2. Representation theory: generalities
2.5. Z -graded Lie algebras and Verma modules
2.5.1. Z-graded Lie algebras
Let us show some general results about representations of Z-graded Lie algebras – particularly ofnondegenerate Z-graded Lie algebras. This is a notion that encompasses many of the concrete Lie algebras that we want to study (among others, A, A0, W and Vir), and thus by proving the properties of nondegenerate Z-graded Lie algebras now we can avoid proving them separately in many different cases.
Definition 2.5.1. A Z-graded Lie algebra is a Lie algebra g with a decomposition g= L
n∈Z
gn(as a vector space) such that [gn,gm]⊆gn+m for alln, m∈Z. The family (gn)n∈
Z is called the grading of this Z-graded Lie algebra.44
Of course, every Z-graded Lie algebra automatically is a Z-graded vector space (by way of forgetting the Lie bracket and only keeping the grading). Note that ifg= L
n∈Z
gn is a Z-graded Lie algebra, then L
n<0
gn,g0 and L
n>0
gn are Lie subalgebras ofg.
Example 2.5.2. We defined a grading on the Heisenberg algebraAin Definition 2.2.6.
This makes A into a Z-graded Lie algebra. Also, A0 is a Z-graded Lie subalgebra of A.
Example 2.5.3. We make the Witt algebra W into a Z-graded Lie algebra by using the grading (W[n])n∈
Z, where W[n] =hLni for every n∈Z.
We make the Virasoro algebra Vir into a Z-graded Lie algebra by using the grading (Vir [n])n∈
Z, where Vir [n] =
hLni, if n 6= 0;
hL0, Ci, if n= 0 for every n∈Z.
44Warning: Some algebraists use the words “Z-graded Lie algebra” to denote a Z-graded Lie su-peralgebra, where the even homogeneous components constitute the even part and the odd ho-mogeneous components constitute the odd part. This is nothow we understand the notion of a
“Z-graded Lie algebra” here. In particular, for us, aZ-graded Lie algebragshould satisfy [x, x] = 0 for allx∈g(not just forxlying in even homogeneous components).
Definition 2.5.4. AZ-graded Lie algebra g= L
n∈Z
gn is said to benondegenerate if (1)the vector space gn is finite-dimensional for everyn∈Z;
(2)the Lie algebra g0 is abelian;
(3) for every positive integern, for genericλ ∈g∗0, the bilinear form gn×g−n → C, (a, b)7→ λ([a, b]) is nondegenerate. (“Generic λ” means “λ lying in some dense open subset of g∗0 with respect to the Zariski topology”. This subset can depend on n.)
Note that condition (3)in Definition 2.5.4 implies that dim (gn) = dim (g−n) for all n∈Z.
Here are some examples:
Proposition 2.5.5. TheZ-graded Lie algebrasA,A0,W and Vir are nondegenerate (with the gradings defined above).
Proposition 2.5.6. Letgbe a finite-dimensional simple Lie algebra. The following is a reasonable (although non-canonical) way to define a grading on g:
Using a Cartan subalgebra and the roots of g, we can present the Lie algebragas a Lie algebra with generatorse1,e2,...,em,f1,f2,...,fm,h1,h2,...,hm (the so-called Chevalley generators) and some relations (among them the Serre relations). Then, we can define a grading on g by setting
deg (ei) = 1, deg (fi) = −1 and deg (hi) = 0 for all i∈ {1,2, ..., m}, and extending this grading in such a way that gbecomes a graded Lie algebra. This grading is non-canonical, but it makes g into a nondegenerate graded Lie algebra.
Proposition 2.5.7. If g is a finite-dimensional simple Lie algebra, then the loop algebrag[t, t−1] and the affine Kac-Moody algebrabg=g[t, t−1]⊕CK can be graded as follows:
Fix Chevalley generators for g and grade g as in Proposition 2.5.6. Now let θ be the maximal root of g, i. e., the highest weight of the adjoint representation of g. Let eθ and fθ be the root elements corresponding to θ. The Coxeter number of g is defined as deg (eθ) + 1, and denoted by h. Now let us grade bg by setting degK = 0 and deg (atm) = dega+mh for every homogeneous a ∈ g and every m ∈ Z. This grading satisfies deg (fθt) = 1 and deg (eθt−1) = −1. Moreover, the mapg[t, t−1]→g[t, t−1], x7→xtis homogeneous of degreeh; this is often informally stated as “degt=h” (although t itself is not an element ofbg). It is easy to see that the elements of bg of positive degree span n+⊕tg[t].
The graded Lie algebra bg is nondegenerate. The loop algebra g[t, t−1], however, is not (with the grading defined in the same way).
If g is a Z-graded Lie algebra, we can write g=M
n∈Z
gn =M
n<0
gn⊕g0⊕M
n>0
gn.
We denote L
n<0
gn by n− and we denote L
n>0
gn by n+. We also denote g0 by h. Then, n−, n+ and h are Lie subalgebras of g, and the above decomposition rewrites as g =
n−⊕h⊕n+ (but this is, of course, not a direct sum of Lie algebras). This is called the triangular decomposition of g.
It is easy to see that when g is a Z-graded Lie algebra, the universal enveloping algebraU(g) canonically becomes aZ-graded algebra.45
2.5.2. Z-graded modules
Definition 2.5.8. Letg be a Lie algebra over a field k. Let M be a g-module. Let U be a vector subspace of g. Let N be a vector subspace of M. Then, U * N will denote the k-linear span of all elements of the form u * n with u∈U and n ∈N. (Notice that this notation is analogous to the notation [U, N] which is defined if U and N are both subspaces of g.)
Definition 2.5.9. Letgbe aZ-graded Lie algebra with grading (gn)n∈
Z. AZ-graded g-module means a Z-graded vector space M equipped with a g-module structure such that any i∈Z and j ∈Z satisfy gi * Mj ⊆Mi+j, where (Mn)n∈
Z denotes the grading of M.
The reader can easily check that when g is a Z-graded Lie algebra, and M is a Z -graded g-module, then M canonically becomes a Z-graded U(g)-module (by taking the canonical U(g)-module structure on M and the given Z-grading on M).
Examples of Z-graded g-modules for various Lie algebras g are easy to get by. For example, when g is a Z-graded Lie algebra, then the adjoint representation g itself is aZ-graded g-module. For two more interesting examples:
Example 2.5.10. The action of the Heisenberg algebraAon theµ-Fock representation Fµ makes Fµ into a Z-graded A-module (i. e., it maps A[i]⊗Fµ[j] to Fµ[i+j] for all i ∈ Z and j ∈ Z). Here, we are using the Z-grading on Fµ defined in Definition 2.2.7. (If we would use the alternative Z-grading on Fµ defined in Remark 2.2.8, then the action of A onFµ would still make Fµ into a Z-gradedA-module.)
The action of A0 on the Fock module F makes F into a Z-gradedA0-module.
Example 2.5.11. Letα ∈Cand β∈C. The Vir-module Vα,β defined in Proposition 2.3.2 becomes a Z-graded Vir-module by means of the grading (Vα,β[n])n∈
Z, where Vα,β[n] =hv−ni for every n∈Z.
Let us formulate a graded analogue of Lemma 2.2.12:
Lemma 2.5.12. Let V be a Z-graded A0-module with grading (V [n])n∈
Z. Let u ∈ V [0] be such that aiu = 0 for all i > 0, and such that Ku = u. Then, there exists a Z-graded homomorphism η : F → V of A0-modules such that η(1) = u.
(This homomorphism η is unique, although we won’t need this.)
Proof of Lemma 2.5.12. Let η be the map F → V which sends every polynomial P ∈ F = C[x1, x2, x3, ...] to P (a−1, a−2, a−3, ...)·u ∈ V. 46 Just as in the Second
45In fact,U(g) is defined as the quotient of the tensor algebraT(g) by a certain ideal. Whengis a Z-graded Lie algebra, this ideal is generated by homogeneous elements, and thus is a graded ideal.
46Note that the term P(a−1, a−2, a−3, ...) denotes the evaluation of the polynomial P at (x1, x2, x3, ...) = (a−1, a−2, a−3, ...). This evaluation is a well-defined element of U(A0), since the elementsa−1,a−2,a−3,... ofU(A0) commute.
proof of Lemma 2.2.12, we can show that η is an A0-module homomorphism F → V such that η(1) =u. Hence, in order to finish the proof of Lemma 2.5.12, we only need to check that η is a Z-graded map.
If A is a set, then NAfin will denote the set of all finitely supported maps A →N. Let n∈Z and P ∈F [n]. Then, we can write the polynomial P in the form
P = X
(i1,i2,i3,...)∈N{1,2,3,...}
fin ;
1i1+2i2+3i3+...=−n
λ(i1,i2,i3,...)xi11xi22xi33... (35)
for some scalarsλ(i1,i2,i3,...) ∈C. Consider these λ(i1,i2,i3,...). From (35), it follows that P (a−1, a−2, a−3, ...) = X
(i1,i2,i3,...)∈N{1,2,3,...}
fin ;
1i1+2i2+3i3+...=−n
λ(i1,i2,i3,...) ai−11 ai−22 ai−33 ...
| {z }
∈U(A0)[i1(−1)+i2(−2)+i3(−3)+...]
(since every positive integerksatisfies a−k∈A0[−k]⊆U(A0)[−k] and thusaik−k∈U(A0)[ik(−k)])
∈ X
(i1,i2,i3,...)∈N{1,2,3,...}
fin ;
1i1+2i2+3i3+...=−n
λ(i1,i2,i3,...)U(A0)
i1(−1) +i2(−2) +i3(−3) +...
| {z }
=−(1i1+2i2+3i3+...)=n (since 1i1+2i2+3i3+...=−n)
= X
(i1,i2,i3,...)∈N{1,2,3,...}
fin ;
1i1+2i2+3i3+...=−n
λ(i1,i2,i3,...)U(A0) [n]⊆U(A0) [n]
(since U(A0) [n] is a vector space). By the definition of η, we have η(P) =P (a−1, a−2, a−3, ...)
| {z }
∈U(A0)[n]
· u
|{z}
∈V[0]
∈U(A0) [n]·V [0]⊆V [n]
(since V is a Z-graded A0-module and thus a Z-graded U(A0)-module). Now forget that we fixed n and P. We have thus shown that every n ∈ Z and P ∈ F [n] satisfy η(P)⊆V [n]. In other words, every n ∈Z satisfies η(F[n])⊆V [n]. In other words, η is Z-graded. This proves Lemma 2.5.12.
And here is a graded analogue of Lemma 2.2.18:
Lemma 2.5.13. Let V be a graded A-module with grading (V [n])n∈
Z. Let µ∈C. Let u∈ V [0] be such thataiu= 0 for all i > 0, such thata0u=µu, and such that Ku =u. Then, there exists a Z-graded homomorphism η : Fµ → V of A-modules such thatη(1) =u. (This homomorphismηis unique, although we won’t need this.) The proof of Lemma 2.5.13 is completely analogous to that of Lemma 2.5.12, but this time using Lemma 2.2.18 instead of Lemma 2.2.12.
2.5.3. Verma modules
Definition 2.5.14. Letgbe aZ-graded Lie algebra (not necessarily nondegenerate).
Let us work with the notations introduced above. Let λ ∈h∗.
LetCλ denote the (h⊕n+)-module which, as a C-vector space, is the free vector space with basis vλ+
(thus, a 1-dimensional vector space), and whose (h⊕n+ )-action is given by
hvλ+=λ(h)vλ+ for every h∈h;
n+vλ+= 0.
The Verma highest-weight module Mλ+ of (g, λ) is defined by Mλ+=U(g)⊗U(h⊕n+)Cλ.
The element 1⊗U(h⊕n+)vλ+ofMλ+will still be denoted byv+λ by abuse of notation, and will be called the defining vector of Mλ+. SinceU(g) andCλ are gradedU(h⊕n+ )-modules, their tensor product U(g)⊗U(h⊕n+)Cλ =Mλ+ becomes graded as well.
LetCλ denote the (h⊕n−)-module which, as a C-vector space, is the free vector space with basis vλ−
(thus, a 1-dimensional vector space), and whose (h⊕n− )-action is given by
hvλ−=λ(h)vλ− for every h∈h;
n−vλ−= 0.
(Note that we denote this (h⊕n−)-module by Cλ, although we already have de-noted an (h⊕n+)-module by Cλ. This is ambiguous, but misunderstandings are unlikely to occur since these modules are modules over different Lie algebras, and their restrictions to h are identical.)
The Verma lowest-weight module Mλ− of (g, λ) is defined by Mλ−=U(g)⊗U(h⊕n−)Cλ.
The element 1⊗U(h⊕n−)vλ−ofMλ−will still be denoted byv−λ by abuse of notation, and will be called the defining vector of Mλ−. SinceU(g) andCλ are gradedU(h⊕n− )-modules, their tensor product U(g)⊗U(h⊕n−)Cλ =Mλ− becomes graded as well.
We notice some easy facts about these modules:
Proposition 2.5.15. Letg be a Z-graded Lie algebra (not necessarily nondegener-ate). Let us work with the notations introduced above. Let λ ∈h∗.
(a)As a gradedn−-module,Mλ+ =U(n−)v+λ; more precisely, there exists a graded n−-module isomorphism U(n−)⊗Cλ →Mλ+ which sends everyx⊗t ∈U(n−)⊗Cλ
to xtv+λ. The Verma module Mλ+ is concentrated in nonpositive degrees:
Mλ+ =M
n≥0
Mλ+[−n] ; Mλ+[−n] =U(n−) [−n]vλ+ for every n≥0.
Also, if dimgj <∞ for all j ≤ −1, we have X
n≥0
dim Mλ+[−n]
qn= 1
Q
j≤−1
(1−q−j)dimgj.
(b)As a gradedn+-module,Mλ− =U(n+)vλ−; more precisely, there exists a graded n+-module isomorphism U(n+)⊗Cλ →Mλ− which sends everyx⊗t ∈U(n+)⊗Cλ
to xtv−λ. The Verma module Mλ− is concentrated in nonnegative degrees:
Mλ− =M
n≥0
Mλ−[n] ; Mλ−[n] =U(n+) [n]vλ− for every n≥0.
Also, if dimgj <∞ for all j ≥1, we have X
n≥0
dim Mλ−[n]
qn= 1
Q
j≥1
(1−qj)dimgj.
Proof of Proposition 2.5.15. (a) Let ρ : U(n−)⊗CU(h⊕n+) → U(g) be the C -vector space homomorphism defined by
ρ(α⊗β) =αβ for all α ∈U(n−) and β ∈U(h⊕n+)
(this is clearly well-defined). By Corollary 2.4.2 (applied to a = n−, b = h⊕n+ and c = g), this ρ is an isomorphism of filtered47 vector spaces, of left U(n−)-modules and of right U(h⊕n+)-modules. Also, it is a graded linear map48 (this is clear from its definition), and thus an isomorphism of graded vector spaces (because if a vector space isomorphism of graded vector spaces is a graded linear map, then it must be an isomorphism of graded vector spaces49). Altogether, ρ is an isomorphism of graded filtered vector spaces, of leftU(n−)-modules and of right U(h⊕n+)-modules. Hence, Mλ+= U(g)
| {z }
∼=U(n−)⊗CU(h⊕n+) (by the isomorphismρ)
⊗U(h⊕n+)Cλ ∼= (U(n−)⊗CU(h⊕n+))⊗U(h⊕n+)Cλ
∼=U(n−)⊗C U(h⊕n+)⊗U(h⊕n+)Cλ
| {z }
∼=Cλ
∼=U(n−)⊗Cλ as graded U(n−) -modules.
This gives us a graded n−-module isomorphism U(n−)⊗Cλ → Mλ+ which is easily seen to send everyx⊗t∈U(n−)⊗Cλ to xtvλ+. Hence, Mλ+ =U(n−)v+λ. Since n− is concentrated in negative degrees, it is clear that U(n−) is concentrated in nonpositive degrees. Hence, U(n−)⊗Cλ is concentrated in nonpositive degrees, and thus the same
47Filtered by the usual filtration on the universal enveloping algebra of a Lie algebra. This filtration does not take into account the grading onn−,h⊕n+ andg.
48Here wedo take into account the grading on n−,h⊕n+ andg.
49If you are wondering why this statement is more than a blatantly obvious tautology, let me add some clarifications:
Agraded linear map is a morphism in the category of graded vector spaces. What I am stating here is that if a vector space isomorphism between graded vector spaces is at the same time a morphism in the category of graded vector spaces, then it must be anisomorphismin the category of graded vector spaces. This is very easy to show, but not a self-evident tautology. In fact, the analogous assertion about filtered vector spaces (i. e., the assertion that if a vector space isomorphism between filtered vector spaces is at the same time a morphism in the category of filtered vector spaces, then it must be anisomorphismin the category of filtered vector spaces) is wrong.
holds for Mλ+ (since Mλ+ ∼= U(n−)⊗Cλ as graded U(n−)-modules). In other words, Mλ+= L
n≥0
Mλ+[−n].
Since the isomorphismU(n−)⊗Cλ →Mλ+ which sends every x⊗t∈U(n−)⊗Cλ to xtv+λ is graded, it sends U(n−) [−n]⊗Cλ = (U(n−)⊗Cλ) [−n] to Mλ+[−n] for every n≥0. Thus, Mλ+[−n] =U(n−) [−n]v+λ for every n≥0. Hence,
dim Mλ+[−n]
= dim U(n−) [−n]v+λ
= dim (U(n−) [−n]) = dim (S(n−) [−n]) because U(n−)∼=S(n−) as graded vector spaces
(by the Poincar´e-Birkhoff-Witt theorem)
for every n≥0. Hence, if dimgj <∞for all j ≤ −1, then X
n≥0
dim Mλ+[−n]
qn=X
n≥0
dim (S(n−) [−n])qn= 1 Q
j≤−1
(1−q−j)dim((n−)j) = 1 Q
j≤−1
(1−q−j)dimgj. This proves Proposition 2.5.15(a).
(b) The proof of part (b) is analogous to that of (a).
This proves Proposition 2.5.15.
We have already encountered an example of a Verma highest-weight module:
Proposition 2.5.16. Letgbe the Lie algebraA0. Consider the Fock moduleF over the Lie algebra A0. Then, there is a canonical isomorphismM1+ →F ofA0-modules (where 1 is the element of h∗ which sends K to 1) which sends v+1 ∈M1+ to 1∈F. First proof of Proposition 2.5.16. As we showed in the First proof of Lemma 2.2.12, there exists a homomorphism ηF,1 : IndACK⊕A0 +
0 C → F of A0-modules such that ηF,1(1) = 1. In the same proof, we also showed that this ηF,1 is an isomor-phism. We thus have an isomorphismηF,1 : IndA0
CK⊕A+0 C→F ofA0-modules such that ηF,1(1) = 1. Since
IndACK⊕A0 +
0 C=U(A0)⊗U(CK⊕A+0)C=U(g)⊗U(h⊕n+)C1
since A0 =g, CK =h, A+0 =n+ and C=C1
=M1+, and since the element 1 of IndA0
CK⊕A+0 Cis exactly the elementv+1 ofM1+, this rewrites as follows: We have an isomorphismηF,1 :M1+→F of A0-modules such that ηF,1 v+1
= 1. This proves Proposition 2.5.16.
Second proof of Proposition 2.5.16. It is clear from the definition ofv1+thataiv1+= 0 for all i > 0, and that Kv+1 = v1+. Applying Lemma 2.2.12 to u = v1+ and V = M1+, we thus conclude that there exists a homomorphism η :F →M1+ of A0-modules such that η(1) =v+1.
On the other hand, sinceM1+ =U(g)⊗U(h⊕n+)C1 (by the definition ofM1+), we can define an U(g)-module homomorphism
M1+→F, α⊗U(h⊕n+)z 7→αz.
Since g = A0, this is an U(A0)-module homomorphism, i. e., an A0-module homo-morphism. Denote this homomorphism by ξ. We are going to prove that η and ξ are mutually inverse.
Since v+1 = 1⊗U(h⊕n+)1, we have ξ v1+
=ξ 1⊗U(h⊕n+)1
= 1·1 (by the definition of ξ)
= 1.
Since v1+ = η(1), this rewrites as ξ(η(1)) = 1. In other words, (ξ◦η) (1) = 1.
Since the vector 1 generates the A0-module F (because Lemma 2.2.10 yields P = P (a−1, a−2, a−3, ...)
| {z }
∈U(A0)
·1 ∈ U(A0)·1 for every P ∈ F), this yields that the A0-module homomorphisms ξ◦η : F → F and id : F → F are equal on a generating set of the A0-module F. Thus, ξ◦η = id.
Also, (η◦ξ) v1+
= η
ξ v1+
| {z }
=1
= η(1) = v1+. Since the vector v1+ generates M1+ as an A0-module (because M1+ =U(g)⊗U(h⊕n+)C1 =U(A0)⊗U(h⊕n+)C1), this yields that theA0-module homomorphisms η◦ξ :M1+→M1+ and id :M1+ →M1+ are equal on a generating set of theA0-module M1+. Thus, η◦ξ = id.
Since η◦ξ= id andξ◦η = id, the mapsξ andηare mutually inverse, so that ξis an isomorphismM1+ →F of A0-modules. We know that ξ sends v+1 to ξ v1+
= 1. Thus, there is a canonical isomorphism M1+ → F of A0-modules which sends v1+ ∈ M1+ to 1∈F. Proposition 2.5.16 is proven.
In analogy to the Second proof of Proposition 2.5.16, we can show:
Proposition 2.5.17. Let g be the Lie algebra A. Let µ ∈ C. Consider the µ-Fock module Fµ over the Lie algebra A. Then, there is a canonical isomorphism M1,µ+ →Fµ of A-modules (where (1, µ) is the element of h∗ which sends K to 1 and a0 to µ) which sends v1,µ+ ∈M1,µ+ to 1∈Fµ.
2.5.4. Degree-0 forms
We introduce another simple notion:
Definition 2.5.18. LetV andW be twoZ-graded vector spaces over a fieldk. Let β :V ×W →k be a k-bilinear form. We say that the k-bilinear form β has degree 0 (or, equivalently, is a degree-0 bilinear form) if and only if it satisfies
β(Vn×Wm) = 0 for all (n, m)∈Z2 satisfying n+m 6= 0 .
(Here, Vndenotes then-th homogeneous component ofV, andWmdenotes them-th homogeneous component of W.)
It is straightforward to see the following characterization of degree-0 bilinear forms:
Remark 2.5.19. Let V and W be two Z-graded vector spaces over a field k. Let β :V ×W →kbe ak-bilinear form. LetB be the linear mapV ⊗W →kinduced by the k-bilinear map V ×W →k using the universal property of the tensor product.
Consider V ⊗W as a Z-graded vector space (in the usual way in which one defines a grading on the tensor product of two Z-graded vector spaces), and considerk as a Z-graded vector space (by letting the whole field k live in degree 0).
Then, β has degree 0 if and only if B is a graded map.