Classical limit of quantum Borcherds-Bozec algebras
aa r X i v : . [ m a t h . R T ] J a n CLASSICAL LIMIT OF QUANTUM BORCHERDS-BOZEC ALGEBRAS
ZHAOBING FAN, SEOK-JIN KANG, YOUNG ROCK KIM ∗ , AND BOLUN TONG Abstract.
Let g be a Borcherds-Bozec algebra, U ( g ) be its universal enveloping algebraand U q ( g ) be the corresponding quantum Borcherds-Bozec algebra. We show that theclassical limit of U q ( g ) is isomorphic to U ( g ) as Hopf algebras. Thus U q ( g ) can beregarded as a quantum deformation of U ( g ). We also give explicit formulas for thecommutation relations among the generators of U q ( g ). Introduction
The quantum Borcherds-Bozec algebras were introduced by T. Bozec in his research ofperverse sheaves theory for quivers with loops [1, 2, 3]. They can be treated as a furthergeneralization of quantum generalized Kac-Moody algebras. Even though they use thesame Borcherds-Cartan data, the construction of the quantum groups are quite different.More precisely, the quantum Borcherds-Bozec algebras have more generators and defin-ing relations than quantum generalized Kac-Moody algebras. For each simple root α i withimaginary index, there are infinitely many generators e il , f il ( l ∈ Z > ) whose degrees are l multiples of α i and − α i . Bozec deals with these generators by treating them as similarpositions as divided powers θ ( l ) i in Lusztig algebras.Bozec gave the general definition of Lusztig sheaves for arbitrary quivers (possibly withmultiple loops) and constructed the canonical basis for the positive half of a quantumBorcherds-Bozec algebra in terms of simple perverse sheaves (cf. [15]). In [2], he studiedthe crystal basis theory for quantum Borcherds-Bozec algebras. He defined the notion ofKashiwara operators and abstract crystals, which provides an important framework forKashiwara’s grand-loop argument (cf. [11]). He also gave a geometric construction of thecrystal for the negative half of a quantum Borcherds-Bozec algebra based on the theory ofLusztig perverse sheaves associated to quivers with loops (cf. [12, 9]), and gave a geometric Mathematics Subject Classification.
Key words and phrases. quantum Borcherds-Bozec algebra, classical limit, commutation relationBorcherds-Bozec algebra. ∗ Corresponding author. All authors have equal contributions. realization of generalized crystals for the integrable highest weight representations via
Nakajima’s quiver varieties (cf. [16, 10]).For a Kac-Moody algebra g , G. Lusztig showed that the integrable highest weight mod-ule L over U ( g ) can be deformed to those integrable highest weight module L over U q ( g )in such a way that the dimensions of weight spaces are invariant under the deformation(cf. [14, 6]). Let A = Q [ q, q − ] be the Laurent polynomial rings, Lusztig constructed a A -subalgebra U A of U q ( g ) generated by divided powers and k ± i , and defined a U A -submodule L A of L . He proved that F ⊗ A L A is isomorphic to L as U ( g )-modules, where F = A /I and I is the ideal of A generated by ( q − U q ( g ) is a deformation of U ( g ) as a Hopf algebra and show that a highest weight U ( g )-module admits a deformation to a highest weight U q ( g )-module. They used the A -formof U q ( g ) and highest weight U q ( g )-module, where A is the localization of Q [ q ] at the ideal( q − A = Q [ q, q − ] ⊆ A .In this paper, we study the classical limit theory of quantum Borcherds-Bozec algebras.We first review some basic notions of Borcherds-Bozec algebras and quantum Borcherds-Bozec algebras. For their representation theory, the readers may refer to [7, 8]. As weshow in Appendix, the commutation relations between e il and f jk are rather complicated.For the aim of classical limit, we need another set of generators. Thanks to Bozec, thereexists an alternative set of primitive generators in U q ( g ), which we denote by s il and t il .They satisfy a simpler set of commutation relations s il t jk − t jk s il = δ ij δ lk τ il ( K li − K − li )for some constants τ il ∈ Q ( q ). Using Lusztig’s approach, we prove that these generatorsalso satisfy the Serre-type relations (cf. [13, Chapter 1]).In Section 3, we define the A -form of quantum Borcherds-Bozec algebras and theirhighest weight representations. We show that the triangular decomposition of U q ( g ) carriesover to A -form. In Section 4, we study the process of taking the limit q →
1. Let U = Q ⊗ A U A be a Q -algebra, where U A is the A -form of U q ( g ). We prove thatthe classical limit U of U q ( g ) is isomorphic to the universal enveloping algebra U ( g )as Hopf algebras, and when we take the classical limit, the Verma module and highestweight modules of U q ( g ) tend to those Verma module and highest weight modules of U ( g ),respectively. Finally, we give the concrete commutation relations between the generators e il and f jk of U q ( g ) in Appendix, they have an interesting combinatorial structure. Acknowledgements . Z. Fan is partially supported by the NSF of China grant 11671108and the Fundamental Research Funds for the central universities GK2110260131. S.-J.Kang was supported by Hankuk University of Foreign Studies Research Fund. Y. R. Kim
LASSICAL LIMIT OF QUANTUM BORCHERDS-BOZEC ALGEBRAS 3 was supported by the Basic Science Research Program of the NRF (Korea) under grantNo. 2015R1D1A1A01059643.1.
Borcherds-Bozec algebras
Let I be an index set possibly countably infinite. An integer-valued matrix A = ( a ij ) i,j ∈ I is called an even symmetrizable Borcherds-Cartan matrix if it satisfies the following con-ditions:(i) a ii = 2 , , − , − , . . . ,(ii) a ij ∈ Z ≤ for i = j ,(iii) there is a diagonal matrix D = diag( r i ∈ Z > | i ∈ I ) such that DA is symmetric.Set I re := { i ∈ I | a ii = 2 } , the set of real indices and I im := { i ∈ I | a ii ≤ } , the setof imaginary indices . We denote by I iso := { i ∈ I | a ii = 0 } the set of isotropic indices .A Borcherds-Cartan datum consists of(a) an even symmetrizable Borcherds-Cartan matrix A = ( a ij ) i,j ∈ I ,(b) a free abelian group P ∨ = (cid:0)L i ∈ I Z h i (cid:1) ⊕ (cid:0)L i ∈ I Z d i (cid:1) , the dual weight lattice ,(c) h = Q ⊗ Z P ∨ , the Cartan subalgebra ,(d) P = { λ ∈ h ∗ | λ ( P ∨ ) ⊆ Z } , the weight lattice ,(e) Π ∨ = { h i ∈ P ∨ | i ∈ I } , the set of simple coroots ,(f) Π = { α i ∈ P | i ∈ I } , the set of simple roots , which is linearly independent over Q and satisfies α j ( h i ) = a ij , α j ( d i ) = δ ij for all i, j ∈ I. (g) for each i ∈ I , there is an element Λ i ∈ P such thatΛ i ( h j ) = δ ij , Λ i ( d j ) = 0 for all i, j ∈ I. The Λ i ( i ∈ I ) are called the fundamental weights .We denote by P + := { λ ∈ P | λ ( h i ) ≥ i ∈ I } the set of dominant integral weights . The free abelian group Q := L i ∈ I Z α i is called the root lattice . Set Q + = P i ∈ I Z ≥ α i and Q − = − Q + . For β = P k i α i ∈ Q + , we define its hight to be ht( β ) := P k i . ZHAOBING FAN, SEOK-JIN KANG, YOUNG ROCK KIM, AND BOLUN TONG
There is a non-degenerate symmetric bilinear form ( , ) on h ∗ satisfying( α i , λ ) = r i λ ( h i ) for all λ ∈ h ∗ , and therefore we have ( α i , α j ) = r i a ij = r j a ji for all i, j ∈ I. For i ∈ I re , we define the simple reflection ω i ∈ GL ( h ∗ ) by ω i ( λ ) = λ − λ ( h i ) α i for λ ∈ h ∗ . The subgroup W of GL ( h ∗ ) generated by ω i ( i ∈ I re ) is called the Weyl group of g . Onecan easily verify that the symmetric bilinear form ( , ) is W -invariant.Let I ∞ := ( I re × { } ) ∪ ( I im × Z > ). For simplicity, we will often write i for ( i,
1) if i ∈ I re . Definition 1.1.
The
Borcherds-Bozec algebra g associated with a Borcherds-Cartan da-tum ( A, P, Π , P ∨ , Π ∨ ) is the Lie algebra over Q generated by the elements e il , f il (( i, l ) ∈ I ∞ ) and h with defining relations(1.1) [ h, h ′ ] = 0 for h, h ′ ∈ h , [ e ik , f jl ] = k δ ij δ kl h i for i, j ∈ I, k, l ∈ Z > , [ h, e jl ] = lα j ( h ) e jl , [ h, f jl ] = − lα j ( h ) f jl , (ad e i ) − la ij ( e jl ) = 0 for i ∈ I re , i = ( j, l ) , (ad f i ) − la ij ( f jl ) = 0 for i ∈ I re , i = ( j, l ) , [ e ik , e jl ] = [ f ik , f jl ] = 0 for a ij = 0 . Let U ( g ) be the universal enveloping algebra of g . Since we have the following equationsin U ( g ) (ad x ) m ( y ) = m X k =0 ( − k (cid:18) mk (cid:19) x m − k yx k for x, y ∈ U ( g ) , m ∈ Z ≥ , we obtain the presentation of U ( g ) with generators and relations given below. Proposition 1.2.
The universal enveloping algebra U ( g ) of g is an associative algebraover Q with unity generated by e il , f il (( i, l ) ∈ I ∞ ) and h subject to the following defining LASSICAL LIMIT OF QUANTUM BORCHERDS-BOZEC ALGEBRAS 5 relations(1.2) hh ′ = h ′ h for h, h ′ ∈ h ,e ik f jl − f jl e ik = k δ ij δ kl h i for i, j ∈ I, k, l ∈ Z > ,he jl − e jl h = lα j ( h ) e jl , hf jl − f jl h = − lα j ( h ) f jl , − la ij X k =0 ( − k (cid:18) − la ij k (cid:19) e i − la ij − k e jl e ki = 0 for i ∈ I re , i = ( j, l ) , − la ij X k =0 ( − k (cid:18) − la ij k (cid:19) f i − la ij − k f jl f ki = 0 for i ∈ I re , i = ( j, l ) ,e ik e jl − e jl e ik = f ik f jl − f jl f ik = 0 for a ij = 0 . The universal enveloping algebra U ( g ) has a Hopf algebra structure given by(1.3) ∆( x ) = x ⊗ ⊗ x,ε ( x ) = 0 ,S ( x ) = − x for x ∈ g , where ∆ : U ( g ) → U ( g ) ⊗ U ( g ) is the comultiplication, ε : U ( g ) → Q is the counit, and S : U ( g ) → U ( g ) is the antipode.Furthermore, by the Poincar´e-Brikhoff-Witt Theorem, the universal enveloping algebraalso has the triangular decomposition (1.4) U ( g ) ∼ = U − ( g ) ⊗ U ( g ) ⊗ U + ( g ) , where U + ( g ) (resp. U ( g ) and U − ( g )) be the subalgebra of U ( g ) generated by the elements e il (resp. h and f il ) for ( i, l ) ∈ I ∞ .In [7], Kang studied the representation theory of the Borcherds-Bozec algebras. We listsome results that we will use later. Proposition 1.3. [7](a) Let λ ∈ P + and V ( λ ) = U ( g ) v λ be the irreducible highest weight g -module. Thenwe have(1.5) f λ ( h i )+1 i v λ = 0 for i ∈ I re ,f il v λ = 0 for ( i, l ) ∈ I ∞ with λ ( h i ) = 0 . (b) Every highest weight g -module with highest weight λ ∈ P + satisfying (1.5) isisomorphic to V ( λ ). ZHAOBING FAN, SEOK-JIN KANG, YOUNG ROCK KIM, AND BOLUN TONG quantum Borcherds-Bozec algebras Let q be an indeterminate and set q i = q r i , q ( i ) = q ( αi,αi )2 . Note that q i = q ( i ) if i ∈ I r e . For each i ∈ I r e and n ∈ Z ≥ , we define[ n ] i = q ni − q − ni q i − q − i , [ n ] i ! = n Y k =1 [ k ] i , (cid:20) nk (cid:21) i = [ n ] i ![ k ] i ![ n − k ] i !Let F = Q ( q ) h f il | ( i, l ) ∈ I ∞ i be the free associative algebra over Q ( q ) generated bythe symbols f il for ( i, l ) ∈ I ∞ . By setting deg f il = − lα i , F become a Q − -graded algebra.For a homogeneous element u in F , we denote by | u | the degree of u , and for any A ⊆ Q − ,set F A = { x ∈ F | | x | ∈ A } .We define a twisted multiplication on F ⊗ F by( x ⊗ x )( y ⊗ y ) = q − ( | x | , | y | ) x y ⊗ x y , and equip F with a co-multiplication δ defined by δ ( f il ) = X m + n = l q − mn ( i ) f im ⊗ f in for ( i, l ) ∈ I ∞ . Here, we understand f i = 1 and f il = 0 for l < Proposition 2.1. [1, 2] For any family ν = ( ν il ) ( i,l ) ∈ I ∞ of non-zero elements in Q ( q ),there exists a symmetric bilinear form ( , ) L : F × F → Q ( q ) such that(a) ( x, y ) L = 0 if | x | 6 = | y | ,(b) (1 , L = 1,(c) ( f il , f il ) L = ν il for all ( i, l ) ∈ I ∞ ,(d) ( x, yz ) L = ( δ ( x ) , y ⊗ z ) L for all x, y, z ∈ F .Here, ( x ⊗ x , y ⊗ y ) L = ( x , y ) L ( x , y ) L for any x , x , y , y ∈ F .From now on, we assume that(2.1) ν il ∈ q Z ≥ [[ q ]] for all ( i, l ) ∈ I ∞ . Then, the bilinear form ( , ) L is non-degenerate on F ( i ) = L l ≥ F − lα i for i ∈ I im \ I iso . LASSICAL LIMIT OF QUANTUM BORCHERDS-BOZEC ALGEBRAS 7
Let b U be the associative algebra over Q ( q ) with generated by the elements q h ( h ∈ P ∨ )and e il , f il (( i, l ) ∈ I ∞ ) with defining relations(2.2) q = , q h q h ′ = q h + h ′ for h, h ′ ∈ P ∨ q h e jl q − h = q lα j ( h ) e jl , q h f jl q − h = q − lα j ( h ) f jl for h ∈ P ∨ , ( j, l ) ∈ I ∞ , − la ij X k =0 ( − k (cid:20) − la ij k (cid:21) i e i − la ij − k e jl e ki = 0 for i ∈ I re , i = ( j, l ) , − la ij X k =0 ( − k (cid:20) − la ij k (cid:21) i f i − la ij − k f jl f ki = 0 for i ∈ I re , i = ( j, l ) ,e ik e jl − e jl e ik = f ik f jl − f jl f ik = 0 for a ij = 0 . We extend the grading by setting | q h | = 0 and | e il | = lα i .The algebra b U is endowed with the co-multiplication ∆ : b U → b U ⊗ b U given by(2.3) ∆( q h ) = q h ⊗ q h , ∆( e il ) = X m + n = l q mn ( i ) e im ⊗ K − mi e in , ∆( f il ) = X m + n = l q − mn ( i ) f im K ni ⊗ f in . where K i = q h i i ( i ∈ I ).Let b U ≤ be the subalgebra of b U generated by f il and q h , for all ( i, l ) ∈ I ∞ and h ∈ P ∨ ,and b U + be the subalgebra generated by e il for all ( i, l ) ∈ I ∞ . In [1], Bozec showed thatone can extended ( , ) L to a symmetric bilinear form ( , ) L on b U satisfying(2.4) ( q h , L = 1 , ( q h , f il ) L = 0 , ( q h , K j ) L = q − α j ( h ) , ( x, y ) L = ( ω ( x ) , ω ( y )) L for all x, y ∈ b U + , where ω : b U → b U is the involution defined by ω ( q h ) = q − h , ω ( e il ) = f il , ω ( f il ) = e il for h ∈ P ∨ , ( i, l ) ∈ I ∞ . For any x ∈ b U , we shall use the Sweedler’s notation, and write∆( x ) = X x (1) ⊗ x (2) . ZHAOBING FAN, SEOK-JIN KANG, YOUNG ROCK KIM, AND BOLUN TONG
Following the Drinfeld double process, we define ˜ U as the quotient of b U by the relations(2.5) X ( a (1) , b (2) ) L ω ( b (1) ) a (2) = X ( a (2) , b (1) ) L a (1) ω ( b (2) ) for all a, b ∈ b U ≤ Definition 2.2.
Given a Borcherds-Cartan datum (
A, P, Π , P ∨ , Π ∨ ), the quantum Borcherds-Bozec algebra U q ( g ) is defined to be the quotient algebra of ˜ U by the radical of ( , ) L restricted to ˜ U − × ˜ U + .Let U + (resp. U − ) be the subalgebra of U q ( g ) generated by e il (resp. f il ) for all ( i, l ) ∈ I ∞ . We will denote by U the subalgebra of U q ( g ) generated by q h for all h ∈ P ∨ . It iseasy to see that q h ( h ∈ P ∨ ) is a Q ( q )-basis of U .In [8], Kang and Kim showed that the co-multiplication ∆ : b U → b U ⊗ b U passes downto U q ( g ) and with this, U q ( g ) becomes a Hopf algebra. They also proved the quantumBorcherds-Bozec algebra has a triangular decomposition . Theorem 2.3. [8] The the quantum Borcherds-Bozec algebra U q ( g ) has the followingtriangular decomposition:(2.6) U q ( g ) ∼ = U − ⊗ U ⊗ U + . By the defining relation (2.5), we obtain complicated commutation relations between e il and f jk for ( i, l ) , ( j, k ) ∈ I ∞ . We will derive explicit formulas for these complicatedcommutation relations in Appendix A. But, as we already see in (1.2), the commutationrelations in the universal enveloping algebra U ( g ) of Borcherds-Bozec algebra g are rathersimple(2.7) e ik f jl − f jl e ik = k δ ij δ kl h i for i, j ∈ I, k, l ∈ Z > . Thanks to Bozec, there exists another set of generators in U q ( g ) called primitive genera-tors . They satisfy a simpler set of commutation relations, and we shall prove that thesegenerators also satisfy all the defining relations of U q ( g ) described in (2.2).We denote by C l (resp. P l ) the set of compositions (resp. partitions) of l , and denoteby η : U q ( g ) → U q ( g ) the Q -algebra homomorphism defined by(2.8) η ( e il ) = e il , η ( f il ) = f il , η ( q h ) = q − h , η ( q ) = q − for h ∈ P ∨ , ( i, l ) ∈ I ∞ . As usual, let S : U q ( g ) → U q ( g ) and ǫ : U q ( g ) → Q ( q ) be the antipode and the counit of U q ( g ), respectively. Then, we have the following proposition. Proposition 2.4. [1, 2] For any i ∈ I i m and l ≥
1, there exist unique elements t il ∈ U −− lα i and s il = ω ( t il ) such that LASSICAL LIMIT OF QUANTUM BORCHERDS-BOZEC ALGEBRAS 9 (1) Q ( q ) h f il | l ≥ i = Q ( q ) h t il | l ≥ i and Q ( q ) h e il | l ≥ i = Q ( q ) h s il | l ≥ i ,(2) ( t il , z ) L = 0 for all z ∈ Q ( q ) h f i , · · · , f il − i ,( s il , z ) L = 0 for all z ∈ Q ( q ) h e i , · · · , e il − i (3) t il − f il ∈ Q ( q ) h f ik | k < l i and s il − e il ∈ Q ( q ) h e ik | k < l i ,(4) η ( t il ) = t il , η ( s il ) = s il ,(5) δ ( t il ) = t il ⊗ ⊗ t il , δ ( s il ) = s il ⊗ ⊗ s il ,(6) ∆( t il ) = t il ⊗ K li ⊗ t il , ∆( s il ) = s il ⊗ K − li + 1 ⊗ s il ,(7) S ( t il ) = − K − li t il , S ( s il ) = − s il K li .If we set τ il = ( t il , t il ) L = ( s il , s il ) L , we have the following commutation relations in U q ( g )(2.9) s il t jk − t jk s il = δ ij δ lk τ il ( K li − K − li ) . Assume that i ∈ I i m and let c = ( c , · · · , c m ) be an element in C l or in P l . We set t i, c = m Y j =1 t ic j and s i, c = m Y j =1 s ic j . Notice that { t i, c | c ∈ C l } is a basis of F − lα i .For i ∈ I iso and c , c ′ ∈ P l , if c = c ′ , then by induction, we have( t i, c , t i, c ′ ) L = ( s i, c , s i, c ′ ) L = 0 . For i ∈ I im \ I iso and c , c ′ ∈ C l , if the partitions obtained by rearranging c and c ′ are notequal, then we have ( t i, c , t i, c ′ ) L = ( s i, c , s i, c ′ ) L = 0 . For each i ∈ I r e , we also use the notation t i and s i . Here we set t i = f i , s i = e i . Sometimes, we simply write t i (resp. s i ) instead of t i (resp. s i ) in this case. By mimickingDefinition 1 . .
13 in [13], we have the following definition.
Definition 2.5.
For every ( i, l ) ∈ I ∞ , we define the linear maps e ′ i,l , e ′′ i,l : F → F by(2.10) e ′ i,l (1) = 0 , e ′ i,l ( t jk ) = δ ij δ lk and e ′ i,l ( xy ) = e ′ i,l ( x ) y + q l ( | x | ,α i ) xe ′ i,l ( y )(2.11) e ′′ i,l (1) = 0 , e ′′ i,l ( t jk ) = δ ij δ lk and e ′′ i,l ( xy ) = q l ( | y | ,α i ) e ′′ i,l ( x ) y + xe ′′ i,l ( y )for any homogeneous elements x, y in F . Proposition 2.6. (a) For any x, y ∈ F , we have( t il y, x ) L = τ il ( y, e ′ i,l ( x )) L , ( yt il , x ) L = τ il ( y, e ′′ i,l ( x )) L (b) The maps e ′ i,l and e ′′ i,l preserve the radical of ( , ) L .(c) Let x ∈ U − , we have(i) If e ′ i,l ( x ) = 0 for all ( i, l ) ∈ I ∞ , then x = 0.(ii) If e ′′ i,l ( x ) = 0 for all ( i, l ) ∈ I ∞ , then x = 0. Proof. (a) For any homogeneous element x ∈ F . We first show that(2.12) δ ( x ) = t il ⊗ e ′ i,l ( x ) + X w =( i,l ) t w ⊗ y w , where if w = ( j , l ) ... ( j r , l r ) is a word in I ∞ , t w = t ( j ,l ) · · · t ( j r ,l r ) and y w is an elementin F depending on w .Since e ′ i,l is a linear map, it is enough to check (2.12) by assuming that x is a monomialin t jk . Fix ( i, l ) ∈ I ∞ . We use induction on the number of t il that appears in x . If x contains no t il , then e ′ i,l ( x ) = 0 and there is no term of the form t il ⊗ − . Now assumethat x contains t il , then we can write x = x t il x for some monomials x , x such that x doesn’t contains t il . So we have(2.13) e ′ i,l ( x ) = e ′ i,l ( x t il x ) = q l ( | x | ,α i ) x e ′ i,l ( t il x ) = q l ( | x | ,α i ) x [ x + q l ( − lα i ,α i ) t il e ′ i,l ( x )] . On the other hand(2.14) δ ( x ) = δ ( x )( t il ⊗ ⊗ t il ) δ ( x ) . By induction hypothesis, the term t il ⊗ − only appear in(2.15) (1 ⊗ x )( t il ⊗ ⊗ x ) + (1 ⊗ x )(1 ⊗ t il )( t il ⊗ e ′ i,l ( x )) , which is equal to(2.16) t il ⊗ q ( | x | ,lα i ) x x + t il ⊗ q − ( | x |− lα i , − lα i ) x t il e ′ i,l ( x )= t il ⊗ q l ( | x | ,α i ) x [ x + q − l ( lα i ,α i ) t il e ′ i,l ( x )] . This shows (2.12).Similarly, we can show that(2.17) δ ( x ) = e ′′ i,l ( x ) ⊗ t il + X w =( i,l ) z w ⊗ t w . Since e ′ i,l and e ′′ i,l are linear maps, the equations (2.12) and (2.17) hold for any x, y ∈ F . LASSICAL LIMIT OF QUANTUM BORCHERDS-BOZEC ALGEBRAS 11
For any c ∈ C il , we have ( t il , t i c ) L = δ ( l ) , c τ il . Thus(2.18) ( t il y, x ) L = τ il ( y, e ′ i,l ( x )) L , ( yt il , x ) L = τ il ( y, e ′′ i,l ( x )) L for any x, y ∈ F .(b) Since τ il = ( t il , t il ) L = 0, our assertion follows.(c) Note that each monomial ends with some t jk ’s. By (a), if e ′′ i,l ( x ) = 0 for all ( i, l ) ∈ I ∞ ,then x belongs to the radial of ( , ) L , which is equal to 0 in U − . (cid:3) For any i ∈ I re and n ∈ N , we set t ( n ) i = t ni [ n ] i ! . By a similar argument as [13, 1.4.2], we have the following Lemma.
Lemma 2.7.
We have(2.19) δ ( t ( n ) i ) = X p + p ′ = n q − pp ′ i t ( p ) i ⊗ t ( p ′ ) i for any i ∈ I re and n ∈ N . Proposition 2.8.
For any i ∈ I re , ( j, l ) ∈ I ∞ , and i = ( j, l ), we have X p + p ′ =1 − la ij ( − p t ( p ) i t jl t ( p ′ ) i = 0in U q ( g ). Proof. If i ∈ I re , we have a ij = α i ,α j )( α i ,α i ) . Set R i, ( j,l ) = X p + p ′ =1 − la ij ( − p t ( p ) i t jl t ( p ′ ) i . By (2.6), we only need to show that e ′′ µ ( R i, ( j,l ) ) = 0 for all µ ∈ I ∞ . It is clear that e ′′ µ ( R i, ( j,l ) ) = 0 if µ = i, ( j, l ) . By the definition of e ′′ i , we have(2.20) e ′′ i ( t ( p ) i t jl t ( p ′ ) i ) = q ( α i , − p ′ α i ) e ′′ i ( t ( p ) i t jl ) t ( p ′ ) i + t ( p ) i t jl e ′′ i ( t ( p ′ ) i )= q − p ′ ( α i ,α i ) q − ( α i ,lα j ) q (1 − p ) i t ( p − i t jl t ( p ′ ) i + q (1 − p ′ ) i t ( p ) i t jl t ( p ′ − i . Thus(2.21) e ′′ i ( R i, ( j,l ) ) = X p + p ′ =1 − la ij ( − p q − p ′ ( α i ,α i ) q − ( α i ,lα j ) q (1 − p ) i t ( p − i t jl t ( p ′ ) i + X p + p ′ =1 − la ij ( − p q (1 − p ′ ) i t ( p ) i t jl t ( p ′ − i = X ≤ p ≤ − la ij ( − p q − (1 − la ij − p )( α i ,α i ) q − ( α i ,lα j ) q (1 − p ) i t ( p − i t jl t (1 − la ij − p ) i + X ≤ p ≤ − la ij ( − p q ( la ij + p ) i t ( p ) i t jl t ( − la ij − p ) i . The coefficient of t ( p ) i t jl t ( − la ij − p ) i in the first sum of (2.21) is(2.22) ( − p +1 q − ( − la ij − p )( α i ,α i ) q − ( α i ,lα j ) q − pi = ( − p +1 q ( l αi,αj )( αi,αi ) + p )( α i ,α i ) − l ( α i ,α j )+( − p ) ( αi,αi )2 = ( − p +1 q l ( α i ,α j )+ p ( αi,αi )2 = ( − p +1 q ( la ij + p ) i . Hence, we have e ′′ i ( R i, ( j,l ) ) = 0.By the definition of e ′′ jl , we have(2.23) e ′′ jl ( t ( p ) i t jl t ( p ′ ) i ) = q − l ( α j ,p ′ α i ) e ′′ jl ( t ( p ) i t jl ) t ( p ′ ) i = q − l ( α j ,p ′ α i ) t ( p ) i t ( p ′ ) i . So(2.24) e ′′ jl ( R i, ( j,l ) ) = X ≤ p ′ ≤ − la ij ( − (1 − la ij − p ′ ) q − l ( α j ,p ′ α i ) t (1 − la ij − p ′ ) i t ( p ′ ) i . By [13, 1.3.4] , we obtain X ≤ p ′ ≤ − l αi,αj )( αi,αi ) ( − (1 − l αi,αj )( αi,αi ) − p ′ ) q − l ( α j ,p ′ α i ) " − l α i ,α j )( α i ,α i ) p ′ i = 0 . Hence, we get e ′′ jl ( R i, ( j,l ) ) = 0. This finishes the proof. (cid:3) LASSICAL LIMIT OF QUANTUM BORCHERDS-BOZEC ALGEBRAS 13
By the above arguments, we have primitive generators t il (( i, l ) ∈ I ∞ ) in U − of degree − lα i and s il (( i, l ) ∈ I ∞ ) in U + of degree lα i satisfying(2.25) s il t jk − t jk s il = δ ij δ lk τ il ( K li − K − li ) , − la ij X k =0 ( − k (cid:20) − la ij k (cid:21) i t i − la ij − k t jl t ki = 0 for i ∈ I re , i = ( j, l ) . By using the involution ω , we get(2.26) − la ij X k =0 ( − k (cid:20) − la ij k (cid:21) i s i − la ij − k s jl s ki = 0 for i ∈ I re , i = ( j, l ) . Since t il (resp. s il ) can be written as a homogeneous polynomial of f ik (resp. e ik ) for k ≤ l , we have(2.27) q h t jl q − h = q − lα j ( h ) t jl , q h s jl q − h = q lα j ( h ) s jl for h ∈ P ∨ , ( j, l ) ∈ I ∞ , and(2.28) [ t ik , t jl ] = [ s ik , s jl ] = 0 for a ij = 0 . A -form of the quantum Borcherds-Bozec algebras We consider the localization of Q [ q ] at the ideal ( q − A = { f ( q ) ∈ Q ( q ) | f is regular at q = 1 } = { g/h | g, h ∈ Q [ q ] , h (1) = 0 } Let J be the unique maximal ideal of the local ring A , which is generated by ( q − A / J ∼ −→ Q , f ( q ) + J f (1) . Note that, for i ∈ I r e , [ n ] i and (cid:20) nk (cid:21) i are elements of Z [ q, q − ] ⊆ A . For any h ∈ P ∨ , n ∈ Z , we formally define ( q h ; n ) q = q h q n − q − ∈ U . Definition 3.1.
We define the A - form , denote by U A of the quantum Borcherds-Bozecalgebra U q ( g ) to be the A -subalgebra generated by the elements s il , T il , q h and ( q h ; 0) q ,for all ( i, l ) ∈ I ∞ and h ∈ P ∨ , where(3.2) T il = 1 τ il q i − t il for ( i, l ) ∈ I ∞ . Let U + A (resp. U − A ) be the A -subalgebra of U A generated by the elements s il (resp. T il ) for ( i, l ) ∈ I ∞ , and U A be the subalgebra of U A generated by q h and ( q h ; 0) q for( h ∈ P ∨ ). Lemma 3.2. (a) ( q h ; n ) q ∈ U A for all n ∈ Z and h ∈ P ∨ .(b) K li − K − li q i − ∈ U A . Proof.
It is straightforward to check that(3.3) ( q h ; n ) q = q n ( q h ; 0) q + q n − q − ,K li − K − li q i − q − q i − K − li ) K li − q − . The lemma follows. (cid:3)
The next proposition shows that the triangular decomposition (2.6) of U q ( g ) carriesover to its A -form. Proposition 3.3.
We have a natural isomorphism of A -modules(3.4) U A ∼ = U − A ⊗ U A ⊗ U + A induced from the triangular decomposition of U q ( g ). Proof.
Consider the canonical isomorphism ϕ : U q ( g ) ∼ −→ U − ⊗ U ⊗ U + given by multi-plication. By (2.25) and (2.27), we have the following commutation relations(3.5) s il ( q h ; 0) q = ( q h ; − lα i ( h )) q s il , ( q h ; 0) q T il = T il ( q h ; − lα i ( h )) q ,s il T jk − T jk s il = δ ij δ lk K li − K − li q i − . Combining with (3.2), we can see that the image of ϕ lies inside U − A ⊗ U A ⊗ U + A . (cid:3) LASSICAL LIMIT OF QUANTUM BORCHERDS-BOZEC ALGEBRAS 15
The representation theory of quantum Borcherds-Bozec algebras has been studied byKang and Kim in [8]. In the following sections, we will use some notions defined in [8],which are similar to those in classical representation theory of quantum groups.Fix λ ∈ P , let V q be a highest weight U q ( g )-module with highest weight λ and highestweight vector v λ . Then we have the A -form for the highest weight modules. Definition 3.4.
The A - form of V q is defined to be the U A -module V A = U A v λ .By the definition of highest weight module and V A , it is easy to see that V A = U − A v λ .The highest weight U q ( g )-module V q has the weight space decomposition(3.6) V q = M µ ≤ λ V qµ , where V qµ = { v ∈ V q | q h v = q µ ( h ) v for all h ∈ P ∨ } . For each µ ∈ P , we define the weightspace ( V A ) µ = V A ∩ V qµ . The following proposition shows that V A also has the weightspace decomposition. Proposition 3.5. V A = L µ ≤ λ ( V A ) µ Proof.
The proof is the same as [5, Proposition 3.3.6]. (cid:3)
Proposition 3.6.
For each µ ∈ P , the weight space ( V A ) µ is a free A -module withrank A ( V A ) µ = dim Q ( q ) V qµ . Proof.
We first show that ( V A ) µ is finite generated as an A -module. Since we have V A = U − A v λ , every element in V A is a polynomial of T il with coefficients in A . Assumethat λ = µ + α for some α ∈ Q + . Then for each v ∈ A with weight µ , v must be a A -linear combination of { T i l · · · T i p l p v λ | l α l + · · · l p α l p = α } , which is a finite set.Let { T ζ v λ } be a Q ( q )-basis of V qµ , where T ζ are monomials in T il . The set { T ζ v λ } certainly belongs to ( V A ) µ and is also A -linearly independent. So we have rank A ( V A ) µ ≥ dim Q ( q ) V qµ . Let { u , · · · , u p } be an A -linearly independent subset of ( V A ) µ . Consider a Q ( q )-linear dependence relation c ( q ) u + · · · + c p ( q ) u p = 0 , c k ( q ) ∈ Q ( q ) for k = 1 , · · · , p. Multiplying some powers of ( q −
1) if needed, we may assume that all c k ( q ) ∈ A , whichimplies that c k ( q ) = 0 for all k = 1 , · · · , p . Hence u , · · · , u p are linearly independent over Q ( q ) and rank A ( V A ) µ ≤ dim Q ( q ) V qµ , which completes the proof. (cid:3) Corollary 3.7.
The Q ( q )-linear map ϕ : Q ( q ) ⊗ A V A → V q given by c ⊗ v cv is anisomorphism. Classical limit of quantum Borcherds-Bozec algebras
Define the Q -linear vector spaces(4.1) U = ( A / J ) ⊗ A U A ∼ = U A / J U A ,V = ( A / J ) ⊗ A V A ∼ = V A / J V A . Then V is naturally a U -module. Consider the natural maps(4.2) U A → U = U A / J U A ,V A → V = V A / J V A . The passage under these maps is referred to as taking the classical limit. We will denoteby x the image of x under the classical limit. Notice that q is mapped to 1 under thesemaps.For each µ ∈ P , set V µ = ( A / J ) ⊗ A ( V A ) µ . Then we have Proposition 4.1. (a) V = L µ ≤ λ V µ .(b) For each µ ∈ P , dim Q V µ = rank A ( V A ) µ = dim Q ( q ) V qµ .Let h ∈ U denote the classical limit of the element ( q h ; 0) q ∈ U A . As in [5], we havethe following lemma. Lemma 4.2. (i) For all h ∈ P ∨ , we have q h = 1.(ii) For any h, h ′ ∈ P ∨ , h + h ′ = h + h ′ . Hence, we have nh = nh for n ∈ Z .Define the subalgebras U = Q ⊗ U A and U ± = Q ⊗ U ± A . The next theorem shows thatwe can define a surjective homomorphism from the universal enveloping algebra U ( g ) to U , and as a U ( g )-module, V is a highest weight module with highest weight λ ∈ P andhighest weight vector v λ . Theorem 4.3. (a) The elements s il , T il (( i, l ) ∈ I ∞ ) and h ( h ∈ P ∨ ) satisfy the defining relations of U ( g ). Hence there exists a surjective Q -algebra homomorphism ψ : U ( g ) → U sending e il to s il , f il to T il , h to h . In particular, the U -module V has a U ( g )-module structure.(b) For each µ ∈ P , h ∈ P ∨ , the element h acts on V µ as scalar multiplication by µ ( h ). So V µ is the µ -weight space of the U ( g )-module V . LASSICAL LIMIT OF QUANTUM BORCHERDS-BOZEC ALGEBRAS 17 (c) As a U ( g )-module, V is a highest weight module with highest weight λ ∈ P andhighest weight vector v λ . Proof. (a) Since K li − K − li q i − q − q i − K − li ) K li − q − K li − K − li q i − r i · · lr i h i = lh i . By (2.25), we have the following equation in U s il T jk − T jk s il = δ ij δ lk lh i , and it is the same as the commutation relations in U ( g ).Since we have q h s jl = q lα j ( h ) s jl q h , q h T jl = q − lα j ( h ) T jl q h for h ∈ P ∨ , ( j, l ) ∈ I ∞ , we get q h − q − s il = s il q lα i ( h ) q h − q − q h − q − s il − s il q h − q − s il q lα i ( h ) − q − q h . Thus hs il − s il h = lα i ( h ) s il . Similarly, we have h T il − T il h = − lα i ( h ) T il . It is easy to check the commutation relations(4.4) [ T ik , T jl ] = [ s ik , s jl ] = 0 for a ij = 0 . For i ∈ I re , we have [ n ] i = n and (cid:20) nk (cid:21) i = (cid:18) nk (cid:19) . Hence the remaining Serre relations follow.(b) For v ∈ ( V A ) µ and h ∈ P ∨ , we have ( q h ; 0) q v = q µ ( h ) − q − v . Hence when we take theclassical limit, we obtain hv = µ ( h ) v .(c) As a U ( g )-module, by (2), we have hv λ = hv λ = λ ( h ) v λ in V for all h ∈ P ∨ . Foreach ( i, l ) ∈ I ∞ , s il v λ is zero. Therefore, V = U − v λ = U − ( g ) v λ and hence V is a highestweight module with highest weight λ ∈ P and highest weight vector v λ . (cid:3) Combining Proposition 4.1 (b) and Theorem 4.3 (b), we have ch V = ch V q . For adominant integral weight λ ∈ P + , the irreducible highest weight U q ( g )-module V q ( λ ) hasthe following property. Proposition 4.4. [8] Let λ ∈ P + and V q ( λ ) be the irreducible highest weight modulewith highest weight λ and highest weight vector v λ . Then the following statements hold.(a) If i ∈ I re , then f λ ( h i )+1 i v λ = 0.(b) If i ∈ I im and λ ( h i ) = 0, then f ik v λ = 0 for all k > U q ( g )-module V q ( λ ) is isomorphic to the irreducible highest U ( g )-module V ( λ ). Theorem 4.5. If λ ∈ P + and V q is the irreducible highest weight U q ( g )-module V q ( λ )with highest weight λ , then V is isomorphic to the irreducible highest weight module V ( λ ) over U ( g ) with highest weight λ . Proof.
By Proposition 4.4, if i ∈ I re , then T λ ( h i )+1 i v λ = 0; if i ∈ I im and λ ( h i ) = 0, then T ik v λ = 0 for all k >
0. Therefore, V is a highest weight U q ( g )-module with highestweight λ and highest weight vector v λ satisfying:(a) If i ∈ I re , then f λ ( h i )+1 i v λ = T λ ( h i )+1 i v λ = 0.(b) If i ∈ I im and λ ( h i ) = 0, then f ik v λ = T ik v λ = 0 for all k > V ∼ = V ( λ ) by Proposition 1.3. (cid:3) By Proposition 4.1 (b), the character of V q ( λ ) is the same as the character of V ( λ ),which is given by (see, [7, 3])(4.5) ch V ( λ ) = P w ∈ W ǫ ( w ) e w ( λ + ρ ) − ρ w ( S λ ) Q α ∈ ∆ + (1 − e − α ) dim g α = P w ∈ W P s ∈ F λ ǫ ( w ) ǫ ( s ) e w ( λ + ρ − s ) − ρ Q α ∈ ∆ + (1 − e − α ) dim g α . Theorem 4.6.
The classical limit U of U q ( g ) is isomorphic to the universal envelopingalgebra U ( g ) as Q -algebras. Proof.
By Theorem 4.3 (a), we already have an epimorphism ψ : U ( g ) ։ U sending e il to s il , f il to T il , h to h , respectively. So it is sufficient to show that ψ is injective.We first show that the restriction ψ of ψ to U ( g ) is an isomorphism of U ( g ) onto U .Note that ψ is certainly surjective. Since χ = { h i | i ∈ I } ∪ { d i | i ∈ I } is a Z -basis ofthe free Z -lattice P ∨ , it is also a Q -basis of the Cartan subalgebra h . Thus any element of LASSICAL LIMIT OF QUANTUM BORCHERDS-BOZEC ALGEBRAS 19 U ( g ) may be written as a polynomial in χ . Suppose g ∈ Ker ψ . Then, for each λ ∈ P ,we have 0 = ψ ( g ) · v λ = λ ( g ) v λ , where v λ is a highest weight vector of a highest weight U q ( g )-module of highest weight λ and λ ( g ) denotes the polynomial in { λ ( x ) | x ∈ χ } corresponding to g . Hence, we have λ ( g ) = 0 for every λ ∈ P . Since we may take any integer value for λ ( x )( x ∈ χ ), g must bezero, which implies that ψ is injective.Next, we show that the restriction of ψ to U − ( g ), denote by ψ − , is an isomorphism of U − ( g ) onto U − . Suppose Ker ψ − = 0, and take a non-zero element u = P a ζ f ζ ∈ Ker ψ − ,where a ζ ∈ Q and f ζ are monomials in f il ’s ( i, l ) ∈ I ∞ . Let N be the maximal length ofthe monomials f ζ in the expression of u and choose a dominant integral weight λ ∈ P + such that λ ( h i ) > N for all i ∈ I . The kernel of the U − ( g )-module homomorphism ϕ : U − ( g ) → V given by x ψ ( x ) · v λ is the left ideal of U − ( g ) generated by f λ ( h i )+1 i ( i ∈ I re )and f il for i ∈ I im with λ ( h i ) = 0. Because of the choice of λ , it is generated by f λ ( h i )+1 i for all i ∈ I re .Therefore, u = P a ζ f ζ / ∈ Ker ϕ . That is, ψ − ( u ) · v λ = ψ ( u ) · v λ = 0, which is acontradiction. Therefore, Ker ψ − = 0 and U − ( g ) is isomorphic to U − .Similarly, we have U + ( g ) ∼ = U +1 . Hence, by the triangular decomposition, we have thelinear isomorphisms U ( g ) ∼ = U − ( g ) ⊗ U ( g ) ⊗ U + ( g ) ∼ = U − ⊗ U ⊗ U +1 ∼ = U , where the last isomorphism follows from Proposition 3.3. It is easy to see that this iso-morphism is actually an algebra isomorphism. (cid:3) We now show that U inherits a Hopf algebra structure from that of U q ( g ). It sufficesto show that U A inherits the Hopf algebra structure from that of U q ( g ). Since we have(4.6) ∆( T il ) = T il ⊗ K li ⊗ T il , ∆( s il ) = s il ⊗ K − li + 1 ⊗ s il , ∆( q h ) = q h ⊗ q h ,S ( T il ) = − K − li T il , S ( s il ) = − s il K li , S ( q h ) = q − h ,ǫ ( T il ) = ǫ ( s il ) = 0 , ǫ ( q h ) = 1 , we get(4.7) ∆(( q h ; 0) q ) = q h ⊗ q h − ⊗ q − q h ; 0) q ⊗ q h ⊗ ( q h ; 0) q ,S (( q h ; 0) q ) = ( q − h ; 0) q ,ǫ (( q h ; 0) q ) = 0 . Hence the maps ∆ : U A → U A ⊗ U A , ǫ : U A → A , and S : U A → U A are all well-definedand U A inherits a Hopf algebra structure from that of U q ( g ).Let us show that the Hopf algebra structure of U coincides with that of U ( g ) underthe isomorphism we have been considering. Taking the classical limit of the equations in(4.6) and in (4.7), we have(4.8) ∆( T il ) = T il ⊗ ⊗ T il , ∆( s il ) = s il ⊗ ⊗ s il , ∆( h ) = h ⊗ ⊗ h,S ( T il ) = − T il , S ( s il ) = − s il , S ( h ) = − h,ǫ ( T il ) = ǫ ( s il ) = ǫ ( h ) = 0 . This coincides with (1.3). Therefore, we have the following corollary.
Corollary 4.7.
The classical limit U of U q ( g ) inherits a Hopf algebra structure from thatof U q ( g ) so that U and U ( g ) are isomorphic as Hopf algebras over Q .Since U − ( g ) ∼ = U − , by the same argument in [5, Theorem 3.4.10], we have the followingtheorem when we take the classical limit on the Verma module over U q ( g ). Theorem 4.8. [5] If λ ∈ P and V q is the Verma module M q ( λ ) over U q ( g ) with highestweight λ , then its classical limit V is isomorphic to the Verma module M ( λ ) over U ( g )with highest weight λ . LASSICAL LIMIT OF QUANTUM BORCHERDS-BOZEC ALGEBRAS 21
Appendix
A.We shall provide an explicit commutation relations for e ik and f jl , for ( i, k ) , ( j, l ) ∈ I ∞ in U q ( g ). Recall that, we have the co-multiplication formulas∆( f il ) = X m + n = l q − mn ( i ) f im K ni ⊗ f in . Then, the defining relation (2.5) yields the following lemma.
Lemma A.1. [8] For any i, j ∈ I and k, l ∈ Z > , we have(a) If i = j , then e ik and f jl are commutative.(b) If i = j , we have the following relations in U q ( g ) for all k, l > X m + n = kn + s = l q n ( m − s )( i ) ν in e is f im K − ni = X m + n = kn + s = l q − n ( m − s )( i ) ν in f im e is K ni . Since we have K ni e im K − ni = q nm ( i ) e im ,K ni f im K − ni = q − nm ( i ) f im . We can modify the equations (A.1) as the following form(A.2) X m + n = kn + s = l q n ( s − m )( i ) ν in K − ni e is f im = X m + n = kn + s = l q n ( m − s )( i ) ν in K ni f im e is . If i ∈ I re , then k = l = 1 and m = s , so there are only one commutation relation in thiscase(A.3) e i f i + ν i K − i = f i e i + ν i K i . If i ∈ I im (we omit the notation “ i ” in this case for simplicity), we first assume that k = l . By (A.2), we have(A.4) k = l = 1 , e f + ν K − = f e + ν K,k = l = 2 , e f + ν K − e f + ν K − = f e + ν Kf e + ν K , · · · k = l = n, e n f n + ν K − e n − f n − + · · · + ν n − K − n e f + ν n K − n = f n e n + ν Kf n − e n − + · · · + ν n − K n − f e + ν n K n . By direct calculation, we can write e n f n − f n e n in the following way e n f n − f n e n = α f n − e n − + α f n − e n − + · · · + α n − f e + α n , where(A.5) α = ν ( K − K − ) ,α = ν ( K − K − ) − ν K − α = ν ( K − K − ) − ν K − ( K − K − ) ,α = ν ( K − K − ) − ν K − α − ν K − α = ν ( K − K − ) − ν ν K − ( K − K − ) + ( ν − ν ν ) K − ( K − K − ) , · · · α n = ν n ( K n − K − n ) − ν K − α n − − ν K − α n − − · · · − ν n − K − ( n − α . If m ∈ N and c = ( c , · · · , c d ) is a composition of m (i.e. c ∈ C m ), then we set ν c = Q dk =1 ν k and k c k = d .By induction, we have(A.6) e n f n = n X p =1 ( p X r =1 (cid:2) ν r ϑ p − r K r − p ( K r − K − r ) (cid:3)) f n − p e n − p + f n e n , where ϑ m = P c ∈C m ( − k c k ν c . For example, ϑ = ν − ν ν + 2 ν ν + ν − ν .Next, we assume that k − l = t , then m − s = t . By (A.2), we get l X n =0 q − nt ( i ) ν n K − n e l − n f k − n = l X n =0 q nt ( i ) ν n K n f k − n e l − n . Hence, we have e l f k + q − t ( i ) ν K − e l − f k − + · · · + q − ( l − t ( i ) ν l − K − ( l − e f t +1 + q − lt ( i ) ν l K − l f t = f k e l + q t ( i ) ν Kf k − e l − + · · · + q ( l − t ( i ) ν l − K ( l − f t +1 e + q lt ( i ) ν l K l f t . We substitute K by q t ( i ) K in formula (A.6) and obtain(A.7) e l f k = l X p =1 ( p X r =1 h ν r ϑ p − r ( q t ( i ) K ) r − p (( q t ( i ) K ) r − ( q t ( i ) K ) − r ) i) f k − p e l − p + f k e l . LASSICAL LIMIT OF QUANTUM BORCHERDS-BOZEC ALGEBRAS 23
Finally, we assume that l − k = t , then s − m = t . By (A.2), we get k X n =0 q nt ( i ) ν n K − n e l − n f k − n = k X n =0 q − nt ( i ) ν n K n f k − n e l − n . Hence, we have e l f k + q t ( i ) ν K − e l − f k − + · · · + q ( l − t ( i ) ν l − K − ( l − e t +1 f + q lt ( i ) ν l K − l e t = f k e l + q − t ( i ) ν Kf k − e l − + · · · + q − ( l − t ( i ) ν l − K ( l − f e t +1 + q − lt ( i ) ν l K l e t . We substitute K by q − t ( i ) K in formula (A.6) and obtain(A.8) e l f k = k X p =1 ( p X r =1 h ν r ϑ p − r ( q − t ( i ) K ) r − p (( q − t ( i ) K ) r − ( q − t ( i ) K ) − r ) i) f k − p e l − p + f k e l . Combine the formulas (A.6), (A.7), and (A.8), we have the following statement.
Proposition A.2.
For i ∈ I im , we have the following commutation relations for all k, l > e il f ik − f ik e il = min { k,l } X p =1 ( p X r =1 h ν ir ϑ i,p − r ( q k − l ( i ) K i ) r − p (( q k − l ( i ) K i ) r − ( q k − l ( i ) K i ) − r ) i) f i,k − p e i,l − p . Where ϑ i,p − r = P c ∈C p − r ( − k c k ν i c . References [1] T. Bozec,
Quivers with loops and perverse sheaves , Math. Ann. (2015), 773-797.[2] T. Bozec,
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Harbin Engineering University, Harbin, China
E-mail address : [email protected] Korea Research Institute of Arts and Mathematics, Asan-si, Chungcheongnam-do, 31551,Korea
E-mail address : [email protected] Graduate School of Education, Hankuk University of Foreign Studies, Seoul, 02450, Ko-rea
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