Factorizable Module Algebras

0.39 MB·

by  Arkady Berenstein

FACTORIZABLE MODULE ALGEBRAS
7 ARKADYBERENSTEINANDKARLSCHMIDT
1
0
2

r
a Abstract. Theaimofthispaperistointroduceandstudyalargeclassofg-modulealgebraswhichwecall
M factorizable by generalizing the Gauss factorization of (square or rectangular) matrices. This class includes
coordinate algebras of corresponding reductive groups G, their parabolicsubgroups, basic affine spaces and

2 many others. It turns out that products of factorizable algebras are also factorizable and it is easy to
create a factorizable algebra our of virtually any g-module algebra. We also have quantum versions of all

] these constructions in the category of Uq(g)-module algebras. Quite surprisingly, our quantum factorizable
T algebrasarenaturallyactedonbythequantizedenvelopingalgebraUq(g∗)ofthedualLiebialgebrag∗ ofg.
R
.
h
t Contents
a
m
1. Introduction 1
[ 2. Definitions, Notation, and Results 3

2.1. Quantum factorization 4
2
v 2.2. Classical factorization 8
8 3. Proofs 11
9 3.1. Proof of Theorem 2.3 11
7
3.2. Proof of Theorem 2.4 12
5
3.3. Proof of Proposition 2.5 13
0
. 3.4. Proof of Corollary 2.6 14
1
3.5. Proof of Theorem 1.7 14
0
7 3.6. Proofs of Theorems 2.8 and 2.11 16
1 3.7. Proof of Theorem 2.13 23
: 3.8. Proof of Proposition 2.16 25
v
i 3.9. Proof of Proposition 2.17 25
X
3.10. Proofs of Theorems 2.23 and 2.26 25
r 3.11. Proof of Proposition 2.29 30
a
References 30
1. Introduction
The aim of this paper is to introduce and study a class of g-module algebras which we call factorizable by
generalizing the Gauss factorization of (square or rectangular) matrices.
Moreprecisely,letgbeacomplexsemisimpleLiealgebra. Givenacommutativeg-modulealgebraA (i.e.,
0
g actsby derivations),we saythat acommutative g-modulealgebraAis factorizable over A ifthere exists a
0
g-equivariantembedding of algebras ϕ:A ֒→A such that the µ◦(1⊗ϕ):A+⊗A →A is an isomorphism
0 0
of vector spaces (where µ denotes multiplication in A and A+ stands for the subalgebra of highest vectors
in A). Factorizable algebras abound in “nature” with A = M∗, the graded dual of the Verma module M
0 0 0
(actually, M∗ has a commutative algebra structure isomorphic to C[U] and such that g acts by derivations,
0
where U is the maximal unipotent subgroup of the corresponding Lie group G). Our first result shows that
the natural objects in the representation theory of g are factorizable.
Theorem 1.1. Up toa localization, thealgebra A=C[G/U]⊗n is factorizable over A =C[U] for any n≥1.
0
1
2 ARKADYBERENSTEINANDKARLSCHMIDT
Remark 1.2. If we replace C[G/U] with A =C[x ,...,x ] and let g=sl (C), then up to localization by
1 1 m m
principle minors, A⊗n = C[Mat ] (n ≥ m) is factorizable over C[U−] where U− is the group of all lower
1 m,n m m
unitriangular matrices in SL (C). This recovers the Gauss factorization of m×n matrices.
m
It turns out that factorizability of module algebras is not difficult to establish and reproduce.
Theorem 1.3. For any complex semisimple Lie algebra g, we have:
(a) (Theorems 2.23 & 2.26) Let A be g-module algebra such that A contains a subalgebra isomorphic to
C[U], invariant under the g-action and let A+ denote the subalgebra of all highest weight vectors in A. Then
the algebra A′ =A+⊗C[U] has a natural g-module algebra structure, such that A′ is factorizable over C[U].
(b) The assignments A 7→ A′ define a functor R from the category A of g-module algebras containing
g
C[U] as a g-module subalgebra to the category C of g-module algebras which are factorizable over C[U] (a full
g
subcategory of A ).
g
We can think of R as a “remembering” functor because it is right adjoint to the “forgetful” functor
F : C → A . Clearly, the composition R◦F is isomorphic to the identity functor on C . Now A has a
g g g g
natural multiplication such that for A,B ∈ A , the tensor product A⊗B is a g-module algebra, plus the
g
embedding C[U]=1⊗C[U]⊂A⊗B.
Proposition 1.4. (Proposition 2.29) C is closed under the natural multiplication on A .
g g
However, neither category is monoidal because each lacks a unit object. It is possible to fix the issue in
bothcategoriesbytensoringoverC[U]ratherthanoverC(seethediscussioninRemark2.30,wherewemake
this more concrete).
It turns out that we can build factorizable algebras over C[U] out of b -module algebras,where b is the
− −
lower Borel subalgebra of g. In fact, all factorizable algebras can be obtained this way.
Main Theorem 1.5. (Theorems 2.23 & 2.28) For any semisimple Lie algebra g, the assignments A7→A+
defines a functor P from A to thecategory b −ModAlg of b -module algebras. Moreover, the composition
g − −
P ◦F is an equivalence of categories C →b −ModAlg.
g −
Remark 1.6. Informally speaking, the theorem asserts that the “forgetful” highest vector functor A7→A+
f
from Ag to AlgC, in fact, “remembers almost everything.”
The functor P from Theorem 1.5 is highly nontrivial: it involves a quite mysterious b action on A such
−
thatA+ isab -invariantsubalgebra(seeSection2.2fordetails). Namely,theactionoftheCartansubalgebra
−
of b is inherited from that of g, but the action of the nilpotent generators f of b is given by the formula
− i −
(1.1) f ⊲a=f (a)−h (a)x
i i i i
for a∈A, where h is the i-th Cartan generator and x is the i-th “near-diagonalelement” in C[U] (⊂A).
i i
It is not difficult to show that f ⊲A+ ⊂A+, but it is much harder to prove that the operators f ⊲ satisfy
i i
the Serre relations (Theorem 2.23).
It turns out that all of the above results (including the mysterious Serre relations) can be quantized as
well. Namely, we replace g with its quantized enveloping algebra U (g) and proceed as follows. Given a
q
U (g)-module algebra A , we say that a U (g)-module algebra A is factorizable over A if there exists a
q 0 q 0
U (g)-equivariant embedding of algebras ϕ : A ֒→ A such that the map µ◦(1⊗ϕ) : A+ ⊗A → A is an
q 0 0
isomorphism of vector spaces. By quantizing our default choice of A in the classicalcase, we will now focus
0
on A =C [U], the quantized coordinate algebra of U (which is isomorphic to U (n )).
0 q q +
Theorem1.7. Uptoalocalization, thealgebraA=C [G/U]⊗n (thebraidedn-foldtensorpower ofC [G/U])
q q
is factorizable over A =C [U] for any n≥1.
0 q
Remark 1.8. If we replace C [G/U] with A = C [x ,...,x ], the algebra of q-polynomials and let g =
q 1 q 1 m
sl (C), then the braided n-th tensor power A⊗n = C [Mat ] (n ≥ m) is factorizable over C [U ], up to
m 1 q m,n q m
localization by principle minors. This recovers the Gauss factorization of quantum m×n matrices (see, for
example, [2]).
Asintheclassicalcase,factorizabilityofmodulealgebrasinthequantumcaseisalsoeasytoestablishand
reproduce.
FACTORIZABLE MODULE ALGEBRAS 3
Theorem 1.9. For any complex semisimple Lie algebra g, we have:
(a) (Theorems 2.8 & 2.11) Let A be a U (g)-module algebra such that A contains a subalgebra isomorphic
q
to C [U], invariant under the action of U (g) and let A+ denote the subalgebra of all highest vectors in A.
q q
The vector space A′ =A+⊗C [U] has the structureof a U (g)-module algebra and is factorizable over C [U].
q q q
(b) The assignments A7→A′ definea functor R from the category Aq of U (g)-modulealgebras containing
q g q
C [U] as a U (g)-module subalgebra to the category Cq of U (g)-module algebras which are factorizable over
q q g q
C [U] (a full subcategory of Aq).
q g
We can think of R as a “remembering” functor because it is right adjoint to the “forgetful” functor
q
F : Cq → Aq. Clearly, the composition R ◦F is the identity functor on Cq. In order to tensor multiply
q g g q q g
objectsofthesecategories,weneedto“trim”itabit. Namely,weconsiderthefullsubcategoryAq consisting
g
ofweight modulealgebrasinAq satisfyingsomeadditionalmildconditions(seeSection2.1). Itturnsoutthat
g
Aq has a natural braided tensor product which we denote by ⊗ (see, e.g. [5] and Section 2.1 below). Similar
g
to the classical case, for A,B ∈ Aq, A⊗B is naturally in Aq with an embedding C [U] = 1⊗C [U] ⊂ A⊗B.
g g q q
As in the classical case, this natural multiplication of course lacks a unit object.
Since Cq is a full subcategory of Aq, we can define Cq as the intersection of Cq and Aq inside Aq.
g g g g g g
Proposition 1.10. (Proposition 2.16) Cq is closed under the braided tensor multiplication in Aq.
g g
That is, the category Cq of factorizable U (g)-weight module algebras is “almost” monoidal but it lacks a
g q
unit object as in the classical case.
Similar to the classical case (Theorem 1.5), we can build factorizable module algebras over C [U] out of
q
some module algebras. However, unlike the expected U (b )-module algebras, we will deal with U (g∗)-
q − q
module algebras, where g∗ is the dual Lie bialgebra of g and all factorizable algebras are obtained this way.
Main Theorem 1.11. (Theorems 2.8 & 2.13) For any semisimple Lie algebra g, the assignments A7→A+
defines a functor P from Aq to the category U (g∗)−ModAlg of U (g∗)-module algebras. Moreover, the
q g q q
composition P ◦F is an equivalence of categories Cq→U (g∗)−ModAlg.
q q g q
Remark 1.12. Inthe spiritofRemark1.6,the theoremassertsthatthe assignmentA7→A+ is the forgetful
f
functor which “remembers almost everything.”
Remark 1.13. We firmly believe that the emergence of the Lie bialgebra g∗ here is not a mere coincidence,
but rather a manifestation of the “semiclassical story” behind the quantum one. We plan to investigate it
in a separate publication, when all relevant objects are Poisson algebras with a compatible action of the
Poisson-Lie group G, where its Poisson-Lie dual G∗ emerges naturally.
The functor P from Theorem 1.11 is highly nontrivial: it involves a quite mysterious U (g∗) action on A
q q
such that A+ is a U (g∗)-invariant subalgebra (see Section 2.1 for details). Namely, the Cartan subalgebra
q
action of U (g∗) is inherited from that of U (g), but the action of the generators F and F of U (g∗) is
q q i,1 i,2 q
given by the formulas
x a−K−1(a)x x K−1(a)−ax
(1.2) F ⊲a=F (a)− i i i, F ⊲a= i i i
i,1 i q −q−1 i,2 q −q−1
i i i i
fora∈A+,whereKi isthei-thCartangeneratorofUq(g),qi =qdi,andxi isthei-thgeneratorofCq[U]⊂A.
Asintheclassicalcase,themostnon-trivialpartoftheproofofthemaintheoremistoshowthatoperators
F ✄ and F ✄ given by (1.2) satisfy the quantum Serre relations, which we establish in Theorem 2.8 (and,
i,1 i,2
which is not difficult, but still surprising, in (1.2), all operators F ✄ commute with all F ✄).
i,1 i,2
2. Definitions, Notation, and Results
In this section, we will recall (or introduce) the relevant definitions and notation necessary to present our
mainresults,whichwillalsobeincluded. Webeginbydefiningthemainobjectofstudyinthispaper: module
algebras.
Definition 2.1. Let k be an arbitrary field and H a k-bialgebra. A k-algebra A is called an H-module
algebra if it is an H-module, multiplication (−)·(−) : A⊗kA → A is a homomorphism of H-modules, and
h(1)=ǫ(h)for h∈H, whereǫ is the counit ofH. Thatis to say,ifwe denote (in sumlessSweedler notation)
4 ARKADYBERENSTEINANDKARLSCHMIDT
∆(h)=h ⊗h forh∈H,thenfora,b∈A,h(a·b)=h (a)·h (b)andh(1)=ǫ(h). Ahomomorphismof
(1) (2) (1) (2)
H-modulealgebrasisahomomorphismofH-modulesandalgebras. WewilldenotethecategoryofH-module
algebras by H-ModAlg.
IfgisaLiealgebraoverk,thenwewillshorten“U(g)-modulealgebra”and“U(g)−ModAlg”to“g-module
algebra” and “g-ModAlg”,respectively.
Let g be a semisimple complex Lie algebra, with I×I symmetrizable Cartan matrix C =(c ) and fixed
i,j
choice of symmetrizers, (d ) , for C (i.e. d c = d c for i,j ∈ I) such that min{d | i ∈ I} = 1. g
i i∈I i i,j j j,i i
has Cartan subalgebra h. h∗ has basis {α | i ∈ I}, the simple roots of the associated root system. Let
i
(·,·) be the symmetric bilinear form on h∗ satisfying (α ,α ) = d c . As usual, we set α∨ := 2αi .
i j i i,j i (αi,αi)
Λ={λ∈h∗ | (α∨,λ)∈Z ∀i∈I} is the set of integralweights. ω ∈Λ is the i-thfundamental weight,which
i i
satisfies(α∨,ω )=δ fori,j ∈I. W is the Weyl groupofg, generatedby the simple reflections{s | i∈I},
i j i,j i
and has longest element w . Given w ∈ W, R(w) is the set of all i = (i ,i ,...,i ) ∈ {1,2,...,|I|}m such
0 1 2 m
that s s ···s is a reduced expression for w.
i1 i2 im
2.1. Quantum factorization. Throughout this section, all tensor products will be taken over C(q) unless
otherwise specified and written −⊗− rather than −⊗C(q)−.
U (g) is the quantum enveloping algebra of g, generated by elements {K±1,E ,F | i ∈ I} subject to the
q i i i
relations
K K =K K ; K K−1 =K−1K =1;
i j j i i i i i
K E K−1 =qci,jE ; K F K−1 =q−ci,jF ;
i j i i j i j i i j
1−ci,j 1−ci,j
(−1)kE(k)E E(1−ci,j−k) = (−1)kF(k)F F(1−ci,j−k) =0 if i6=j;
i j i i j i
k=0 k=0
X X
qm−q−m
where yi(n) = (n)1qi!yin, (n)qi! =(1)qi(2)qi···(n)qi, (m)qi = qii−qi−i 1 , and qi =qdi. Uq(g) is a Hopf algebra
with comultiplication (∆), counit (ǫ), and antipode (S) given on generators by
∆(K±1)=K±1⊗K±1; ǫ(K±1)=1 S(K±1)=K∓1;
i i i i i i
∆(E )=E ⊗K +1⊗E ; ǫ(E )=0; S(E )=−E K−1;
i i i i i i i i
∆(F )=F ⊗1+K−1⊗F ; ǫ(F )=0; S(F )=−K F .
i i i i i i i i
K is the Hopf subalgebra of U (g) generated by all K (as a C(q)-algebra), while U (b ) (respectively
q i q +
U (b )) is the Hopf subalgebra of U (g) generated by all K±1 and E (respectively K±1 and F ). We will
q − q i i i i
assume henceforth that all U (b )-modules are integrable. In other words, if M is a U (b )-module, we will
q + q +
assume that ∀x∈M,i∈I, ∃n≥0 such that En(x)=0. Note that every U (g)-module is a U (b )-module,
i q q +
so we are assuming these are “bounded above”. We do not assume the same for the action of F . For a
i
U (b )-module M, we designate M+ :={m∈M | E (m)=0 ∀i∈I}, the set of “highest vectors”. If A is a
q + i
U (b )-module algebra, then A+ is a subalgebra.
q +
Suppose M is a U (b )-module. For each i ∈ I and x ∈ M \{0}, set ℓ (x) = max{ℓ ∈ Z | Eℓ(x) 6= 0}
q + i ≥0 i
andE(top)(x)=E(ℓi(x))(x). Givenw ∈W oflengthm,i∈R(w), andx∈M\{0},wealsousetheshorthand
i i
E(top)(x)=E(top)E(top)···E(top)(x)
i im im−1 i1
and define ν (x)=(a ,a ,...,a ) by the following:
i 1 2 m
a =ℓ (E(top)E(top)···E(top)(x)).
k ik ik−1 ik−2 i1
Lastly, for j=(j ,...,j )∈Zm , we set E(j) :=E(jm)E(jm−1)···E(j1).
1 m ≥0 i im im−1 i1
Definition 2.2. Let A be a U (b )-module algebra, w ∈ W, and i ∈ R(w). If E(top)(x) ∈ A+ for all
q + i
x∈A\{0}, then we say A is i-adapted. We say a basis B for A is an i-adapted basis if
FACTORIZABLE MODULE ALGEBRAS 5
(1) E(top)(b)=1 for b∈B.
i
(2) The restriction of ν to B is an injective map B ֒→Zm , where m is the length of w.
i ≥0
Ifthere existsany w∈W andi∈R(w) so thatAis i-adapted,thenwe saymore generallythatA is adapted.
It turns out that if if A possesses an i-adapted basis for some i∈R(w) and is a “large enough” U (b )-
0 q +
module subalgebra of A, then A is factorizable over A . The following theorem makes this precise.
0
Theorem 2.3. Let A be a U (b )-module algebra. Given some w ∈ W and i ∈ R(w), let A be a U (b )-
q + 0 q +
module subalgebra of A possessing an i-adapted basis B. Denote by µ : A+ ⊗A → A the restriction of the
0
multiplication of A to these U (b )-module subalgebras inside A. Then
q +
(1) µ is an injective homomorphism of U (b )-modules.
q +
(2) µ is an isomorphism if and only if A is i-adapted and ν (A\{0})=ν (A \{0}).
i i 0
We willproveTheorem2.3in Section3.1. Theorem2.3demonstratesa closerelationshipbetweenbeing i-
adaptedandbeingfactorizableoveraU (b )-modulesubalgebrapossessingani-adaptedbasis. Thefollowing
q +
theorem explores this relationship from a different angle.
Theorem2.4. Letw ∈W andi∈R(w)andsupposeAisani-adaptedU (b )-modulealgebrawithaU (b )-
q + q +
module subalgebra A . Then µ : A+ ⊗A → A as in Theorem 2.3 is an isomorphism of U (b )-modules if
0 0 q +
and only if A possesses an i-adapted basis and ν (A \{0})=ν (A\{0}).
0 i 0 i
Theorem2.4isprovedinSection3.2. WenowrestrictourfocustoaspecificU (g)-modulealgebra,namely
q
C [U]. As a C(q)-algebra, C [U] is generated by the set {x | i∈I}, subject to the quantum Serre relations:
q q i
1−ci,j
(−1)kx(k)x x(1−ci,j−k) =0 if i6=j.
i j i
k=0
X
The U (g)-module structure on C [U] is summarized in the following equations:
q q
K±1(x )=q∓ci,jx ∀i,j ∈I
i j i j
E (x )=δ ∀i,j ∈I
i j i,j
x x−K−1(x)x
F (x)= i i i ∀i∈I,x∈C [U].
i q −q−1 q
i i
The first author and A. Zelevinsky observed in ([1], Proposition 3.5) that C [U] possesses a basis Bdual
q
such that, for i ∈ R(w ), the restriction of ν to Bdual is injective. Note that they use the notation A in
0 i
place of C [U] and view it only as a U (n )-module. As hinted by the notation, Bdual is the so-called “dual
q q +
canonical basis”. In Section 3.3, we prove the following proposition.
Proposition 2.5. Given any i∈R(w ), Bdual is an i-adapted basis for C [U].
0 q
Combining Proposition 2.5 with Theorem 2.3, we are led to the following corollary, though it does still
require some proof.
Corollary 2.6. Let A be a U (g)-module algebra containing C [U] as a U (g)-module subalgebra. If there
q q q
existsaU (g)-modulealgebraA′ containingAasaU (g)-modulesubalgebra,suchthatA′ isgeneratedby(A′)+
q q
asaU (g)-modulealgebra, thenµ:A+⊗C [U]→Aasin Theorem 2.3is anisomorphism ofU (b )-modules.
q q q +
Corollary2.6willbeprovedinSection3.4andprovidesuswiththemeanstoproveTheorem1.7,whichwe
do in Section 3.5. In the meantime, there are two families of quantities that arose in the proof of Corollary
2.6:
K−2(a)−a
x a−K (a)x and F (a)+x i
i i i i i q −q−1
i i
where i ∈ I and a ∈ A′. These quantities are equally valid to consider for a ∈ A, without the assumed
presence of A′. If a ∈ A+, then both of these quantities are also in A+. It is therefore natural to ask what
K−2−1
relationsthe families of operatorsL −R K andF +L i satisfy, where L (respectively R ) represents
i i i i iqi−qi−1 i i
left (respectivelyright)multiplication by x . Orto putit another way,do these operatorsindicate the action
i
of a known algebra which is somehow related to U (g)? We can answer in the affirmative. It can be proved
q
6 ARKADYBERENSTEINANDKARLSCHMIDT
thatbothofthefamiliesofoperatorsobservedinfactsatisfythequantumSerrerelationsandthetwofamilies
“almost”commute with eachother. This resembles anactionofU (g∗), which we now define for the reader’s
q
convenience.
Definition 2.7. U (g∗) is a Hopf algebra, generated by {K±1,F ,F | i ∈ I} subject to the following
q i i,1 i,2
relations for i,j ∈I and k ∈{1,2}:
K±1K±1 =K±1K±1; K K−1 =K−1K =1;
i j j i i i i i
F F =F F ; K F K−1 =q−ci,jF ;
i,1 j,2 j,2 i,1 i j,k i i j,k
1−ci,j
(−1)kF(ℓ)F F(1−ci,j−ℓ) =0 if i6=j.
i,k j,k i,k
ℓ=0
X
The comultiplication (∆), counit (ǫ), and antipode (S) are given on generators by
∆(K±1)=K±1⊗K±1; ǫ(K±1)=1; S(K±1)=K∓1;
i i i i i i
∆(F )=F ⊗1+K−1⊗F ; ǫ(F )=0; S(F )=−K F ;
i,1 i,1 i i,1 i,1 i,1 i i,1
∆(F )=F ⊗K−1+1⊗F ; ǫ(F )=0; S(F )=−F K .
i,2 i,2 i i,2 i,2 i,2 i,2 i
AftersometweakingandcombiningofouroperatorswiththeinheritedCartanaction,weseethatouroper-
atorsreallydoindicatethepresenceofaU (g∗)-modulealgebrastructure. Thefollowingtheoremsummarizes
q
and is proved in Section 3.6.
Theorem 2.8. Let A be a U (g)-module algebra containing C [U] as a U (g)-module subalgebra. Then A is
q q q
a U (g∗)-module algebra with action given by
q
x a−K−1(a)x x K−1(a)−ax
K±1⊲a=K±1(a), F ⊲a=F (a)− i i i, F ⊲a= i i i.
i i i,1 i q −q−1 i,2 q −q−1
i i i i
In particular, the subalgebra A+ is invariant under this action of U (g∗) and is therefore a U (g∗)-module
q q
subalgebra.
Theorem 2.8 is in some sense a statement about the existence of a functor. To make this precise, we
introduce a category whose objects bear properties similar to those found in Theorem 2.3.
Definition 2.9. Let Cq be the category whose objects consist of pairs (A,ϕ ), where
g A
• A is an adapted U (g)-module algebra such that ν (A\{0})=ν (C [U]\{0}) for all i∈R(w ).
q i i q 0
• ϕ :C [U]֒→A is an embedding of U (g)-module algebras.
A q q
A morphism (A,ϕ ) → (B,ϕ ) in Cq is a homomorphism of U (g)-module algebras ψ : A → B such that
A B g q
ψ◦ϕ =ϕ .
A B
Given a homomorphism of U (g)-module algebras ψ : A → B, it follows that ψ(A+) ⊆ B+, so ψ| may
q A+
bethoughtofasamapofC(q)-algebrasA+ →B+. IfψisamorphisminCq,(A,ϕ )→(B,ϕ ),thenactually
g A B
ψ| is a homomorphism of U (g∗)-module algebras. As a consequence, we have the following corollary.
A+ q
Corollary 2.10. There is a functor (−)+ : Cq → U (g∗)−ModAlg (denoted P ◦F in Section 1) which
g q q q
assignstoanobject(A,ϕ )ofCq itssubalgebraofhighest vectorsA+,equippedwiththeU (g∗)-modulealgebra
A g q
structure of Theorem 2.8. (−)+ is given on morphisms by restriction.
Theorem2.3stronglysuggeststhat(−)+ mightactuallybeanequivalenceofcategories. Infact,thisisthe
case,but in order to describe a quasi-inverse,we need the following theoremwhich describes a U (g)-module
q
algebra structure on A⊗C [U] if A is a U (g∗)-module algebra.
q q
FACTORIZABLE MODULE ALGEBRAS 7
Theorem 2.11. If A is a U (g∗)-module algebra, then A⊗C [U] has the structureof a U (g)-module algebra
q q q
satisfying the following formulas:
(1⊗x )(a⊗1)=K (a)⊗x +(q −q−1)F K (a)⊗1
i i i i i i,2 i
K±1⊲(a⊗x)=K±1(a)⊗K±1(x)
i i i
E ⊲(a⊗x)=a⊗E (x)
i i
K (a)−K−1(a)
F ⊲(a⊗x)=(F (a)+F K (a))⊗x+ i i ⊗x x+K−1(a)⊗F (x).
i i,1 i,2 i q −q−1 i i i
i i
Theorem 2.11 will be proved in Section 3.6. As a result, we have the following corollary.
Corollary 2.12. There is a functor (−)⊗C [U]:U (g∗)−ModAlg→Cq which assigns to a U (g∗)-module
q q g q
algebra A, the pair (A⊗C [U],1⊗id), where A⊗C [U] is given the U (g)-module structureof Theorem 2.11.
q q q
(−)⊗C [U] is given on morphisms by ψ 7→ψ⊗id.
q
The following theorem says that (−)⊗C [U] is the promised quasi-inverse for (−)+.
q
Theorem 2.13. (−)+ :Cq →U (g∗)−ModAlg and(−)⊗C [U]:U (g∗)−ModAlg→Cq arequasi-inverse
g q q q g
equivalences of categories.
Theorem 2.13 is proved in section 3.7. Now, it is well-known that if A and B are U (g)-weight module
q
algebras, then so is the braided tensor product A⊗B. A⊗B has multiplication given by (a⊗b)(a′ ⊗b′) =
aR (a′)⊗R (b)b′, where R is the universal R-matrix for U (g) and we use sumless Sweedler notation
(2) (1) q
R=R ⊗R . The R-matrix is of the form
(1) (2)
←
R=qPi,jdi(C−1)i,jHi⊗Hj eq(qα−α2−qα−1)Eα⊗Fα!
α>0
Y
(see Section 3.3 in [5] for details). We define a subcategory of Cq on which this tensor product will make
g
sense.
Definition 2.14. Let Cq be the full subcategory of Cq whose objects consist of pairs (A,ϕ ), where A is
g g A
additionally assumed to be a U (g)-weight module algebra.
q
The following proposition is then clear.
Proposition 2.15. (−)+ and (−)⊗C [U] restrict to equivalences between Cq and U (g∗)−WModAlg, the
q g q
category of U (g∗)-weight module algebras.
q
If (A,ϕ ) and (B,ϕ ) are objects of Cq, then we already saw that A⊗B is also a U (g)-weight module
A B g q
algebra. Furthermore, it is obvious that 1⊗ϕ and ϕ ⊗1 are injections C [U]֒→A⊗B. However,it is not
B A q
immediately obvious that A⊗B is adapted with ν (A⊗B\{0})= ν (C [U]\{0}) ∀i ∈R(w ). Nevertheless,
i i q 0
this is the case, which the following proposition asserts.
Proposition 2.16. If (A,ϕ ) and (B,ϕ ) are objects of Cq, then (A⊗B,1⊗ϕ ) and (A⊗B,ϕ ⊗1) are
A B g B A
objects of Cq as well.
g
Proposition 2.16 is proved in Section 3.8. Proposition 2.15 allows us to turn Proposition 2.16 into a
statement about U (g∗)-module algebras. We define two “fusion” products on the category of U (g∗)-weight
q q
module algebras, namely the following:
A∗B :=((A⊗C [U])⊗(B⊗C [U]),1⊗1⊗1⊗id)+
q q
A⋆B :=((A⊗C [U])⊗(B⊗C [U]),1⊗id⊗1⊗1)+
q q
The reader may be bothered that the objects (A⊗B,1⊗ϕ ) and (A⊗B,ϕ ⊗1) are not (necessarily at
B A
least) isomorphic despite having equal underlying U (g)-module algebras. An attempt to force a common
q
quotient leads to the discovery of an interesting U (g∗)-module algebra structure on A⊗B if A and B are
q
U (g∗)-weight module algebras.
q
8 ARKADYBERENSTEINANDKARLSCHMIDT
Proposition 2.17. Let A and B be U (g∗)-weight module algebras. Then the C(q)-vector space A⊗B has
q
the structure of a U (g∗)-weight module algebra satisfying the following equations
q
(a⊗b)(a′⊗b′)=qPi,jdi(C−1)i,j(α∨i,|b′|)(α∨j,|a′|)aa′⊗bb′
K±1✄(a⊗b)=K±1(a)⊗K±1(b)
i i i
F ✄(a⊗b)=K−1(a)⊗F (b)
i,1 i i,1
F ✄(a⊗b)=F (a)⊗K−1(b).
i,2 i,2 i
for weight vectors a,a′ ∈A, b,b′ ∈B, and i∈I, where |·| indicates weight.
Proposition 2.17 is proved in Section 3.9 and induces a fusion product on Cq:
g
(A,ϕ )⋄(B,ϕ ):=(((A,ϕ )+⊗(B,ϕ )+)⊗C [U],1⊗1⊗id).
A B A B q
2.2. Classical factorization. Throughout this section, all tensor products will be taken over C unless
otherwise specified and written −⊗− rather than −⊗C−. We also assume henceforth that every algebra is
commutative unless otherwise stated.
ThesemisimplecomplexLiealgebragwithCartanmatrixC =(c )isgeneratedbyelements{e ,f ,h |i∈
i,j i i i
I} subject to the following relations:
[h ,h ]=0; [e ,f ]=δ h ;
i j i j i,j i
[h ,e ]=c e ; [h ,f ]=c f ;
i j i,j j i j i,j j
(ad e )1−ci,j(e )=(ad f )1−ci,j(f )=0
i j i j
where as usual (ad x)(y) = [x,y] for x,y ∈ g. The universal enveloping algebra U(g) of g is a (noncom-
mutative) Hopf algebra on the same generators and relations, where [x,y] = xy −yx for x,y ∈ U(g). The
comultiplication of U(g) is given on generators by
∆(x)=x⊗1+1⊗x
for x∈{e ,f ,h | i∈I}.
i i i
n (respectivelyb )is the Lie subalgebraofg generatedbyall e (respectivelyh andf ). We willassume
+ − i i i
henceforth that all n -modules are integrable. In other words, if M is an n -module, we will assume that
+ +
∀x∈M,i∈I,∃n≥0suchthaten(x)=0. Notethateveryg-moduleisalsoan -module,soweareassuming
i +
these are “bounded above” as well. We do not assume the same for the action of f . For a g-module M, we
i
designate M+ :={m∈M | e (m)=0 ∀i∈I}, the set of “highest vectors”. If A is a g-module algebra, then
i
A+ is a subalgebra. For n∈Z and i∈I, we will use the notation e(n) = 1en.
≥0 i n! i
SupposeM isann -modulealgebra. Foreachi∈I andx∈M\{0},setℓ (x)=max{ℓ∈Z |eℓ(x)6=0}
+ i ≥0 i
and e(top)(x)=e(ℓi(x))(x). Given w ∈W of length m, i∈R(w), and x∈M \{0}, we also use the shorthand
i i
e(top)(x)=e(top)e(top) ···e(top)(x)
i im im−1 i1
and define ν (x)=(a ,a ,...,a ) by the following:
i 1 2 m
a =ℓ (e(top)e(top)···e(top)(x)).
k ik ik−1 ik−2 i1
Lastly, for j=(j ,...,j )∈Zm , we set e(j) :=e(jm)e(jm−1)···e(j1).
1 m ≥0 i im im−1 i1
Definition2.18. LetAbeann -modulealgebra,w ∈W,andi∈R(w). Ife(top)(x)∈A+ forallx∈A\{0},
+ i
then we say A is i-adapted. We say a basis B for A is an i-adapted basis if
(top)
(1) e (b)=1 for b∈B.
i
(2) The restriction of ν to B is an injective map B ֒→Zm , where m is the length of w.
i ≥0
Ifthere existsany w∈W andi∈R(w) so thatAis i-adapted,thenwe saymore generallythatA is adapted.
As in the quantum case, if A possesses an i-adapted basis for some i ∈ R(w) and is a “large enough”
0
n -module subalgebra of A, then A is factorizable over A . The following theorem makes this precise.
+ 0
FACTORIZABLE MODULE ALGEBRAS 9
Theorem 2.19. Let A be an n -module algebra. Given some w ∈ W and i ∈ R(w), let A be an n -
+ 0 +
module subalgebra of A possessing an i-adapted basis B. Denote by µ : A+ ⊗A → A the restriction of the
0
multiplication of A to these n -module subalgebras inside A. Then
+
(1) µ is an injective homomorphism of n -modules.
+
(2) µ is an isomorphism if and only if A is i-adapted and ν (A\{0})=ν (A \{0}).
i i 0
Theorem 2.19 demonstrates a close relationship between being i-adapted and being factorizable over an
n -module subalgebra possessing an i-adapted basis. The following theorem explores this relationship from
+
a different angle.
Theorem 2.20. Let w ∈ W and i ∈ R(w) and suppose A is an i-adapted n -module algebra with an n -
+ +
module subalgebra A . Then A is factorizable over A if and only if A possesses an i-adapted basis and
0 0 0
ν (A \{0})=ν (A\{0}).
i 0 i
The proofsofTheorems2.19and 2.20are nearlyidenticalto those ofTheorems 2.3and2.4 (respectively),
so we do not replicate them here. We now restrict our focus to a specific g-module algebra, C[U]. Actually,
C[U] is a specialization of C [U] to q = 1. It is a (commutative) Poisson algebra with Poisson generators
q
{x | i∈I}, subject to the relations
i
ε(i,j,1−c )=0 if i6=j,
i,j
where ε(i,j,n) is defined inductively by ε(i,j,0)=x , ε(i,j,n+1)={x ,ε(i,j,n)}−d (c +2n)x ε(i,j,n).
j i i i,j i
The g-module structure on C[U] is summarized in the following equations:
h (x )=−c x ∀i,j ∈I
i j i,j j
h ({x,y})={h (x),y}+{x,h (y)} ∀i∈I, x,y ∈C[U]
i i i
e (x )=δ ∀i,j ∈I
i j i,j
e ({x,y})={e (x),y}+{x,e (y)}+d e (x)h (y)−d h (x)e (y) ∀i∈I, x,y ∈C[U]
i i i i i i i i i
1 1
f (x)= {x ,x}+ x h (x) ∀i∈I, x∈C[U].
i i i i
2d 2
i
Remark 2.21. ComparingthedefiningrelationsofC[U]withtheactionoff thereon,oneseesthatf (x )=
i i i
−x2 and ε(i,j,n) = (2d )nfn(x ), so that f1−ci,j(x ) = 0 if i 6= j. Observe also that {x ,x} = 2d f (x)−
i i i j i j i i i
d x h (x) for x∈C[U] and i∈I. It follows that C[U] is generated as a g-module algebra by {x | i∈I}.
i i i i
It is known that C[U] possesses an i-adapted basis for every i∈R(w ) (in fact, this follows from the fact
0
that C [U] does). Combining this fact with Theorem 2.19 leads us to the following corollary.
q
Corollary 2.22. Let A be a g-module algebra containing C[U] as a g-module subalgebra. µ:A+⊗C[U]→A
as in Theorem 2.19 is an isomorphism of n -modules if and only if there exists a g-module algebra A′ which
+
contains A as a g-module subalgebra and is generated by (A′)+ as a g-module algebra.
The proof of the “if” part of Corollary 2.22 is very similar to the proof of Corollary 2.6 so we do not
reproduce it here. The “only if” part will be a very easy consequence of the discussion at the end of Section
2.2, so we will address it there. Also, just as in the quantum setting, Corollary 2.22 leads to a proof of
Theorem 1.1, which is nearly identical to that of Theorem 1.7, so we won’t include it here. We do, however,
note that instead of two families of quantities as arose in the proof of Corollary 2.6, only one arises in the
proof of the “if” part of Corollary 2.22:
f (a)−h (a)x
i i i
where i ∈ I and a ∈ A′. As in the quantum case, if a ∈ A+, then f (a)−h (a)x ∈ A+. Once again it is
i i i
natural to ask what relations the family of operators f −m h satisfies, where m denotes multiplication by
i i i i
x . Or to put it another way, do these operators indicate the action of a known algebra or Lie algebra which
i
is somehow related to g? Again, we can answer in the affirmative, which the following theorem summarizes.
Theorem 2.23. Let A be a g-module algebra containing C[U] as a g-module subalgebra. Then A has another
structure of a b -module algebra with action given by the formulas
−
h ✄a=h (a), f ✄a=f (a)−h (a)x .
i i i i i i
In particular, the subalgebra A+ is invariant under this b action and is therefore a b -module subalgebra.
− −
10 ARKADYBERENSTEINANDKARLSCHMIDT
Theorem2.23isprovedinSection3.10. Asinthequantumcase,Theorem2.23isinsomesenseastatement
abouttheexistenceofafunctor. Tomakethisprecise,weintroduceacategorywhoseobjectsbearproperties
similar to those found in Theorem 2.19.
Definition 2.24. Let C be the category whose objects consist of pairs (A,ϕ ), where
g A
• A is an adapted g-module algebra such that ν (A\{0})=ν (C[U]\{0}) for all i∈R(w );
i i 0
• ϕ :C[U]֒→A is an embedding of g-module algebras.
A
A morphism (A,ϕ ) → (B,ϕ ) in C is a homomorphism of g-module algebras ψ : A → B such that
A B g
ψ◦ϕ =ϕ .
A B
Given a homomorphism of g-module algebras ψ : A → B, it follows that ψ(A+) ⊆ B+, so ψ| may be
A+
thought of as a map of C-algebras A+ → B+. If ψ is a morphism in C , (A,ϕ ) → (B,ϕ ), then actually
g A B
ψ| is ahomomorphismofb -module algebras,wherethe b -modulestructureis theone giveninTheorem
A+ − −
2.23. As a consequence, we have the following corollary.
Corollary 2.25. There is a functor (−)+ :C →b −ModAlg (denoted P ◦F in Section 1) which assigns
g −
to an object (A,ϕ ) of C its subalgebra of highest vectors A+, equipped with the b -module algebra structure
A g −
of Theorem 2.23. (−)+ is given on morphisms by restriction.
Again, we hope for a quasi-inverse functor for (−)+, but we must first state the following theorem.
Theorem 2.26. IfA is a b -module algebra, then A⊗C[U]has thestructureof ag-modulealgebra satisfying
−
the following formulas:
(1⊗x)(a⊗1)=a⊗x
h ✄(a⊗x)=h (a)⊗x+a⊗h (x)
i i i
e ✄(a⊗x)=a⊗e (x)
i i
f ✄(a⊗x)=f (a)⊗x+h (a)⊗x x+a⊗f (x).
i i i i i
Theorem 2.26 will be proved in Section 3.10. The following corollary is almost immediate.
Corollary 2.27. There is a functor (−)⊗C[U]:b −ModAlg→C which assigns to a b -module algebra
− g −
A, the pair (A⊗C[U],1⊗id), where A⊗C[U] is given the g-module algebra structure of Theorem 2.26.
(−)⊗C[U] is given on morphisms by ψ 7→ψ⊗id.
As promised, (−)⊗C[U] is the desired quasi-inverse for (−)+.
Theorem 2.28. (−)+ : C → b − ModAlg and (−)⊗ C[U] : b −ModAlg → C are quasi-inverse
g − − g
equivalences of categories.
The proof of Theorem 2.28 is very similar to that of 2.13, so we do not include it here. Now, it is
well-known that if A and B are g-module algebras, then so is A⊗B. A⊗B has the na¨ıve multiplication
(a⊗b)(a′⊗b′) = aa′⊗bb′. So if (A,ϕ ) and (B,ϕ ) are objects of C , then A⊗B is a g-module algebra.
A B g
Furthermore,itisobviousthat1⊗ϕ andϕ ⊗1areinjectionsC[U]֒→A⊗B. However,itisnotimmediately
B A
obviousthat A⊗B is adapted with ν (A⊗B\{0})=ν (C[U]\{0})∀i∈R(w ). Nevertheless,this is the case
i i 0
and the following proposition asserts as much and is proved in Section 3.11.
Proposition 2.29. If (A,ϕ ) and (B,ϕ ) are objects of C , then (A⊗B,1⊗ϕ ) and (A⊗B,ϕ ⊗1) are
A B g B A
objects of C as well.
g
Theorem2.28andProposition2.29allowus to define two“fusion”productsonthe categoryofb -module
−
algebras,namely the following:
A∗B :=((A⊗C[U])⊗(B⊗C[U]),1⊗1⊗1⊗id)+
A⋆B :=((A⊗C[U])⊗(B⊗C[U]),1⊗id⊗1⊗1)+
Unfortunately,asinthequantumcase,theobjects(A⊗B,1⊗ϕ )and(A⊗B,ϕ ⊗1)arenot(necessarily
B A
atleast)isomorphic,despite having equalunderlying g-module algebras. However,givenb -module algebras
−