PRESENTING QUEER SCHUR SUPERALGEBRAS

Transcription

1 PRESENTING QUEER SCHUR SUPERALGEBRAS JIE DU AND JINKUI WAN Abstract. Associated to the two types of finite dimensiona simpe superagebras, there are the genera inear Lie superagebra and the queer Lie superagebra. The universa enveoping agebras of these Lie superagebras act on the tensor spaces of the natura representations and, thus, define certain finite dimensiona quotients, the Schur superagebras and the queer Schur superagebra. In this paper, we introduce the quantum anaogue of the queer Schur superagebra and investigate the presentation probem for both the queer Schur superagebra and its quantum anaogue.. Introduction A superspace V is a vector space over a fied endowed with a Z 2 -grading (or a parity structure): V = V 0 V, where an eement in V 0 is caed even, whie an eement in V is caed odd. A superagebra A is a Z 2 -graded (associative) agebra with over a fied. Thus, the underying space of A is a superspace A = A 0 A and the mutipication satisfies A i A j A i+j, for i, j Z 2. It is nown (see, e.g., []) that a finite dimensiona simpe superagebra over the compex fied C is either isomorphic to the (fu) matrix superagebra M = M m+n (C) with even part M 0 = {( A 0 0 B) A Mm (C), B M n (C) } and odd part M = {( 0 C D 0) C Mm,n (C), D M n,m (C) }, or isomorphic to the queer matrix superagebra Q = {( ) A B B A A, B Mn (C) } with even part Q 0 = {( ) A 0 0 A A Mn (C) } and odd part Q = {( 0 B B 0) B Mn (C) }. Associated to a superagebra A, there is a Lie superagebra A equipped with the super bracet product (or super commutator) defined by [x, y] := xy ( )ˆx ŷ yx, where x, y A are homogeneous eements and ẑ = i if z A i. Thus, the two type simpe superagebras M and Q give rise to two Lie superagebras g(m n) := M, the genera inear Lie superagebra, and q(n) := Q, the queer Lie superagebra. If V denotes the natura representation of g(m n) (resp., q(n)), then the tensor product V r is a representation of the universa enveoping agebra U(g(m n)) (resp., U(q(n))). The image of U(g(m n)) (resp., U(q(n))) in End(V r ) is caed the Schur superagebra (resp. queer Schur superagebra or Schur superagebra of type Q, foowing []). The Schur superagebras and their representations were introduced and investigated by severa authors incuding Donin [4] and Brundan Kujawa [2] amost over ten years ago. Recenty, the study of quantum Schur superagebras has made substantia progress; see [5, 9, 0, 6, 7]. In particuar, in [0], E Turey and Kujawa provided a presentation Date: Juy, Mathematics Subject Cassification. Primary: 20G05, 20G43. Secondary: 7B37. Supported by ARC DP and NSFC-003. The research was carried out whie Wan was visiting the University of New South Waes during the year The hospitaity and support of UNSW are gratefuy acnowedged.

2 2 DU AND WAN of the Schur superagebras and their quantum anaogues, which generaizes the wor of Doty and Giaquinto [5] for (quantum) Schur agebras. It is nown that the queer Lie superagebra q(n) differs drasticay from the basic cassica Lie superagebras. For exampe, the Cartan subagebra of q(n) is not purey even and there is no invariant biinear form on q(n). On the other hand, q(n) behaves in many aspects as the Lie agebra g(n). In particuar, there exists a beautifu anaogue of the Schur-Wey duaity discovered by Sergeev [8], often referred as Sergeev duaity. In [6], Oshansi constructed a quantum deformation U q (q(n)) of the universa enveoping agebra U(q(n)) and estabished a quantum anaog of the Sergeev duaity in the generic case. The queer Schur superagebra was introduced and studied by Brundan and Keshchev [], and then they determined the irreducibe projective representations of the symmetric group S r via Sergeev duaity. In this paper, we wi introduce the quantum anaogue of queer Schur superagebras and wi foow the wors [5] and [8] to determine a presentation for the queer Schur superagebra and its quantum anaogue, which was stated as an interesting probem in [0]. In particuar, in the quantum case we aso estabish the existence of Z-form for the quantum superagebra U q (q(n)). This is based on a engthy but straightforward cacuation of the commutation formuas for the divided powers of root vectors. We organise the paper as foows. We first investigate a presentation of the queer Schur superagebra in the first three sections. More precisey, we study in 2 the basics of the queer Lie superagebra and its universa enveoping superagebra, and estabish the commutation formuas for divided powers of root vectors and a Kostat Z-form in 3. A presentation for the queer Schur superagebra is given in 4. From 5 onwards, we investigate the quantum case. We start in 5 with the Oshansi presentation (via a certain matrix in End(V ) 2 satisfying the quantum Yang-Baxter equation) and the Drinfed Jimbo type presentation for the quantum queer superagebra and introduce a quantum root vectors. We compute a commutation formuas for these vectors in 6 and for those with higher order in 7. A Lusztig type form for the quantum queer superagebra is introduced and certain quotients are investigated in 8. Finay, we sove the presentation probem for the quantum queer Schur superagebra in the ast section. Throughout the paper, et Z 2 = {0, }. We wi use a twofod meaning for Z 2. We wi regard Z 2 as an abeian group when we use it to describe a superspace. However, for a matrix or an n-tupe with entries in Z 2, we wi regard it as a subset of Z. 2. The queer Lie superagebra q(n) and the associated Schur superagebra Q(n, r) The ground fied in this section is the fied Q of rationa numbers. It is nown that the genera inear Lie superagebra g(n n) consists of matrices of the form ( ) A B, (2.0.) C D where A, B, C, D are arbitrary n n matrices, and the rows and coumns of (2.0.) are abeed by the set I(n n) = {, 2,..., n,, 2,..., n}. For i, j I(n n), denote by E g(n n) the matrix unit with at the (i, j) position and 0 esewhere. The set {E i, j I(n n)} is a basis of g(n n) and the Z 2 -grading on

3 PRESENTING QUEER SCHUR SUPERALGEBRAS 3 g(n n) is defined via Ê = 0 if ij > 0 and Ê = if ij < 0. Then the Lie bracet in g(n n) is given by [E, E, ] = δ j E i, ( )Ê Ê, δ i E,j. (2.0.2) The queer Lie superagebra, denoted by g = q(n), is the subagebra of the genera inear Lie superagebra g(n n) consisting of matrices of the form ( ) A B, (2.0.3) B A where A and B are arbitrary n n matrices. The even (resp., odd) part g 0 (resp., g ) consists of those matrices of the form (2.0.3) with B = 0 (resp., A = 0). We fix h to be the standard Cartan subagebra of g consisting of matrices of the form (2.0.3) with A, B being arbitrary diagona. Then the agebra h 0 has a basis {h,..., h n } and h has a basis {h,..., h n }, where h i = E i,i + E i, i, h ī = E i, i + E i,i. (2.0.4) Fix the trianguar decomposition g = n h n +, where n + (resp., n ) is the subagebra of g which consists of matrices of the form (2.0.3) with A, B being arbitrary upper trianguar (resp., ower trianguar) matrices. Observe that the even subagebra h 0 of h can be identified with the standard Cartan subagebra of g(n) via the natura isomorphism q(n) 0 = g(n). (2.0.5) Let {ɛ i i =,..., n} be the basis for h 0 dua to the standard basis {h i i =,..., n} for h 0 and we define a biinear form (, ) on h 0 via (ɛ i, ɛ j ) := ɛ j (h i ) = δ ij. (2.0.6) For α h 0, et g α = {x g [h, x] = α(h)x for a h h 0 }. Then we have the root superspace decomposition g = h α Φ g α with the root system Φ = {α := ɛ i ɛ j i j n}. The set of positive roots corresponding to the Bore subagebra b = h n + is Φ + = {α i < j n}. Observe that each root superspace g α has dimension vector (, ) for α Φ. Let α i = α i,i+ for i n. Then the root space g αi is spanned by {e i, e ī } with whie the root space g αi e i = E i,i+ + E i, i, e ī = E i, i + E i,i+, (2.0.7) is spanned by {f i, f ī } with f i = E i+,i + E i, i, f ī = E i+, i + E i,i. (2.0.8) Moreover, α = α i + + α j for a i < j. Let P := n i=zɛ i, (resp., P 0 = n i=nɛ i ) be the weight attice (reps., positive weight attice) of q(n). The universa enveoping superagebra U = U(q(n)) is obtained from the tensor agebra T (q(n)) by factoring out the idea generated by the eements [u, v] u v + ( )û ˆv v u The dimension vector of a superspace V is the tupe dimv = (dim V 0, dim V ).

4 4 DU AND WAN for u g i, v g j with i, j Z 2, where [u, v] denotes the Lie bracet of u, v in q(n). It inherits a Z 2 -grading from q(n). Proposition 2. ([, Proposition.], cf.([3])). The universa enveoping superagebra U(q(n)) is the associative superagebra over Q generated by even generators h i, e j, f j, and odd generators h ī, e j, f j, with i n and j n subject to the foowing reations: (QS) [h i, h j ] = 0, [h i, h j] = 0, [h ī, h j] = δ ij 2h i ; (QS2) [h i, e j ] = (ɛ i, α j )e j, [h i, e j] = (ɛ i, α j )e j, [h i, f j ] = (ɛ i, α j )f j, [h i, f j] = (ɛ i, α j )f j; (QS3) [h ī, e j ] = (ɛ { i, α j )e j, [h ī, f j ] = (ɛ i, α j )f j, ej, if i = j or j +, [h ī, e j] = [h ī, f j] = (QS4) [e i, f j ] = δ ij (h i h i+ ), [e ī, f j] = δ ij (h i + h i+ ), [e ī, f j ] = δ ij (h ī h i+ ), [e i, f j] = δ ij (h ī h i+ ); { fj, if i = j or j +, 0, otherwise; (QS5) [e i, e j] = [e ī, e j] = [f i, f j] = [f ī, f j] = 0 for i j, [e i, e j ] = [f i, f j ] = 0 for i j >, [e i, e i+ ] = [e ī, e i+ ], [e i, e i+ ] = [e ī, e i+ ], [f i+, f i ] = [f i+, f ī ], [f i+, f ī ] = [f i+, f i ]; (QS6) [e i, [e i, e j ]] = [e ī, [e i, e j ]] = 0, [f i, [f i, e j ]] = [f ī, [f i, f j ]] = 0 for i j =. Let U + (resp. U 0 and U ) be the subagebra of U = U(q(n)) generated by the eements e i, e ī (resp. h j, h j, and f i, f ī ), where i n, j n. Then, simiar to Lie agebras, we have the Poincaré-Birhoff-Witt (PBW) Theorem and trianguar decomposition as foows; see [3, Theorem.32], [, (.3)], [7, Theorem 2.]. Proposition 2.2. the set () Suppose that {z,..., z p } is a homogeneous basis for q(n). Then {z a z a 2 2 z ap p a,..., a p N, a i Z 2 if z i is odd, i p} is a basis for U(q(n)). (2) The agebra U(q(n)) has the trianguar decomposition U(q(n)) = U U 0 U +. Let V be the vector superspace over Q with dimension vector dimv = (n, n). Fix a basis {v,..., v n } for V 0 and a basis {v,..., v n } for V, respectivey. Then there is a natura action of the agebra g(n n) on V given by eft mutipication, that is, E v = δ j v i (2.2.) for i, j, I(n n). The restriction to the agebra q(n) impies that V naturay affords a representation of U = U(q(n)). As U(q(n)) admits a comutipication U(q(n)) U(q(n)) U(q(n)) given on eements of q(n) by x x + x, we have that the r-fod tensor product V r of the natura modue V aso affords a U(q(n))-modue. Let φ r denote the corresponding superagebra homomorphism: φ r : U(q(n)) End Q (V r ). Define the queer Schur superagebra (aso nown as Schur superagebra of type Q, cf. []) to be Q(n, r) = φ r (U(q(n))), (2.2.2) that is, the image of φ r. Therefore, Q(n, r) can be viewed as a quotient of U(q(n)).

5 PRESENTING QUEER SCHUR SUPERALGEBRAS 5 Simiar to Schur agebras associated to g(n), there is another way to define the queer Schur superagebra Q(n, r) via the nown Schur-Sergeev duaity as foows. Denote by C r the Cifford superagebra generated by odd eements c,..., c r subject to the reations c 2 i =, c i c j = c j c i, i j r. (2.2.3) Denote by H c r = C r S r the so-caed Sergeev superagebra, which is generated by the even eements s,..., s r and the odd eements c,..., c r subject to (2.2.3), and the additiona reations: s 2 i =, s i s j = s j s i, i, j r, i j >, s i s i+ s i = s i+ s i s i+, i r 2, s i c i = c i+ s i, s i c j = c j s i, i r, j r, j i, i +. Then by [8, Lemma 2], we have a representation (ψ r, V r ) of H c r on V r defined by ψ r (s )(v j v j v j+ v jr ) = ( )ˆv j ˆv j+ vj v j+ v j v jr, ψ r (c )(v j v j v jr ) = ( ) (ˆv j +...+ˆv j ) v j... v j J V (v j )... v jr, for a j,..., j r I(n n), r, r, where J V End(V ) satisfies J V (v a ) = v a and J V (v a ) = v a for a n. Then a cassica resut [8, Theorem 3] of Sergeev says Q(n, r) = End H c r (V r ), (2.2.4) which was introduced and studied in []. In particuar, we note that Q(n, r) is naturay a subsuperagebra of the Schur superagebra associated to the Lie supeagebra g(n n). 3. Commutation formuas for root vectors and Kostant Z-form For α = ɛ i ɛ j Φ with i j n, we introduce the root vectors in q(n) as foows: x x α = E + E i, j, x x α = E i, j + E. (3.0.5) Ceary, x α is even and x α is odd for α Φ. Observe that the even eement x α for α Φ correspond to the usua root vectors in g(n) under the identification (2.0.5). Moreover the set {x α, x α α Φ} {h i, h ī i n} is a homogeneous basis for q(n). By (3.0.5) and (2.0.2), a direct cacuation gives rise to the foowing super commutator formuas. {, if j = ; Lemma 3.. For α, α, Φ satisfying α + α, Φ, et ε = ε ;, =, if i =. Then we have in q(n) () [x, x, ] = (2) [x, x, ] = (3) [ x, x, ] = h i h j, if α + α, = 0; εx β, if β = α + α, Φ; h ī h j, if α + α, = 0; ε x β, if β = α + α, Φ; h i + h j, if α + α, = 0; x β, if β = α + α, Φ; (4) [h, x ] = (ɛ, α )x, [h, x ] = (ɛ, α ) x, [h, x ] = (ɛ, α ) x, [h, x ] = (ɛ, α ) x.

6 6 DU AND WAN By Lemma 3.(3) and (QS), we have for α Φ and i n: x 2 α = 0, h 2 ī = h i. (3..) For α Φ and i n, N, we introduce the foowing eements: ( ) ( ) x () α = x α!, hi hi =, and = h i(h i ) (h i + ) 0! ( ). It is nown that q(n) can naturay be viewed as a subspace of the universa enveoping agebra U(q(n)) and moreover, for any homogeneous u, v q(n), we have uv ( )û ˆv vu = [u, v] in U(q(n)), where [u, v] means the Lie bracet of u, v in q(n). Then by Lemma 3., we have the foowing. Proposition 3.2. Maintain the notation defined in Lemma 3. and et m, s N. Then the foowing hods in U(q(n)): min(m,s) ( ) x (s), x(m) + x (s t) hi h j m s + 2t, x (m t) t, if α + α, = 0; () x (m) t= x(s), = min(m,s) x (s), x(m) + ε t x (s t), x (t) β x(m t), if β = α + α, Φ; t= x (s), x(m), otherwise. x, x (m) (2) x (m) + (h ī h j)x (m ) x x (m 2), if α + α, = 0; x, = x, x (m) + ε x β x (m ), if β = α + α, Φ; x, x (m) otherwise. (3) x x, =, x, x + (h i + h j ), if α + α, = 0; x, x + x β, if β = α + α, Φ; x, x, otherwise. (4) x (m) h = h x (m) (ɛ, α )x x (m ), x h = h x (ɛ, α ) x. (5) x (m) h = (h m(ɛ, α ))x (m), x h = (h (ɛ, α )) x. Proof. Since the even root vectors x α can be identified with the usua root vectors in g(n) under the identification (2.0.5), part () foows from the cassica case (cf. [5, (5.a-c)]). Now suppose α + α, = 0. Then i =, j =. By Lemma 3.(2), the foowing hods: x m x j,i = x j,i x m + = x j,i x m + m d=0 m d=0 d=0 x d (h ī h j) x m d ( d (hī h j)x d t=0 ) x t 2x x d t x m d m ( ) = x j,i x m + (hī h j)x d 2dx x d x m d = x j,i x m + m(h ī h j)x m m(m )x x m 2, where the second and third equaities are due to the facts x (h ī h j) = (h ī h j)x 2 x and x x = x x by Lemma 3.. Hence, the first case of part (2) hods. Simiary, the other cases can be verified.

7 PRESENTING QUEER SCHUR SUPERALGEBRAS 7 Define the Kostant Z-form U Z to be the Z-subagebra of U generated by { () e i, f () i, e ī, f ī i n, N } { ( ) h i, h ī }. i n, N Denote by U + Z (resp. U Z ) the Z-subagebra of U generated by e() i, e ī (resp. f () i, f ī ), where ( ) i n and N. Let UZ 0 be the Z-subagebra of U generated by hi, h ī ( i n, N). Remar 3.3. Let G = {x (s), x, h ī s N, i j n}. If we introduce the degree for the generators x (s), x, h ī, ( h i deg(x (s) ij ) = s j i, deg( x ij) = j i, deg(h ī ) =, deg ) for UZ with ( ) hi = 0, (3.3.) then every commutator ab ( )â ˆbba in Proposition 3.2()-(4), where a, b G beong to different trianguar parts, is a inear combination of monomias in the generators of degree stricty ess. The fact wi be usefu beow. For b N n, set b = b + + b n and define ( ) h n = b For D = (D,..., D n ) Z n 2, set For the subset Z 2 of N, et h D = h D i= ( hi b i ). (3.3.2) h Dn n. M n (N Z 2 ) := M n (N) M n (Z 2 ). For an n n matrix X, et X = X + X 0 + X + be the decomposition of X into ower trianguar, diagona, and upper trianguar parts of X, and for each A = (A 0, A ) M n (N Z 2 ), et A ε = (A ε 0, A ε ) for every ε {+, 0, } and define e A + = i<j n (x (a0 ij ) x a ij ), f A = i<j n (x (a0 ji ) j,i x a ji j,i ), (3.3.3) where A 0 = (a 0 ij), A = (a ij), and the order in the products are defined as foows (cf. [8]). For the jth row (reading to the right) a j,j+,..., a jn of A +, put π j (A + ) = (x (a0 j,j+ ) j,j+ x a j,j+ j,j+ ) (x(a0 jn ) j,n xa jn j,n ) and et e A + = π n (A + ) π (A + ). (3.3.4) Simiary for the jth coumn (reading upwards) a nj,..., a j+,j for A, put and et Then we have the foowing. π j (A ) = (x (a0 nj ) n,j x a nj n,j ) (x(a0 j+,j ) j+,j x a j+,j j+,j ) f A = π (A ) π n (A ). (3.3.5)

8 8 DU AND WAN Proposition 3.4. () As vector spaces, we have U Z = U Z U Z 0 U + Z. (2) The superagebra U Z is a free Z-modue with basis given by the set ) { h fa ( h A 0 A 0 e A + A M n (N Z 2 ) }. (3.4.) 0 Proof. Appying the commutation formuas in Proposition 3.2 yieds the incusion U Z U Z U Z 0U + Z. Thus, the required isomorphism in part () foows from the restriction to U Z of the canonica isomorphism U(q(n)) U U 0 U + in Proposition 2.2(2). Ceary by Propositions 2.2 and 3.2 (and noting (3..)), the subagebra U + Z (resp. U Z ) is spanned by the set {e A + A M n(n Z 2 ), A = A 0 = 0} (resp. {f A A M n (N Z 2 ), A + = A 0 = 0}). Furthermore, by (QS) and (3..) (together with a resut for U Z (q(n) 0 ), we have that U 0 Z is spanned by the set {( h A 0 0) ha 0 A M n(n Z 2 ), A + = A = 0}. Putting together we obtain that U Z is spanned by the set (3.4.). Meanwhie by Proposition 2.2() it is easy to chec that the set (3.4.) is ineary independent. Hence the proposition is proved. 4. Presenting the queer Schur superagebra Q(n, r) Denote by Λ(n, r) the set of compositions of r into n parts, or equivaenty we can view Λ(n, r) as a subset of P 0 in the foowing way: Λ(n, r) = {λ = λ ɛ + + λ n ɛ n P 0 λ + + λ n = r}. For λ Λ(n, r), denote by (λ) the number of nonzero parts in λ, that is, (λ) = {i λ i 0, i n}. Reca that {v,..., v n } and {v,..., v n } are bases for V 0 and V. For a r-tupe j = (j,..., j r ) I(n n) r, we set v j = v j v jr and define wt(j) = (µ,..., µ n ) via µ i = { j = ±i, r}, for a i n. (4.0.2) Then wt(j) Λ(n, r) and the set {v j j I(n n) r } is a basis for V r. Lemma 4.. Suppose j I(n n) r and wt(j) = µ. We have, for i n, () φ r (h i )(v j ) = µ i v j ; (2) φ r (h ī )(v j ) = 0 if µ i = 0. Proof. Firsty, by the definition of the action of q(n) on V, (2.0.4) and (2.2.), we obtain φ (h i )(v j ) = (E i,i + E i, i )(v j ) = δ i, j v j, φ (h ī )(v j ) = (E i, i + E i,i )(v j ) = δ i, j v j. for i n and j I(n n). Hence, by the definition of the homomorphism φ r and the comutipication on U(q(n)) and noting ĥi = 0 and ĥī =, we have φ r (h i )(v j ) = ( φ (h i ) + + φ (h i ) ) (v j v jr ) r = δ i, j v j, = φ r (h ī )(v j ) = ( φ (h ī ) + + φ (h ī ) ) (v j v jr ) r = ( )ˆv j + +ˆv j δi, j v j v j v j v j+ v jr, = for i n. Then by (4.0.2), the emma foows.

9 PRESENTING QUEER SCHUR SUPERALGEBRAS 9 Let I be the idea of the universa enveoping agebra U = U(q(n)) given by I = h + + h n r, h (h ) (h r),..., h n (h n ) (h n r). (4..) Then by (3..) we obtain for i n Define h i (h i ) (h i r) = h 2 ī (h i ) (h i r) I. (4..2) U(n, r) = U/I, U Z (n, r) = U Z /I U Z U 0 (n, r) = U 0 /I U 0, U 0 Z(n, r) = U 0 Z/I U 0 Z. (4..3) For notationa simpicity, we wi denote, by abuse of notation, the images of e i, f i, h i, x α, e A etc. in U(n, r) by the same etters. Proposition 4.2. The homomorphism φ r : U End Q (V r ) satisfies I er φ r. Hence there exists a natura surjective homomorphism φ r : U(n, r) Q(n, r). (4.2.) Proof. Fix an arbitrary j I(n n) r and write wt(j) = µ = (µ,..., µ n ) Λ(n, r). By Lemma 4.(), we have φ r (h + + h n )(v j ) = (µ + + µ n )v j = rv j. This impies φ r (h + + h n r) = 0 (4.2.2) since the set {v j j I(n n) r } is a basis for V r. On the other hand, again by Lemma 4.(), we have φ r (h ī (h i ) (h i r))(v j ) =(µ i ) (µ i r) ( φ r (h ī ) ) (v j ). (4.2.3) Observe that 0 µ i r. If µ i = 0, then we have φ r (h ī )(v j ) = 0 by Lemma 4.(2); otherwise, we have µ i r, which impies (µ i ) (µ i r) = 0. Therefore, by (4.2.3), we deduce that φ r (h ī (h i ) (h i r))(v j ) = 0. This means φ r (h ī (h i ) (h i r)) = 0 (4.2.4) for a i n. Therefore by (4..), (4.2.2) and (4.2.4), the idea I is contained in er φ r and hence the proposition is verified. For λ Λ(n, r), write λ = ( ) h U 0 λ Z(n, r). Proposition 4.3. The foowing hod in UZ 0 (n, r): () The set { λ λ Λ(n, r)} is a set of pairwise orthogona centra idempotents in UZ 0(n, r) and λ Λ(n,r) λ =. (2) ( h b) = 0 for any b N n with b > r. (3) For i n, λ Λ(n, r) and b N n, we have h i λ = λ i λ, ( ) ( ) ( ) h λ h λ = λ, and = ( ) λ λ, b b b b λ Λ(n,r) where ( ( ) λ b) = n λi i=. b i (4) Suppose i n and λ Λ(n, r). If λ i = 0, then h ī λ = 0.

10 0 DU AND WAN (5) The Z-agebra UZ 0 (n, r) is spanned by the set { λ h D λ Λ(n, r), D Z n 2, D i λ i for i n}. Proof. By (4..) and (4..2), we now the eements h + +h n r and h (h ) (h r),..., h n (h n ) (h n r) are contained in the idea I. Then under the identification (2.0.5), there is a natura agebra homomorphism from the agebra T 0 introduced in [5, (4.)] for g(n) to U 0 (n, r), which sends the eement H i in [5] to h i for i n. Therefore, parts ()-(3) foow from [5, Proposition 4.2]. To prove part (4), we observe that the foowing hods for λ Λ(n, r) and i n (h i )(h i 2) (h i r) λ = (λ i )(λ i 2) (λ i r) λ (4.3.) by part (3). Now suppose λ i = 0. Then by (4.3.) we obtain This impies (h i )(h i 2) (h i r) λ = ( ) r r! λ. ( ) r r!h ī λ = h ī (h i )(h i 2) (h i r) λ = 0 since h ī (h i )(h i 2) (h i r) I U 0. Therefore h ī λ = 0. Finay, the fact that the eements ( h b) hd with b N n, D Z n 2 span UZ 0 impies that UZ 0(n, r) is spanned by the set { λh D λ Λ(n, r), D Z n 2}. By part (4), we have λ h D = 0 if there exists i n such that D i > λ i. Therefore part (5) is proved. Proposition 4.4. Suppose i n, α Φ, λ Λ(n, r). Then the foowing commutation formuas hod in U Z (n, r): { { λ+α x x α λ = α, if λ + α Λ(n, r), λ+α x x α λ = α, if λ + α Λ(n, r), { { xα λ x α = λ α, if λ α Λ(n, r), xα λ x α = λ α, if λ α Λ(n, r), h ī λ = λ h ī. Proof. The ast equaity foows from the reation (QS). The proof of the remaining equaities is parae to that of [5, Proposition 4.5] (see Lemma 3.). Let us iustrate by checing in detai the second formua. Suppose α = α = ɛ i ɛ j with i j. Then by Proposition 4.3(3) we obtain ( ) ( ) ( ) h i + x α λ = x α λ i + h i + h h2 hn λ = x α λ i + λ λ 2 λ n ( = h (hi ) ( ) i hj + ( ) ) h x λ i + λ i λ j λ α (4.4.) ( ( ) ( ) hi hj + ( ) ) h = x λ i + λ α. If λ + α Λ(n, r), then λ j = 0 and (4.4.) becomes ( ( ) hi ( ) ) ( ) h h x α λ = x λ i + λ α = x α = 0, λ + ɛ i i λ j

11 PRESENTING QUEER SCHUR SUPERALGEBRAS ( ( ) hj hj by Proposition 4.3(2). If λ+α Λ(n, r), then λ j and Hence, (4.4.) becomes ( ( ) ( ) hi hj ( ) ) h x α λ = x λ i + λ j λ α + ( ) ( ) h h = x α + x α = λ+α x α, λ + ɛ i λ + α as desired. ( ) hj + = λ j λ j ) + λ j ( ( ) ( ) hi hj ( ) ) h x λ i + λ j λ α For A = (A 0, A ) M n (N Z 2 ) with A 0 = (a 0 ij), A = (a ij) and a ij = a 0 ij + a ij, define deg(a) := (a + + a nn) + (a ij + a ij ) j i, ro(a) := ( co(a) := ( χ(a) := (a + n a j,..., j n a j,..., j n a nj ), j n a jn ), j i j n n (a j + a j ), a 22 + j=2 n (a 2j + a j2 ),..., a nn ) j=3 = co(a 0 0) + co(a 0 ) + ro(a + ) + co(a ).. (4.4.2) Let M U Z be the set of monomias m in x (s) α, x α, h, ( h i s ) (α Φ, n, s N). The degree of a monomia m M is defined accordingy by (3.3.). Let Simiar to (3.4.), we define M n (N Z 2 ) = {C M n (N Z 2 ) C 0 0 = 0}. u C = f C h C 0 e C + U Z and u (C,λ) = f C λ h C 0 e C + U Z (n, r) for C M n (N Z 2 ) and λ Λ(n, r). Ceary, by definition, the degree function defined in Remar 3.3 satisfies deg(u C ) = deg(c). By Proposition 4.4, we have in U Z (n, r) u (C,λ) = λ u C = u C λ if u (C,λ) 0, (4.4.3) where λ = λ + ro(c ) co(c ) and λ = λ ro(c + ) + co(c + ). Lemma 4.5. () Let m M. Then, in U Z (n, r), m can be written as a inear combination of u (C,λ) for C M n (N Z 2 ), λ Λ(n, r) such that deg(c) deg(m). (2) Suppose C M n (N Z 2 ). Then we have in U Z (n, r) f C h C 0 e C + = ±e C +h C 0 f C + ζ C (G,λ)u (G,λ), for some ζ C (G,λ) Z, where the summation is over G M n(n Z 2 ) and λ Λ(n, r) such that deg(g) < deg(c).

12 2 DU AND WAN Proof. We appy induction on N = deg(m). The inductive base is cear. Assume now N >. First, by appying a sequence of commutation formuas for generators from different trianguar parts, Remar 3.3 and Proposition 3.2(5) te that we may write m = m m 0 m + + f, where m ε UZ ε and f is a inear combination of monomias of degree stricty ess than N. By induction, f is a inear combination of the basis eements in (3.4.). Now, by the (homogeneous) commutation formuas within a trianguar part, m m 0 m + is a inear combination of basis eements of degree at most N. Hence, in U Z ( ) h m = h b C 0 e C +, C M n(n Z 2 ),b N n ξ m (C,b)f C where ξ(c,b) m Z and ξ(c,b) m = 0 uness deg(c) = deg(u C ) deg(m). Finay, since ( h ( b) = λ λ Λ(n,r) b) λ in U Z (n, r), by Proposition 4.3, proving (). By appying a sequence of commutation formuas for generators from different trianguar parts, we have in U Z f C h C 0 e C + = ±e C +h C 0f C + g, where g is a inear combination of monomias m in the generators for U Z and, by Remar 3.3, deg(m) < deg(e C +h C 0 f C ). Now part (2) foows from part (). Set Given two eements λ = n i λ iɛ i, µ = n i µ iɛ i P +, we define λ µ λ i µ i, for i n. (4.5.) B = {u (C,λ) C M n (N Z 2 ), λ Λ(n, r), χ(c) λ}. Let M n (N Z 2 ) r := {A = (A 0, A ) = ((a 0 ij), (a ij)) M n (N Z 2 ) (a0 ij + a ij) = r}. Then, one can aso chec the foowing hods B = {u A := f A χ(a) ha 0 e A + A M n (N Z 2 ) r }. Proposition 4.6. The agebra U Z (n, r) is spanned by the set B. Proof. Let U Z (n, r) be the Z-submodue of U Z(n, r) spanned by B. Ceary by Proposition 3.4 and (4..3), the agebra U Z (n, r) is spanned by monomias m M. Then by Lemma 4.5 and Proposition 4.4, it suffices to show that u (C,λ) U Z (n, r) for a C M n (N Z 2 ) and λ Λ(n, r). Now fix a C M n (N Z 2 ) and λ Λ(n, r). Write C 0 = (c 0 ij), C = (c ij) and et c ij = c 0 ij + c ij. If χ(c) λ, then u (C,λ) B and we are done. Now assume χ(c) = (χ (C),..., χ n (C)) λ. Then by (4.5.), there exists i n such that λ i < χ i (C). We proceed on deg(c). If deg(c) = 0, then the resut hods as u (C,λ) = λ B. Now assume that deg(c) and et i be the biggest i such that λ i < χ i (C). Let G be the submatrix ((c 0, ) (c, )) i, n of C at the bottom right corner. Then by (3.3.4) and (3.3.5) we can write e C + = e G + m, f C = m f G and h C 0 = h h G 0, where h = i j= hc jj. Observe that f j G λ h = h f G λ by Proposition 3.2(4) and Proposition 4.4. Then we have u (C,λ) = f C λ h C 0 e C + = (m f G ) ( λ h h G 0 ) (e G +m ) = m h (f G λ h G 0 e G +) m. We can assume f G λ 0. Otherwise, we are done. Then by (4.4.3) we have f G λ = λ f G and hence u (C,λ) = m h λ ( fg h G 0 e G + ) m,

13 PRESENTING QUEER SCHUR SUPERALGEBRAS 3 where λ = λ + ro(g ) co(g ). Then appying Lemma 4.5(2) to f G h G 0 e G +, we have u (C,λ) =m h λ ( ± eg +h G 0 f G + ζ G (J,γ)u (J,γ) )m = ± m h λ e G +h G 0 f G m + ζ G (J,γ)m λ h u (J,γ) m, where ζ G (J,γ) = 0 uness deg(j) < deg(g). Observe that deg(g) + deg(m h ) + deg(m ) = deg(c). Note that by (4.4.3), we can appy Lemma 4.5 to each term m λ h u (J,γ) m to obtain the foowing: u (C,λ) = ±m h λ e G +h G 0 f G m +, where is a inear combination of u (C,µ) with deg(c ) < deg(c). By induction, U Z (n, r). Meanwhie we caim that m λ e G +h G 0 f G h m = 0. Indeed, we can assume λ e G Otherwise we are done. Then λ e G + = e G + λ by (4.4.3), where λ = λ ro(g + ) + co(g + ) = λ + ro(g ) co(g ) ro(g + ) + co(g + ). Moreover λ Λ(n, r). Since G is the submatrix of C consisting of the ast n i + rows and coumns, we have λ i = λ i ( n j=i+ c ji ) ( n j=i+ c ij ) = λ i (χ i (C) c ii) = λ i χ i (C) + c ii by (4.4.2). This means λ i < c ii since χ i (C) > λ i, forcing c i,i = and λ i = 0 since λ Λ(n, r). Hence, λ h G 0 = 0 by Proposition 4.3(4), proving the caim. In concusion, u (C,λ) U Z (n, r). We remar that the proof above foows [8]. However, it is possibe to modify the proof of [5] to give an aternative proof. Theorem 4.7. The homomorphism φ r : U(n, r) Q(n, r) in (4.2.) is an isomorphism. In other words, the Schur superagebra Q(n, r) is the associative superagebra generated by even generators h i, e j, f j, and odd generators h ī, e j, f j, with i n and j n subject to the reations (QS)-(QS6) together with the foowing extra reations: (QS7) h + h h n = r; (QS8) h ī (h i ) (h i r) = 0 for i n. Proof. By [, 4], we now that the dimension of the agebra Q(n, r) is equa to the number of monomias of tota degree r in the free supercommutative agebra in n 2 even variabes and n 2 odd variabes. Thus, dim Q(n, r) = M n (N Z 2 ) r. (4.7.) Hence, by (4.7.), Proposition 4.2 and Proposition 4.6 we have dim U(n, r) B = dim Q(n, r) dim U(n, r), which impies dim U(n, r) = B = dim Q(n, r). This forces that the surjective homomorphism φ r is an isomorphism. 2 In the non super case, we have λ e G + = 0 as λ i < χ i(g + ). However, this inequaity may not be true in the super case as the i-th entry of C 0 may not be zero.

14 4 DU AND WAN Coroary 4.8. The set B = { u A A M n (N Z 2 ) r } is a Z-basis for U Z (n, r). In particuar, the set { λ h D λ Λ(n, r), D Z n 2, D i λ i, i n} is Z-basis for UZ 0 (n, r). Coroary 4.9. We have the foowing dimension formuas: () dim Q Q(n, r) = ( ) ( ) r n 2 + n 2 =0 ; r (2) dim Q Q 0 (n, r) = λ Λ(n,r) 2(λ). Using a simiar argument in the proof of [5, Theorem 2.4], we obtain the foowing. Theorem 4.0. The queer Schur superagebra Q(n, r) is the unitary associative superagebra generated by the even eements λ, e j, f j and odd eements h ī, e j, f j for λ Λ(n, r), i n, j n subject to (QS3) and (QS5)-(QS6) as we as the foowing reations: (QS ) λ µ = δ λ,µ λ, λ Λ(n,r) λ =, h ī λ = λ h ī, h ī h j + h jh ī = δ ij λ Λ(n,r) 2λ i λ, h ī λ = 0 if λ i = 0; { { (QS2 λ+αj e ) e j λ = j, if λ + α j Λ(n, r), λ+αj e j, if λ + α e j λ = j Λ(n, r), { { λ αj f f j λ = j, if λ α j Λ(n, r), λ αj f j, if λ α f j λ = j Λ(n, r), { { ej λ e j = λ αj, if λ α j Λ(n, r), e j λ e j = λ αj, if λ α j Λ(n, r), { { fj λ f j = λ+αj, if λ + α j Λ(n, r), f j λ f j = λ+αj, if λ + α j Λ(n, r), 0, otherwise; (QS4 ) e i f j f j e i = δ ij λ Λ(n,r) (λ i λ i+ ) λ, e ī f j + f je ī = δ ij λ Λ(n,r) (λ i + λ i+ ) λ ; e i f j f je i = δ ij (h ī h i+ ), e ī f j f j e ī = δ ij (h ī h i+ ). 5. The quantum queer superagebra U q (q(n)) and its root vectors The quantum queer superagebra is more subte than the enveoping agebra. In the next four sections, we wi investigate the quantum root vectors and their commutation formuas. We then generaize Theorems 4.7 and 4.0 to quantum queer Schur agebras in the ast section. In [6], Oshansi introduced the quantum deformation U q = U q (q(n)) of the universa enveoping agebra U(q(n)) of q(n) as foows. Let the symbo { }, where the dots standard for some inequaities, equa if a these inequaities are satisfied and 0 otherwise. For i, j, I(n n), put ϕ(i, j) = δ i, j sgn(j), p(i, j) = { 0, if ij > 0,, if ij < 0, and θ(i, j, ) = sgn(sgn(i) + sgn(j) + sgn()), where sgn(a) = if a > 0 and sgn(a) = if a < 0 for an arbitrary nonzero integer a.

15 PRESENTING QUEER SCHUR SUPERALGEBRAS 5 Definition 5.. The quantum queer superagebra U q (q(n)) is the associative superagebra over Q(q) generated by L for i, j I(n n) with i j subject to the foowing reations L i,i L i, i = L i, i L i,i =, ( ) p()p(,) q ϕ(j,) L L, + { j < }θ(i, j, )(q q )L i, L,j + {i < j }θ( i, j, )(q q )L i, L, j (5..) = q ϕ(i,) L, L + { < i }θ(i, j, )(q q )L i, L,j + { i < j}θ( i, j, )(q q )L i, L,j. The Z 2 -grading on U q (q(n)) is defined via ˆL = p(i, j) for i, j I(n n) with i j. Foowing [6, Remar 7.3], we introduce the foowing set of generators of U q (q(n)): K i :=L i,i, K i := L i, i, K ī := q q L i,i, E j := q q K j+l j, j, E j := q q K j+l j,j, F j := q q L j,j+k j+, F j := q q L j,j+k j+. (5..2) for i n and j n. Then the agebra U q (q(n)) can be defined via a quantum anaogue of the reations (QS)-(QS6) using (5..) as foows. Proposition 5.2 ([, Theorem 2.] cf. ([2])). The quantum superagebra U q (q(n)) is isomorphic to the unita associative superagebra over Q(q) generated by even generators K ± i, E j, F j and odd generators K ī, E j, F j, for i n, j n, satisfying the foowing reations: (QQ) K i K i = = K i K i, K i K j = K j K i, K i K j = K jk i, Ki 2 K 2 i K ī K j + K jk ī = 2δ ij q 2 q ; 2 (QQ2) K i E j = q (ɛ i,α j ) E j K i, K i E j = q (ɛ i,α j ) E jk i, K i F j = q (ɛ i,α j ) F j K i, K i F j = q (ɛ i,α j ) F jk i ; (QQ3) K ī E i qe i K ī = E ī K i, qk ī E i E i K ī = K i E i, K ī F i qf i K ī = F ī K i, qk ī F i F i K ī = K i F i, K ī E ī + qe ī K ī = E i K i, qk ī E i + E i K ī = K i E i, K ī F ī + qf ī K ī = F i K i, qk ī F i + F i K ī = K i F i, K ī E j E j K ī = K ī F j F j K ī = K ī E j + E jk ī = K ī F j + F jk ī = 0 for j i, i ; (QQ4) E i F j F j E i = δ ij K i K i+ K i K i+ q q, ( Ki K i+ K i E ī F j + F je ī = δ ij E i F j F je i = δ ij (K i+ Kī K i+ K E ī F j F j E ī = δ ij (K i+ K ī K i+ K i ); K ) i+ + (q q )K q q ī K i+ i ), (QQ5) E 2 ī = q q q + q E2 i, F 2 ī = q q q + q F 2 i, E i E j E je i = F i F j F jf i = 0 for i j,,

16 6 DU AND WAN E i E j E j E i = F i F j F j F i = E ī E j + E je ī = F ī F j + F jf ī = 0 for i j >, E i E i+ qe i+ E i = E ī E i+ + qe i+ E ī, E i E i+ qe i+ E i = E ī E i+ qe i+ E ī, qf i+ F i F i F i+ = qf i+ F ī + F ī F i+, qf i+ F i F i F i+ = qf i+ F ī F ī F i+ ; (QQ6) Ei 2 E j (q + q )E i E j E i + E j Ei 2 = 0, Fi 2 F j (q + q )F i F j F i + F j Fi 2 = 0, Ei 2 E j (q + q )E i E j E i + E j Ei 2 = 0, Fi 2 F j (q + q )F i F j F i + F j Fi 2 = 0, where i j =. Remar 5.3. The generators in (5..2) are different from those in [, Theorem 2.]. Actuay, setting E j = K j+ E j, E j = K j+ E j, F j = F j K j+, F j = F jk j+ (5.3.) for j n, then by [2, Remar.2] the eements K ± i, K ī, E j, F j, E j, F j can be identified with q ± i, ī, e j, f j, e j, f j in [, Theorem 2.], respectivey. The reations in Proposition 5.2 are obtained by rewriting the whoe defining reations among q ± i, ī, e j, f j, e j, f j given by [, Theorem 2.] in terms of the eements K ± i, K ī, E j, F j, E j, F j by using (5.3.). For convenience, et us write down some of the reations in terms of the generators E j, F j, E j, F j as foows, which wi be usefu ater on. Lemma 5.4 ([, Theorem 2.]). The foowing hods in U q : for n 2, a n and i, j n with i j >, E +E E E + = E + E E E +, F +F F F + = F + F F F +, K a E i = q (ɛa,αi) E ik a, K a F i = q (ɛa,αi) F i K a, K a E ī = q(ɛa,α i) E ī K a, K a F ī = q (ɛa,α i) F ī K a, E ie j E je i = F i F j F jf i = 0. By [6, 4] (cf. [, (2.9)]), the comutipication : U q (q(n)) U q (q(n)) U q (q(n)) is defined by j (L ) = L i, L,j, (5.4.) =i for i, j I(n n) with i j. Then by (5..2) we have (K i ) = K i K i, (E j ) = E j + E j K j K j+, (F j ) = K j K j+ F (5.4.2) j + F j. for i n and j n. By Proposition 5.2, it is routine to chec that there is an anti-invoution Ω : U q (q(n)) U q (q(n)) given by Ω(q) = q, Ω(K i ) = K i, Ω(K ī ) = K ī, Ω(E j ) = F j, Ω(F j ) = E j, Ω(E j) = F j, Ω(F j) = E j. (5.4.3) for i n and j n. As for Lie agebras, we have the foowing PBW Theorem for U q (q(n)) due to Oshansi. Proposition 5.5 ([6, Theorem 6.2]). Fix an order on L for i, j I(n n) with i < j. Then the set { L m, L mn n,n L m ij m,..., m n Z, m ij N, m ij Z 2 if ˆL } =, (5.5.) i<j I(n n) is a Q(q)-basis of the agebra U q (q(n)).

17 PRESENTING QUEER SCHUR SUPERALGEBRAS 7 Remar 5.6. In [6], a particuar order on the eements L for i, j I(n n) with i < j was chosen to prove the PBW Theorem. Actuay, it is easy to chec that the arguments in the proof of [6, Theorem 6.2] do not depend on the choice of the order. For α = ɛ i ɛ j Φ with i j n, we introduce inductivey the root vectors as foows. For i n, we set X i,i+ = E i, X i+,i = F i, X i,i+ = E ī, X i+,i = F ī. For j i >, we define { Xi, X X α X :=,j qx,j X i,, if i < j, X i, X,j q X,j X i,, if i > j, { Xi, X X α X :=,j qx,j X i,, if i < j, X i, X,j q X,j X i,, if i > j, (5.6.) where is stricty between i and j. It is straightforward to chec that X, X are independent of the choice of. Ceary X are even eements whie X are odd eements. By (5..2) and (QQ2) in Proposition 5.2, one can chec that for α Φ, i n K i X α K i = q (ɛ i,α) X α, K i X α K i = q (ɛ i,α) X α. (5.6.2) Therefore the eements X α and X α can be viewed as the quantum anaog of the root vectors x α and x α introduced in (3.0.5). Meanwhie, by (5.4.3), we obtain Ω(X ) = X j,i, Ω(X ) = X j,i. (5.6.3) for i j n. By (5..2) and (5..), a direct cacuation shows that q L K a, if i = a, K a L = ql K a, if j = a, L K a, otherwise, Ka L = ql Ka, if i = a, q L Ka, if j = a, L Ka, otherwise. (5.6.4) for a n and i, j I(n n) with i j and i j. Lemma 5.7. The foowing hods for i < j n: X = q q K jl j, i, X = q q K jl j,i, X j,i = q q L K j, X j,i = q q L K j.

18 8 DU AND WAN Proof. Suppose i < j n. Reca the generators (5.3.) used in [, Theorem 2.]. Then by [, (2.4)] and Remar 5.3, we have ( j L j, i =( ) j i (q q ) a=i+ ( j L j,i =( ) j i (q q ) ( j L =(q q ) a=i+ ( j L = (q q ) K a a=i+ a=i+ ) K a ) K a ) K a j a=i+ ) j a=i+ j a=i+ adf a(f i ), j a=i+ ade a(e i), ade a(e ī ), adf a(f ī ), (5.7.) where adg a (G i ) := G a G i G i G a, j a=i+ adg a(g i ) := adg j adg i+ (G i ) if j i + 2 and j a=i+ adg a(g i ) = G i if j = i +, for G i = E i, E ī, F i, F ī and G a = E a, F a with i + a j. The first formua in (5.7.) and Lemma 5.4 impy ( j L j, i =( ) j i (q q ) a=i+ ( j =( ) j i (q q ) =K i+ ade i(l j, i ), a=i+ ) K a j a=i+2 ) ( j K a ade i ade a( ade i(e i+)) a=i+2 ) ade a(e i+) then, by (5.6.4) and (5.3.), L j, i =K i+ E il j, i K i+ L j, i E i =K i+ E il j, i ql j, i K i+ E i =E i L j, i ql j, i E i. (5.7.2) Simiary, by (5.7.) and the corresponding equaities in Lemma 5.4, one can prove L j,i =E i L j,i+ ql j,i+ E i, L =L i+,j F i q F i L i+,j, L = L i,j F i q F i L i,j. (5.7.3) We now prove the first formua in the emma by induction on j i. Indeed, if j = i +, then X = E j = q q K jl j, j+

19 PRESENTING QUEER SCHUR SUPERALGEBRAS 9 by (5..2). Now assume j i + 2, then by induction and (5.6.) we have X = X i,i+ X i+,j qx i+,j X i,i+ = E i q q K jl j, i q q q K jl j, i E i = q q K j(e i L j, i ql j, i E i ) (by (5.6.4) as j i + 2) = q q K jl j, i (by (5.7.2)), as desired. By a parae argument, we can prove the remaining three formuas. Fix an order in Φ +, or equivaenty in the set {(i, j) i < j n}. Reca that M n (N Z 2 ) is the set consisting of C M n (N Z 2 ) such that C0 0 = 0. Given C = (C 0, C ) M n (N Z 2 ), we can introduce the eements X C + = i<j n ( X c 0 ij X c ij ), XC = i<j n ( X c 0 ji j,i ) X c ji j,i, KC 0 = Kc K c nn n, where C 0 = (c 0 ij), C = (c ij). For σ = (σ,..., σ n ) Z n, et K σ = K σ K σn n. Proposition 5.8. The set { XC K σ K C 0 X C + C M n(n Z 2 ), σ Z n} (5.8.) is a Q(q)-basis of U q (q(n)). In particuar, if U q (q(n) 0 ) denotes the subagebra generated by E,..., E n, F,..., F n, K ±,..., K ±, then which is a Hopf agebra isomorphism. n U q (q(n) 0 ) = U q (g(n)), (5.8.2) Proof. Suppose C M n (N Z 2 ) and C 0 = (c 0 ij), C = (c ij). By Lemma 5.7 and (5.6.4), the foowing hods ( X C K σ K C 0 X C + =g(c, σ)l m, L mn c n,n L 0 ji Lc ji ) L c, L c nn n,n i<j n i<j n ( c L 0 ) (5.8.3) ij j, i Lc ij j,i, where g(c, σ) Q(q) is nonzero, m j = σ j + j i= (c0 ij + c ij c 0 ji c ji) for j n, and the product is taen with to the fixed order on {(i, j) i < j n}. This impies, up to nonzero scaars, the set (5.8.) actuay coincides with the set (5.5.) where the order on L for i, j I(n n) with i < j is taen to be compatibe with the product on the right hand side of (5.8.3). Then by Proposition 5.5, the first assertion is verified. For the second assertion, we observe that the reations in Proposition 5.2 invoving E i, F i, K j for i n and j n are the same as the standard reations for U q (g(n)). This gives a homomorphism from U q (g(n)) to U q (q(n) 0 ) which is an isomorphism by the first assertion. The ast assertion is cear. Remar 5.9. Under the identification (5.8.2), the eements X α and X α up to scaar mutipication coincides with the root vectors in [5, (3.)]. More precisey, et X α = q i j+ X α and X α = q j i X α for α = ɛ i ɛ j Φ +. Then X α and X α correspond to the root vectors X α and X α in [5, (3.)] (cf. [9]).

20 20 DU AND WAN Let Uq 0 be the subagebra of U q (q(n)) generated by K ± i, K ī for i n, and et U q + (resp. Uq ) be the subagebra of U q (q(n)) generated by the eements E j, E j (resp. F j, F j) for j n. We have reproduced the foowing. Proposition 5.0 ([, Theorem 2.3]). There is a Q(q)-inear isomorphism U q (q(n)) = U q U 0 q U + q. 6. Commutation formuas for quantum root vectors We divide the commutation formuas into four groups which wi be discussed in four cases beow. Each of the first three cases consists of two emmas, deaing with the case of two positive (or negative) roots and the case of one positive and one negative roots. The three cases are divided according to whether the pair of root vectors are even-even, even-odd, or odd-odd. Moreover, by the anti-automorphism Ω given in (5.6.3), it suffices to oo at the commutation formuas of positive root vector X, X (i < j) with others. Case Commutation formuas for two even-even root vectors X X, (i < j, ). Lemma 6.. The foowing hods for i, j,, n satisfying i < j, < : X, X, (i < j < < or i < < < j), q X X, = X, X, (i < < j = or i = < j < ), qx, X + X i,, (i < = j < ), X, X (q q )X,j X i,, (i < < j < ). Proof. Suppose i, j,, n and i < j, <. Using the formua (5..), a straightforward cacuation shows that L, L j, i, (i < j < < or i < < < j), q L j, i L, = L, L j, i, (i < < j = or i = < j < ), L, L j, i (q q )L j, j L, i, (i < = j < ), L, L j, i (q q )L j, L, i, (i < < j < ). Then the emma is proved case-by-case using Lemma 5.7. Let us iustrate by checing in detai the case when i < = j <. In this case, by (5..2), Lemma 5.7 and the above formua, we have X X, = q q K jl j, i q q K L, = (q q ) 2 K jk L j, i L, = (q q ) K ( ) jk 2 L, L j, i (q q )L j, j L, i = (q q ) K L 2, qk j L j, i q q K jl j, j K L, i =qx, X + X i,, where the second equaity and fourth equaity are due to (5.6.4). The remaining cases can be verified simiary and we sip the detai. Observe that we can derive another set of commutation formuas for X and X, by soving for X, X in Lemma 6. and then interchanging (i, j) and (, ). Namey, we

21 PRESENTING QUEER SCHUR SUPERALGEBRAS 2 have X, X, qx X X, =, X, q X, X q X,j, X, X + (q q )X,j X i,, ( < < i < j or < i < j < ), ( < i < = j or = i < < j), ( < i = < j), ( < i < < j). (6..) This together with Lemma 6. gives a compete commutation formuas for even positive root vectors X and X, with i < j, <. Lemma 6.2. Assume that i, j,, n satisfy i < j, >. Then X, X + K ik j K i K j, (i =, j = ), q q X X X, =, X, (i < j < or i < < < j), X, X K K j X,j, (i = < j < ), X, X + K K j X i,, (i < < j = ), X, X + (q q )K K j X,j X i,, (i < < j < ), Proof. Suppose i, j,, n and i < j, >. By (5..), it is easy to chec that L j, i L, = L, L j, i, if i < j < < or i < < < j. Then in this situation, by Lemma 5.7 and (5.6.4), we have X X, = X, X. Simiary, in other cases, again by the formua (5..), we obtain L, L j, i + (q q )(L j,j L i, i L j, j L i,i ), (i =, j = ), L, L j, i + (q q )L j, L i, i, (i = < j < ), L j, i L, = ql, L j, i, (i < = j < ), L, L j, i + (q q )L j,j L, i, (i < < j = ), L, L j, i + (q q )L j, L, i, (i < < j < ). As before, the emma is proved case-by-case using Lemma 5.7. Let us iustrate by checing in detai the case when i < < j =. In this case, by (5..2), Lemma 5.7 and the above formuas we have X X, = q q K jl j, i q q L,K = (q q ) K ( ) 2 j L, L j, i + (q q )L j,j L, i K = (q q ) L,K 2 j q q K L j, i q q K jl j,j K = q q L,K q q K jl j, i + K j K q q K L, i =X, X + K j K X i,, L, i where the third equaity is due to (5.6.4) and the assumption i < < j =. remaining cases can be verified simiary and we sip the detai. The Observe that appying the anti-automorphism Ω to the formuas in Lemma 6.2 and interchanging (i, j) and (, ), one can obtain another set of commutation formuas for

22 22 DU AND WAN X and X, as foows: X, X, X X, = X, X X,j K i K X, X + X i, K i K X, X (q q )X,j X i, K i K ( < i < j or < i < j < ),, ( = i < < j),, ( < i < = j),, ( < i < < j). This together with Lemma 6.2 gives a compete commutation formuas for even positive root vectors X and the even negative root vectors X, with i < j, >. Case 2 Commutation formuas for two even-odd root vectors X X, (i < j, ). Lemma 6.3. Suppose that i, j,, n with i < j and <. () If i =, j = or i < j < < or < < i < j or < i < j <, then X X, = X, X. (2) In other cases, the foowing formuas hod: X, X + (q q )(X,j X i, X,j X i, ), q X, X, qx, X + X i,, qx, X (q q )X,j X i,, X X, = X, X (q q )X,j X i,, q X, X + q (q q )X,j X i,, q X, X q X,j, qx, X, X, X + (q q )X,j X i,, (i < < < j), (i = < j < ), (i < = j < ), (i < < j = ), (i < < j < ), ( = i < < j), ( < i = < j), ( < i < = j), ( < i < < j). Proof. Let i, j,, {, 2,..., n} satisfy i < j and <. As before by (5..), one can chec that if i =, j = or i < j < < or < < i < j or < i < j <, then L j, i L, = L, L j, i. Then part () is proved by Lemma 5.7 and (5.6.4). To prove part (2), again by (5..) we obtain L j, i L, = L, L j, i + (q q )(L j, L, i L j, L,i ), q L, L j, i, L, L j, i (q q )L j, j L,i, ql, L j, i (q q )L j, L,i, L, L j, i (q q )L j, L,i, q L, L j, i + q (q q )L j, L, i, L, L j, i + (q q )L j, L i, i, ql, L j, i, L, L j, i + (q q )L j, L, i, (i < < < j), (i = < j < ), (i < = j < ), (i < < j = ), (i < < j < ), ( = i < < j), ( < i = < j), ( < i < = j), ( < i < < j). As before, part (2) of the emma is proved case-by-case using Lemma 5.7. Let us iustrate by checing in detai the case when = i < < j. In this case, by the above formua and

23 PRESENTING QUEER SCHUR SUPERALGEBRAS 23 Lemma 5.7 we have X X, = q q K jl j, i q q K L, = (q q ) 2 K jk L j, i L, = (q q ) K ( ) jk 2 q L, L j, i + q (q q )L j, L, i ( ) = q K (q q ) 2 L, K j L j, i + q (q q )K j L j, K L, i =q X, X + q (q q )X,j X i,, where the second and fourth equaities are due to (5.6.4). The remaining cases can be verified simiary and we sip the detai. Lemma 6.4. Suppose that i, j,, n with i < j and >. Then () If i < j < or < i < j or < i < j <, then X X, = X, X. (2) In other cases, the foowing formua hods: X, X (K jk i K j K ī ), (i =, j = ), X, X + q(q q )K K (X,jX i, X,j X i, ), (i < < < j), X, X K i K j X,j, (i = < j < ), X X X, =, X + (q q )K K jx i, + K K j X i,, (i < < j = ), X, X + (q q )K K j X,j X i,, (i < < j < ), X, X X,j K i K (q q )X,j K ī K, ( = i < < j), X, X + X i, K i K, ( < i < = j), X, X (q q )K i K X,jX i,, ( < i < < j). Proof. Assume i < j n, < n. By (5..) one can prove L, L j, i, if i < j < < or < < i < j or < i < j <, L j, i L, = ql, L j, i, if i < = j <, q L, L j, i, if < i = < j. Then part () can be proved by (5.6.4) and Lemma 5.7. Otherwise, again by (5..) it is straightforward to chec that the foowing hods L j, i L, = L, L j, i + (q q )(L j,j L i, i L j, j L i,i ), L, L j, i + (q q )(L j, L, i L j, L,i ), L, L j, i + (q q )L j, L i, i, L, L j, i + (q q )(L j,j L, i L j, j L,i ), L, L j, i + (q q )L j, L, i, L, L j, i + (q q )(L j, L i, i L j, L i,i ), L, L j, i (q q )L j, j L,i, L, L j, i (q q )L j, L,i, (i =, j = ), (i < < < j), (i = < j < ), (i < < j = ), (i < < j < ), ( = i < < j), ( < i < = j), ( < i < < j).

24 24 DU AND WAN This together with Lemma 5.7 and (5.6.4) gives rise to part (2). Let us expain in detai the case when i < < < j. In this case, by the above formua and Lemma 5.7 we have X X, = q q K jl j, i q q L,K = (q q ) K ( 2 j L, L j, i + (q q )(L j, L, i L j, L,i ) ) K = (q q ) L,K 2 K jl j, i + qk q q K (K j L j, K L, i K j L j, K L,i ) =X, X + q(q q )K K (X,jX i, X,j X i, ), where the third and fourth equaities are due to (5.6.4). Case 3 Commutation formuas for two odd-odd root vectors X X, (i < j, ). Lemma 6.5. Let i, j,, {, 2,..., n} satisfy i < j and <. Then we have X X, = q q q + q X2, X, X, X, X (q q )(X,j X i, + X,j X i, ), qx, X q(q q )X,j X i,, qx, X + X i,, qx, X (q q )X,j X i,, X, X (q q )X,j X i,, (i =, j = ), (i < j < < ), (i < < < j), (i = < j < ), (i < = j < ), (i < < j = ), (i < < j < ). Proof. Suppose that i, j,, {, 2,..., n} satisfy i < j and <. As before by (5..), we get L j,i L, = q q q + q L2 j, i, L, L j,i, L, L j,i (q q )(L j, L,i + L j, L, i ), ql, L j,i q(q q )L j, L, i, L, L j,i (q q )L j, j L, i, ql, L j,i (q q )L j, L, i, L, L j,i (q q )L j, L, i, (i =, j = ), (i < j < < ), (i < < < j), (i = < j < ), (i < = j < ), (i < < j = ), (i < < j < ). Then the emma is proved case-by-case as before. We wi iustrate by checing in detai the case when i < = j <. In this case, by the above formuas, (5..2) and Lemma 5.7

25 PRESENTING QUEER SCHUR SUPERALGEBRAS 25 we have X X, = q q K jl j,i q q K L, = (q q ) K jk 2 L j,i L, = (q q ) K ( ) jk 2 L, L j,i (q q )L j, j L, i = (q q ) qk L 2, K j L j,i q q K jl j, j K L, i = qx, X + X i,, where the second and fourth equaities are due to (5.6.4). The remaining cases can be verified simiary, and we omit the detai. As before, by soving for X, X in Lemma 6.5 and then interchanging (i, j) and (, ), we obtain X, X, ( < < i < j), X, X + (q q )(X,j X i, X,j X i, ), ( < i < j < ), q X X, = X, X q (q q )X,j X i,, ( = i < < j), q X, X + q X,j, ( < i = < j), q X, X (q q )X,j X i,, ( < i < = j), X, X (q q )X,j X i,, ( < i < < j). This together with Lemma 6.5 gives a compete commutation formua for two even odd root vectors. Lemma 6.6. The foowing hods for i, j,, {, 2,..., n} satisfying i < j and > : () If i =, j =, then X X j,i = X j,i X + K ik j K i K j + (q q )K q q ī K j. (2) In other cases, the foowing formuas hod: X, X, (i < j < ), X, X q(q q )K K (X,jX i, + X,j X i, ), (i < < < j), X X, = X, X + q X,j K i K j q (q q )X,j K ī K j, (i = < j < ), X, X (q q )K K jx i, + K K j X i,, (i < < j = ), X, X (q q )K K j X,j X i,, (i < < j < ). Proof. Suppose that i, j,, {, 2,..., n} satisfy i < j and >. If i =, j =, then by (5..) we have L j,i L = L L j,i + (q q )(L j,j L i,i L j, j L i, i ) (q q )L j,j L i,i. (6.6.)