) for all x g
0
and y M.
26 Nomenclature
Thus, any a g
0
, b g
0
and c M satisfy (η ([a, b])) (c) = ω ([a, b] , c) and
(a * η (b) b * η (a)) (c)
= (a * η (b)) (c)
| {z }
=(η(b))([a,c])
(by the definition of the dual of a g
0
-module)
(b * η (a)) (c)
| {z }
=(η(a))([b,c])
(by the definition of the dual of a g
0
-module)
=
(η (b)) ([a, c])
| {z }
=ω(b,[a,c])
(by (13))
(η (a)) ([b, c])
| {z }
=ω(a,[b,c])
(by (13))
= (ω (b, [a, c])) (ω (a, [b, c]))
= ω
b, [a, c]
|{z}
=[c,a]
+ ω (a, [b, c]) = ω (b, [c, a])
| {z }
=ω([c,a],b)
(since ω is antisymmetric)
+ ω (a, [b, c])
| {z }
=ω([b,c],a)
(since ω is antisymmetric)
= ω ([c, a] , b) ω ([b, c] , a) = ω ([a, b] , c) (by (3)) ,
so that (η ([a, b])) (c) = (a * η (b) b * η (a)) (c). Thus, any a g
0
and b g
0
satisfy
η ([a, b]) = a * η (b) b * η (a). This shows that η is a 1-cocycle, i. e., belongs to
Z
1
(g
0
, M
). Lemma 1.7.10 is proven.
Proof of Theorem 1.7.4. First notice that any a, b, c g satisfy
(14) ([a, b] , c) = ([b, c] , a) = ([c, a] , b)
4
. Moreover,
(15) there exist a, b, c g such that ([a, b] , c) = ([b, c] , a) = ([c, a] , b) 6= 0.
5
This will be used later in our proof; but as for now, forget about these a, b, c.
It is easy to see that the 2-cocycle ω on g [t, t
1
] defined by (12) is not a 2-
coboundary.
6
4
Proof. First of all, any a, b, c g satisfy
([a, b] , c) = (a, [b, c]) (since the form (·, ·) is invariant)
= ([b, c] , a) (since the form (·, ·) is symmetric) .
Applying this to b, c, a instead of a, b, c, we obtain ([b, c] , a) = ([c, a] , b). Hence, ([a, b] , c) = ([b, c] , a) =
([c, a] , b), so that (14) is proven.
5
Proof. Since g is simple, we have [g, g] = g and thus ([g, g] , g) = (g, g) 6= 0 (since the form (·, ·)
is nondegenerate). Hence, there exist a, b, c g such that ([a, b] , c) 6= 0. The rest is handled by (14).
6
Proof. Assume the contrary. Then, this 2-cocycle ω is a coboundary, i. e., there exists a linear
map ξ : g
t, t
1
C such that ω = .
Now, pick some a g and b g such that (a, b) 6= 0 (this is possible since the form (·, ·) is
nondegenerate). Then,
ω
|{z}
=
at, bt
1
= ()
at, bt
1
= ξ
at, bt
1
| {z }
=[a,b]
= ξ ([a, b])
and
ω
|{z}
=
(a, b) = () (a, b) = ξ ([a, b]) ,
so that ω
at, bt
1
= ω (a, b). But by the definition of ω, we easily see that ω
at, bt
1
= 1 (a, b)
|{z}
6=0
6= 0
and ω (a, b) = 0 (a, b) = 0, which yields a contradiction.
1. THE MAIN EXAMPLES 27
Now let us consider the structure of g [t, t
1
]. We have g [t, t
1
] =
L
nZ
gt
n
gt
0
= g.
This is, actually, an inclusion of Lie algebras. So g is a Lie subalgebra of g [t, t
1
], and
gt
n
is a g-submodule of g [t, t
1
] isomorphic to g for every n Z.
Let ω be an arbitrary 2-cocycle on g [t, t
1
] (not necessarily the one defined by (12)).
Let n Z. Then, ω |
g×gt
n
, when considered as a map g (gt
n
)
, belongs to
Z
1
(g, (gt
n
)
) (by Lemma 1.7.10, applied to g, gt
n
and g [t, t
1
] instead of g
0
, M and
g), i. e., is a 1-cocycle. But by Theorem 1.7.9, we have H
1
(g, (gt
n
)
) = 0, so this
rewrites as ω |
g×gt
n
B
1
(g, (gt
n
)
). In other words, there exists some ξ
n
(gt
n
)
such
that ω |
g×gt
n
=
n
. Pick such a ξ
n
. Thus,
ω (a, bt
n
) = (ω |
g×gt
n
)
|
{z }
=
n
(a, bt
n
) = (
n
) (a, bt
n
) = ξ
n
([a, bt
n
]) for all a, b g.
Define a map ξ : g [t, t
1
] C by requiring that ξ |
gt
n
= ξ
n
for every n Z.
Now, let eω = ω . Then,
eω (x, y) = ω (x, y) ξ ([x, y]) for all x, y g
t, t
1
.
Replace ω by eω (this doesn’t change the residue class of ω in H
2
(g [t, t
1
]), since eω
differs from ω by a 2-coboundary). By doing this, we have reduced to a situation when
ω (a, bt
n
) = 0 for all a, b g and n Z.
7
Since ω is antisymmetric, this yields
(16) ω (bt
n
, a) = 0 for all a, b g and n Z.
Now, fix some n Z and m Z. Since ω is a 2-cocycle, the 2-cocycle condition
yields
0 = ω
[a, bt
n
]
| {z }
=[a,b]t
n
, ct
m
+ ω
[ct
m
, a]
| {z }
=[c,a]t
m
=[a,c]t
m
, bt
n
+ ω
[bt
n
, ct
m
]
| {z }
=[b,c]t
n+m
, a
= ω ([a, b] t
n
, ct
m
) + ω ([a, c] t
m
, bt
n
)
|
{z }
=ω(bt
n
,[a,c]t
m
)
+ ω
[b, c] t
n+m
, a
| {z }
=0
(by (16))
= ω ([a, b] t
n
, ct
m
) + ω (bt
n
, [a, c] t
m
) for all a, b, c g.
In other words, the bilinear form on g given by (b, c) 7→ ω (bt
n
, ct
m
) is g-invariant. But
every g-invariant bilinear form on g must be a multiple of our bilinear form (·, ·) (since
g is simple, and thus the space of all g-invariant bilinear forms on g is 1-dimensional
8
).
Hence, there exists some constant γ
n,m
C (depending on n and m) such that
(17) ω (bt
n
, ct
m
) = γ
n,m
· (b, c) for all b, c g.
It is easy to see that
(18) γ
n,m
= γ
m,n
for all n, m Z,
since the bilinear form ω is skew-symmetric whereas the bilinear form (·, ·) is symmetric.
7
But all the ξ-freedom has been used up in this reduction - i. e., if the new ω is nonzero, then the
original ω was not a 2-coboundary. This gives us an alternative way of proving that the 2-cocycle ω
on g
t, t
1
defined by (12) is not a 2-coboundary.
8
and spanned by the Killing form
28 Nomenclature
Now, for any m Z, n Z and p Z, the 2-cocycle condition yields
ω ([at
n
, bt
m
] , ct
p
) + ω ([bt
m
, ct
p
] , at
n
) + ω ([ct
p
, at
n
] , bt
m
) = 0 for all a, b, c g.
Due to
ω
[at
n
, bt
m
]
| {z }
=[a,b]t
n+m
, ct
p
= ω
[a, b] t
n+m
, ct
p
= γ
n+m,p
· ([a, b] , c) (by (17))
and the two cyclic permutations of this identity, this rewrites as
γ
n+m,p
· ([a, b] , c) + γ
m+p,n
· ([b, c] , a) + γ
p+n,m
· ([c, a] , b) = 0.
Since this holds for all a, b, c g, we can use (15) to transform this into
γ
n+m,p
+ γ
m+p,n
+ γ
p+n,m
= 0.
Due to (18), this rewrites as
γ
n,m+p
+ γ
m,p+n
+ γ
p,m+n
= 0.
Denoting by s the sum m + n + p, we can rewrite this as
γ
n,sn
+ γ
m,sm
γ
m+n,smn
= 0.
In other words, for fixed s Z, the function Z C, n 7→ γ
n,sn
is additive. Hence,
γ
n,sn
=
1,s1
and γ
sn,n
= (s n) γ
1,s1
for every n Z. Thus,
(s n) γ
1,s1
= γ
sn,n
= γ
n,sn
(by (18))
=
1,s1
for every n Z
Hence,
1,s1
= 0. Thus, for every s 6= 0, we conclude that γ
1,s1
= 0 and hence
γ
n,sn
= n γ
1,s1
|{z}
=0
= 0 for every n Z. In other words, γ
n,m
= 0 for every n Z and
m Z satisfying n + m 6= 0.
What happens for s = 0 ? For s = 0, the equation γ
n,sn
=
1,s1
becomes
γ
n,n
=
1,1
.
Thus we have proven that γ
n,m
= 0 for every n Z and m Z satisfying n+m 6= 0,
and that every n Z satisfies γ
n,n
=
1,1
.
Hence, the form ω must be a scalar multiple of the form which sends every (f, g) to
Res
t=0
(df, g)
|{z}
scalar-valued 1-form
=
P
iZ
i (f
i
, g
i
). We have thus proven that every 2-cocycle ω is
a scalar multiple of the 2-cocycle ω defined by (12) modulo the 2-coboundaries. Since
we also know that the 2-cocycle ω defined by (12) is not a 2-coboundary, this yields
that the space H
2
(g [t, t
1
]) is 1-dimensional and spanned by the residue class of the
2-cocycle ω defined by (12). This proves Theorem 1.7.4.
2. Representation theory: generalities
2.1. Representation theory: general facts. The first step in the representation
theory of any objects (groups, algebras, etc.) is usually proving some kind of Schur’s
lemma. There is one form of Schur’s lemma that holds almost tautologically: This
is the form that claims that every morphism between irreducible representations is
either 0 or an isomorphism.
9
However, the more often used form of Schur’s lemma is
9
There are also variations on this assertion:
1) Every morphism from an irreducible representation to a representation is either 0 or injective.
2) Every morphism from a representation to an irreducible representation is either 0 or surjective.
2. REPRESENTATION THEORY: GENERALITIES 29
a bit different: It claims that, over an algebraically closed field, every endomorphism
of a finite-dimensional irreducible representation is a scalar multiple of the identity
map. This is usually proven using eigenvalues, and this proof depends on the fact that
eigenvalues exist; this (in general) requires the irreducible representation to be finite-
dimensional. Hence, it should not come as a surprise that this latter form of Schur’s
lemma does not generally hold for infinite-dimensional representations. This makes
this lemma not particularly useful in the case of infinite-dimensional Lie algebras. But
we still can show the following version of Schur’s lemma over C:
Lemma 2.1.1 (Dixmier’s Lemma). Let A be an algebra over C, and let V be an
irreducible A-module of countable dimension. Then, any A-module homomorphism
φ : V V is a scalar multiple of the identity.
This lemma is called Dixmier’s lemma, and its proof is similar to the famous proof
of the Nullstellensatz over C using the uncountability of C.
Proof of Lemma 2.1.1. Let D = End
A
V . Then, D is a division algebra (in fact,
the endomorphism ring of an irreducible representation always is a division algebra).
For any nonzero v V , we have Av = V (otherwise, Av would be a nonzero proper
A-submodule of V , contradicting the fact that V is irreducible and thus does not have
any such submodules). In other words, for any nonzero v V , every element of V
can be written as av for some a A. Thus, for any nonzero v V , any element
φ D is completely determined by φ (v) (because φ (av) = (v) for every a A,
so that the value φ (v) uniquely determines the value of φ (av) for every a A, and
thus (since we know that every element of V can be written as av for some a A)
every value of φ is uniquely determined). Thus, we have an embedding of D into
V . Hence, D is countably-dimensional (since V is countably-dimensional). But a
countably-dimensional division algebra D over C must be C itself
10
, so that D = C,
and this is exactly what we wanted to show. Lemma 2.1.1 is proven.
Note that Lemma 2.1.1 is a general fact, not particular to Lie algebras; however, it
is not as general as it seems: It really makes use of the uncountability of C, not just
of the fact that C is an algebraically closed field of characteristic 0. It would be wrong
if we would replace C by (for instance) the algebraic closure of Q.
Remark 2.1.2. Let A be a countably-dimensional algebra over C, and let V be
an irreducible A-module. Then, V itself is countably dimensional.
Proof of Remark 2.1.2. For any nonzero v V , we have Av = V (by the same
argument as in the proof of Lemma 2.1.1), and thus dim()(Av) = dim(V ). Since
dim()(Av) dim(A), we thus have dim(V ) = dim()(Av) dim(A), so that V has
countable dimension (since A has countable dimension). This proves Remark 2.1.2.
Corollary 2.1.3. Let A be an algebra over C, and let V be an irreducible
A-module of countable dimension. Let C be a central element of A. Then, C |
V
is
a scalar (i. e., a scalar multiple of the identity map).
Both of these variations follow very easily from the definition of “irreducible”.
10
Proof. Indeed, assume the contrary. So there exists some φ D not belonging to C. Then, φ
is transcendental over C, so that C (φ) D is the field of rational functions in one variable φ over
C. Now, C (φ) contains the rational function
1
φ λ
for every λ C, and these rational functions
for varying λ are linearly independent. Since C is uncountable, we thus have an uncountable linearly
independent set of elements of C (φ), contradicting the fact that C (φ) is a subspace of the countably-
dimensional space D, qed.
30 Nomenclature
Proof of Corollary 2.1.3. Since C is central, the element C commutes with any
element of A. Thus, C |
V
is an A-module homomorphism, and hence (by Lemma 2.1.1,
applied to φ = C |
V
) a scalar multiple of the identity. This proves Corollary 2.1.3.
2.2. Representations of the Heisenberg algebra A.
2.2.1. General remarks. Consider the oscillator algebra (aka Heisenberg algebra)
A = ha
i
| i Zi + hKi. Recall that
[a
i
, a
j
] =
i,j
K for any i, j Z;
[K, a
i
] = 0 for any i Z.
Let us try to classify the irreducible A-modules.
Let V be an irreducible A-module. Then, V is countably-dimensional (by Remark
2.1.2, since U (A) is countably-dimensional), so that by Corollary 2.1.3, the endomor-
phism K |
V
is a scalar (because K is a central element of A and thus also a central
element of U (A)).
If K |
V
= 0, then V is a module over the Lie algebra ACK = ha
i
| i Zi. But
since ha
i
| i Zi is an abelian Lie algebra, irreducible modules over ha
i
| i Zi are
1-dimensional (again by Corollary 2.1.3), so that V must be 1-dimensional in this case.
Thus, the case when K |
V
= 0 is not an interesting case.
Now consider the case when K |
V
= k 6= 0. Then, we can WLOG assume that
k = 1, because the Lie algebra A has an automorphism sending K to λK for any
arbitrary λ 6= 0 (this automorphism is given by a
i
7→ λa
i
for i > 0, and a
i
7→ a
i
for
i 0).
We are thus interested in irreducible representations V of A satisfying K |
V
=
1. These are in an obvious 1-to-1 correspondence with irreducible representations of
U (A) (K 1).
Proposition 2.2.1. We have an algebra isomorphism
ξ : U (A) (K 1) D (x
1
, x
2
, x
3
, ...) C [x
0
] ,
where D (x
1
, x
2
, x
3
, ...) is the algebra of differential operators in the variables x
1
, x
2
,
x
3
, ... with polynomial coefficients. This isomorphism is given by
ξ (a
i
) = x
i
for i 1;
ξ (a
i
) = i
x
i
for i 1;
ξ (a
0
) = x
0
.
Note that we are sloppy with notation here: Since ξ is a homomorphism from
U (A) (K 1) (rather than U (A)), we should write ξ (a
i
) instead of ξ (a
i
), etc..
We are using the same letters to denote elements of U (A) and their residue classes in
U (A) (K 1), and are relying on context to keep them apart. We hope that the
reader will forgive us this abuse of notation.
2. REPRESENTATION THEORY: GENERALITIES 31
Proof of Proposition 2.2.1. It is clear
11
that there exists a unique algebra homo-
morphism ξ : U (A) (K 1) D (x
1
, x
2
, x
3
, ...) satisfying
ξ (a
i
) = x
i
for i 1;
ξ (a
i
) = i
x
i
for i 1;
ξ (a
0
) = x
0
.
It is also clear that this ξ is surjective (since all the generators x
i
,
x
i
and x
0
of the
algebra D (x
1
, x
2
, x
3
, ...) C [x
0
] are in its image).
In the following, a map ϕ : A N (where A is some set) is said to be finitely
supported if all but finitely many a A satisfy ϕ (a) = 0. Sequences (finite, infinite,
or two-sided infinite) are considered as maps (from finite sets, N or Z, or occasionally
other sets). Thus, a sequence is finitely supported if and only if all but finitely many
of its elements are zero.
If A is a set, then N
A
fin
will denote the set of all finitely supported maps A N.
By the easy part of the Poincar´e-Birkhoff-Witt theorem (this is the part which states
that the increasing monomials span the universal enveloping algebra
12
), the family
13
Y
iZ
a
n
i
i
· K
m
!
(...,n
2
,n
1
,n
0
,n
1
,n
2
,...)N
Z
fin
, mN
is a spanning set of the vector space U (A). Hence, the family
Y
iZ
a
n
i
i
!
(...,n
2
,n
1
,n
0
,n
1
,n
2
,...)N
Z
fin
is a spanning set of U (A) (K 1), and since this family maps to a linearly indepen-
dent set under ξ (this is very easy to see), it follows that ξ is injective. Thus, ξ is an
isomorphism, so that Proposition 2.2.1 is proven.
Definition 2.2.2. Define a vector subspace A
0
of A by A
0
=
ha
i
| i Z {0}i + hKi.
Proposition 2.2.3. This subspace A
0
is a Lie subalgebra of A, and Ca
0
is also
a Lie subalgebra of A. We have A = A
0
Ca
0
as Lie algebras. Hence,
U (A) (K 1) = U (A
0
Ca
0
) (K 1)
=
(U (A
0
) (K 1))
| {z }
=
D(x
1
,x
2
,x
3
,...)
C [a
0
]
|{z}
=
C[x
0
]
(since K A
0
). Here, the isomorphism U (A
0
) (K 1)
=
D (x
1
, x
2
, x
3
, ...) is
defined as follows: In analogy to Proposition 2.2.1, we have an algebra isomorphism
e
ξ : U (A
0
) (K 1) D (x
1
, x
2
, x
3
, ...)
11
from the universal property of the universal enveloping algebra, and the universal property of
the quotient algebra
12
The hard part says that these increasing monomials are linearly independent.
13
Here,
Q
iZ
a
n
i
i
denotes the product ...a
n
2
2
a
n
1
1
a
n
0
0
a
n
1
1
a
n
2
2
.... (This product is infinite, but still has
a value since only finitely many n
i
are nonzero.)
32 Nomenclature
given by
e
ξ (a
i
) = x
i
for i 1;
e
ξ (a
i
) = i
x
i
for i 1.
The proof of Proposition 2.2.3 is analogous to that of Proposition 2.2.1 (where it is
not completely straightforward).
2.2.2. The Fock space. From Proposition 2.2.3, we know that
U (A
0
) (K 1)
=
D (x
1
, x
2
, x
3
, ...) End (C [x
1
, x
2
, x
3
, ...]) .
Hence, we have a C-algebra homomorphism U (A
0
) End (C [x
1
, x
2
, x
3
, ...]). This
makes C [x
1
, x
2
, x
3
, ...] into a representation of the Lie algebra A
0
. Let us state this as
a corollary:
Corollary 2.2.4. The Lie algebra A
0
has a representation F = C [x
1
, x
2
, x
3
, ...]
which is given by
a
i
7→ x
i
for every i 1;
a
i
7→ i
x
i
for every i 1,
K 7→ 1
(where a
i
7→ x
i
is just shorthand for a
i
7→ (multiplication by x
i
)”). For every
µ C, we can upgrade F to a representation F
µ
of A by adding the condition that
a
0
|
F
µ
= µ ·id.
Definition 2.2.5. The representation F of A
0
introduced in Corollary 2.2.4 is
called the Fock module or the Fock representation. For every µ C, the representa-
tion F
µ
of A introduced in Corollary 2.2.4 will be called the µ-Fock representation
of A. The vector space F itself is called the Fock space.
Let us now define some gradings to make these infinite-dimensional spaces more
manageable:
Definition 2.2.6. Let us grade the vector space A by A =
L
nZ
A[n], where
A[n] = ha
n
i for n 6= 0, and where A[0] = ha
0
, Ki. With this grading, we have
[A[n] , A[m]] A[n + m] for all n Z and m Z. (In other words, the Lie algebra
A with the decomposition A =
L
nZ
A[n] is a Z-graded Lie algebra. The notion of a
Z-graded Lie algebra” that we have just used is defined in Definition 2.5.1.)
Note that we are denoting the n-th homogeneous component of A by A[n] rather
than A
n
, since otherwise the notation A
0
would have two different meanings.
Definition 2.2.7. We grade the polynomial algebra F by setting deg (x
i
) = i
for each i. Thus, F =
L
n0
F [n], where F [n] is the space of polynomials of degree
n, where the degree is our degree defined by deg (x
i
) = i (so that, for instance,
x
2
1
+ x
2
is homogeneous of degree 2). With this grading, dim()(F [n]) is the
number p (n) of all partitions of n. Hence,
X
n0
dim()(F [n]) q
n
=
X
n0
p (n) q
n
=
1
(1 q) (1 q
2
) (1 q
3
) ···
=
1
Q
i1
(1 q
i
)
2. REPRESENTATION THEORY: GENERALITIES 33
in the ring of power series Z [[q]].
We use the same grading for F
µ
for every µ C. That is, we define the grading
on F
µ
by F
µ
[n] = F [n] for every n Z.
Remark 2.2.8. Some people prefer to grade F
µ
somewhat differently from F :
namely, they shift the grading for F
µ
by
µ
2
2
, so that deg 1 =
µ
2
2
in F
µ
, and
generally F
µ
[z] = F
µ
2
2
+ z
(as vector spaces) for every z C. This is a grading
by complex numbers rather than integers (in general). (The advantage of this grading
is that we will eventually find an operator whose eigenspace to the eigenvalue n is
F
µ
[n] = F
µ
2
2
+ n
for every n C.)
With this grading, the equality
P
n0
dim()(F [n]) q
n
=
1
Q
i1
(1 q
i
)
rewrites as
P
nC
dim()(F
µ
[n]) q
n+
µ
2
2
=
q
µ
2
Q
i1
(1 q
i
)
, if we allow power series with complex ex-
ponents. We define a “power series” ch (F
µ
) by
ch (F
µ
) =
X
nC
dim()(F
µ
[n]) q
n+
µ
2
2
=
q
µ
2
Q
i1
(1 q
i
)
.
But we will not use this grading; instead we will use the grading defined in Definition
2.2.7.
Proposition 2.2.9. The representation F is an irreducible representation of A
0
.
Lemma 2.2.10. For every P F , we have
P (a
1
, a
2
, a
3
, ...) ·1 = P in F.
(Here, the term P (a
1
, a
2
, a
3
, ...) denotes the evaluation of the polynomial P
at (x
1
, x
2
, x
3
, ...) = (a
1
, a
2
, a
3
, ...). This evaluation is a well-defined element of
U (A
0
), since the elements a
1
, a
2
, a
3
, ... of U (A
0
) commute.)
Proof of Lemma 2.2.10. For every Q F , let mult Q denote the map F F,
R 7→ QR. (In Proposition 2.2.1, we abused notations and denoted this map simply
by Q; but we will not do this in this proof.) Then, by the definition of ξ, we have
ξ (a
i
) = mult (x
i
) for every i 1.
Since we have defined an endomorphism mult Q End F for every Q F , we thus
obtain a map mult : F End F . This map mult is an algebra homomorphism (since
it describes the action of F on the F -module F ).
34 Nomenclature
Let P F . Since ξ is an algebra homomorphism, and thus commutes with poly-
nomials, we have
ξ (P (a
1
, a
2
, a
3
, ...))
= P (ξ (a
1
) , ξ (a
2
) , ξ (a
3
) , ...) = P (mult (x
1
) , mult (x
2
) , mult (x
3
) , ...)
(since ξ (a
i
) = mult (x
i
) for every i 1)
= mult
P (x
1
, x
2
, x
3
, ...)
| {z }
=P
since mult is an algebra homomorphism,
and thus commutes with polynomials
= mult P.
Thus,
P (a
1
, a
2
, a
3
, ...) ·1 = (mult P ) (1) = P · 1 = P.
This proves Lemma 2.2.10.
Proof of Proposition 2.2.9. 1) The representation F is generated by 1 as a U (A
0
)-
module (due to Lemma 2.2.10). In other words, F = U (A
0
) ·1.
2) Let us forget about the grading on F which we defined in Definition 2.2.7, and
instead, once again, define a grading on F by deg (x
i
) = 1 for every i {1, 2, 3, ...}.
Thus, the degree of a polynomial P F with respect to this grading is what is usually
referred to as the degree of the polynomial P .
If P F and if α · x
m
1
1
x
m
2
2
x
m
3
3
... is a monomial in P of degree deg P , with α 6= 0,
then
m
1
x
1
m
1
!
m
2
x
2
m
2
!
m
3
x
3
m
3
!
...P = α
14
.
14
Proof. Let P F . Let α ·x
m
1
1
x
m
2
2
x
m
3
3
... be a monomial in P of degree deg P , with α 6= 0. Since
the monomial α ·x
m
1
1
x
m
2
2
x
m
3
3
... has degree deg P , we have
deg P = deg (α · x
m
1
1
x
m
2
2
x
m
3
3
...) = m
1
+ m
2
+ m
3
+ ....
For every set A, define N
A
fin
as in the proof of Proposition 2.2.1.
Now, for every (n
1
, n
2
, n
3
, ...) N
{1,2,3,...}
fin
, let β
(n
1
,n
2
,n
3
,...)
be the coefficient of the polynomial
P before the monomial x
n
1
1
x
n
2
2
x
n
3
3
.... Then, β
(n
1
,n
2
,n
3
,...)
= 0 for every (n
1
, n
2
, n
3
, ...) N
{1,2,3,...}
fin
satisfying n
1
+ n
2
+ n
3
+ ... > deg P (because for every (n
1
, n
2
, n
3
, ...) N
{1,2,3,...}
fin
satisfying n
1
+
n
2
+ n
3
+ ... > deg P , the monomial x
n
1
1
x
n
2
2
x
n
3
3
... has degree n
1
+ n
2
+ n
3
+ ... > deg P , and thus the
coefficient of the polynomial P before this monomial must be 0). On the other hand, β
(m
1
,m
2
,m
3
,...)
= α
(since α · x
m
1
1
x
m
2
2
x
m
3
3
... is a monomial in P , and thus the coefficient of the polynomial P before the
monomial x
m
1
1
x
m
2
2
x
m
3
3
... is α).
2. REPRESENTATION THEORY: GENERALITIES 35
On the other hand, recall that β
(n
1
,n
2
,n
3
,...)
is the coefficient of the polynomial P before the
monomial x
n
1
1
x
n
2
2
x
n
3
3
... for every (n
1
, n
2
, n
3
, ...) N
{1,2,3,...}
fin
. Hence,
P =
X
(n
1
,n
2
,n
3
,...)N
{1,2,3,...}
fin
β
(n
1
,n
2
,n
3
,...)
x
n
1
1
x
n
2
2
x
n
3
3
...
=
X
(n
1
,n
2
,n
3
,...)N
{1,2,3,...}
fin
;
n
1
+n
2
+n
3
+...deg P
β
(n
1
,n
2
,n
3
,...)
x
n
1
1
x
n
2
2
x
n
3
3
... +
X
(n
1
,n
2
,n
3
,...)N
{1,2,3,...}
fin
;
n
1
+n
2
+n
3
+...>deg P
β
(n
1
,n
2
,n
3
,...)
| {z }
=0
(since n
1
+n
2
+n
3
+...>deg P )
x
n
1
1
x
n
2
2
x
n
3
3
...
=
X
(n
1
,n
2
,n
3
,...)N
{1,2,3,...}
fin
;
n
1
+n
2
+n
3
+...deg P
β
(n
1
,n
2
,n
3
,...)
x
n
1
1
x
n
2
2
x
n
3
3
... +
X
(n
1
,n
2
,n
3
,...)N
{1,2,3,...}
fin
;
n
1
+n
2
+n
3
+...>deg P
0x
n
1
1
x
n
2
2
x
n
3
3
...
|
{z }
=0
=
X
(n
1
,n
2
,n
3
,...)N
{1,2,3,...}
fin
;
n
1
+n
2
+n
3
+...deg P
β
(n
1
,n
2
,n
3
,...)
x
n
1
1
x
n
2
2
x
n
3
3
...
=
X
(n
1
,n
2
,n
3
,...)N
{1,2,3,...}
fin
;
n
1
+n
2
+n
3
+...deg P ;
(n
1
,n
2
,n
3
,...)6=(m
1
,m
2
,m
3
,...)
β
(n
1
,n
2
,n
3
,...)
x
n
1
1
x
n
2
2
x
n
3
3
... + β
(m
1
,m
2
,m
3
,...)
| {z }
=α
x
m
1
1
x
m
2
2
x
m
3
3
...
since (m
1
, m
2
, m
3
, ...) N
{1,2,3,...}
fin
satisfies m
1
+ m
2
+ m
3
+ ... = deg P
=
X
(n
1
,n
2
,n
3
,...)N
{1,2,3,...}
fin
;
n
1
+n
2
+n
3
+...deg P ;
(n
1
,n
2
,n
3
,...)6=(m
1
,m
2
,m
3
,...)
β
(n
1
,n
2
,n
3
,...)
x
n
1
1
x
n
2
2
x
n
3
3
... + αx
m
1
1
x
m
2
2
x
m
3
3
....
Thus,
m
1
x
1
m
1
!
m
2
x
2
m
2
!
m
3
x
3
m
3
!
...P
=
m
1
x
1
m
1
!
m
2
x
2
m
2
!
m
3
x
3
m
3
!
...
X
(n
1
,n
2
,n
3
,...)N
{1,2,3,...}
fin
;
n
1
+n
2
+n
3
+...deg P ;
(n
1
,n
2
,n
3
,...)6=(m
1
,m
2
,m
3
,...)
β
(n
1
,n
2
,n
3
,...)
x
n
1
1
x
n
2
2
x
n
3
3
... + αx
m
1
1
x
m
2
2
x
m
3
3
...
=
X
(n
1
,n
2
,n
3
,...)N
{1,2,3,...}
fin
;
n
1
+n
2
+n
3
+...deg P ;
(n
1
,n
2
,n
3
,...)6=(m
1
,m
2
,m
3
,...)
β
(n
1
,n
2
,n
3
,...)
m
1
x
1
m
1
!
m
2
x
2
m
2
!
m
3
x
3
m
3
!
... (x
n
1
1
x
n
2
2
x
n
3
3
...) + α
m
1
x
1
m
1
!
m
2
x
2
m
2
!
m
3
x
3
m
3
!
... (x
m
1
1
x
m
2
2
x
m
3
3
...)
| {z }
=
m
1
!
m
1
!
m
2
!
m
2
!
m
3
!
m
3
!
...=1
=
X
(n
1
,n
2
,n
3
,...)N
{1,2,3,...}
fin
;
n
1
+n
2
+n
3
+...deg P ;
(n
1
,n
2
,n
3
,...)6=(m
1
,m
2
,m
3
,...)
β
(n
1
,n
2
,n
3
,...)
m
1
x
1
m
1
!
m
2
x
2
m
2
!
m
3
x
3
m
3
!
... (x
n
1
1
x
n
2
2
x
n
3
3
...) + α.
(19)
36 Nomenclature
Thus, for every nonzero P F , we have 1 U (A
0
) · P
15
. Combined with
1), this yields that for every nonzero P F , the representation F is generated by P
as a U (A
0
)-module (since F = U (A
0
) · 1
|{z}
U(A
0
)·P
U (A
0
) · U (A
0
) · P = U (A
0
) · P ).
Consequently, F is irreducible. Proposition 2.2.9 is proven.
Proposition 2.2.11. Let V be an irreducible A
0
-module on which K acts as 1.
Assume that for any v V , the space C [a
1
, a
2
, a
3
, ...] · v is finite-dimensional, and
the a
i
with i > 0 act on it by nilpotent operators. Then, V
=
F as A
0
-modules.
But now, let (n
1
, n
2
, n
3
, ...) N
{1,2,3,...}
fin
be a sequence satisfying n
1
+ n
2
+ n
3
+ ... deg P
and (n
1
, n
2
, n
3
, ...) 6= (m
1
, m
2
, m
3
, ...). Since n
1
+ n
2
+ n
3
+ ... deg P = m
1
+ m
2
+ m
3
+ ... but
(n
1
, n
2
, n
3
, ...) 6= (m
1
, m
2
, m
3
, ...), it is clear that there exists at least one ` {1, 2, 3, ...} satisfying
n
`
< m
`
. Consider such an `. Since the differential operators
x
1
,
x
2
,
x
3
, ... commute, we have
m
1
x
1
m
1
!
m
2
x
2
m
2
!
m
3
x
3
m
3
!
... =
Q
i∈{1,2,3,...}\{`}
m
i
x
i
m
i
!
!
m
`
x
`
m
`
!
, so that
m
1
x
1
m
1
!
m
2
x
2
m
2
!
m
3
x
3
m
3
!
... (x
n
1
1
x
n
2
2
x
n
3
3
...) =
Y
i∈{1,2,3,...}\{`}
m
i
x
i
m
i
!
m
`
x
`
m
`
!
(x
n
1
1
x
n
2
2
x
n
3
3
...)
=
Y
i∈{1,2,3,...}\{`}
m
i
x
i
m
i
!
m
`
x
`
m
`
!
(x
n
1
1
x
n
2
2
x
n
3
3
...)
| {z }
=0
(since n
`
<m
`
)
=
Y
i∈{1,2,3,...}\{`}
m
i
x
i
m
i
!
(0) = 0.
Now, forget that we fixed (n
1
, n
2
, n
3
, ...) N
{1,2,3,...}
fin
. We have thus proven that every sequence
(n
1
, n
2
, n
3
, ...) N
{1,2,3,...}
fin
satisfying n
1
+ n
2
+ n
3
+ ... deg P and (n
1
, n
2
, n
3
, ...) 6= (m
1
, m
2
, m
3
, ...)
must satisfy
m
1
x
1
m
1
!
m
2
x
2
m
2
!
m
3
x
3
m
3
!
... (x
n
1
1
x
n
2
2
x
n
3
3
...) = 0. Hence, (19) becomes
m
1
x
1
m
1
!
m
2
x
2
m
2
!
m
3
x
3
m
3
!
...P
=
X
(n
1
,n
2
,n
3
,...)N
{1,2,3,...}
fin
;
n
1
+n
2
+n
3
+...deg P ;
(n
1
,n
2
,n
3
,...)6=(m
1
,m
2
,m
3
,...)
β
(n
1
,n
2
,n
3
,...)
m
1
x
1
m
1
!
m
2
x
2
m
2
!
m
3
x
3
m
3
!
... (x
n
1
1
x
n
2
2
x
n
3
3
...)
| {z }
=0
(since n
1
+n
2
+n
3
+...deg P
and (n
1
,n
2
,n
3
,...)6=(m
1
,m
2
,m
3
,...))
+α
=
X
(n
1
,n
2
,n
3
,...)N
{1,2,3,...}
fin
;
n
1
+n
2
+n
3
+...deg P ;
(n
1
,n
2
,n
3
,...)6=(m
1
,m
2
,m
3
,...)
β
(n
1
,n
2
,n
3
,...)
0
| {z }
=0
+α = α,
qed.
15
Proof. Let P F be nonzero. Then, there exist a monomial α · x
m
1
1
x
m
2
2
x
m
3
3
... in P of degree
P with α 6= 0. Consider such a monomial. As shown above, we have
m
1
x
1
m
1
!
m
2
x
2
m
2
!
m
3
x
3
m
3
!
...P = α. But we
know that a
i
A
0
acts as i
x
i
on F for every i 1. Thus,
1
i
a
i
A
0
acts as
x
i
=
x
i
on F for
every i 1. Hence,
1
1
a
1
m
1
m
1
!
1
2
a
2
m
2
m
2
!
1
3
a
3
m
3
m
3
!
...P =
m
1
x
1
m
1
!
m
2
x
2
m
2
!
m
3
x
3
m
3
!
...P = α.
2. REPRESENTATION THEORY: GENERALITIES 37
Before we prove this, a simple lemma:
Lemma 2.2.12. Let V be an A
0
-module. Let u V be such that a
i
u = 0 for
all i > 0, and such that Ku = u. Then, there exists a homomorphism η : F V
of A
0
-modules such that η (1) = u. (This homomorphism η is unique, although we
won’t need this.)
We give two proofs of this lemma. The first one is conceptual and gives us a glimpse
into the more general theory (it proceeds by constructing an A
0
-module Ind
A
0
CK⊕A
+
0
C,
which is an example of what we will later call a Verma highest-weight module in
Definition 2.5.14). The second one is down-to-earth and proceeds by direct construction
and computation.
First proof of Lemma 2.2.12. Define a vector subspace A
+
0
of A
0
by A
+
0
= ha
i
| i positive integeri.
It is clear that the internal direct sum CK A
+
0
is well-defined and an abelian Lie
subalgebra of A
0
. We can make C into an
CK A
+
0
-module by setting
Kλ = λ for every λ C;
a
i
λ = 0 for every λ C and every positive integer i.
Now, consider the A
0
-module Ind
A
0
CK⊕A
+
0
C = U (A
0
)
U
(
CK⊕A
+
0
)
C. Denote the element
1
U
(
CK⊕A
+
0
)
1 U (A
0
)
U
(
CK⊕A
+
0
)
C of this module by 1.
We will