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Simplicity of Kac Modules for the Quantum General Linear Superalgebra

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Simplicity of Kac Modules for the Quantum General Linear Superalgebra SIMPLICITY OF KAC MODULES FOR THE QUANTUM GENERAL LINEAR SUPERALGEBRA RANDALL R. HOLMES1 CHAOWEN ZHANG Abstract. A general necessary and sufficient condition is obtained for a Kac module of the quantum general linear superalgebra to be simple. More explicit conditions are then obtained by considering separately the case where the quantum parameter is not a root of unity and the case where it is a root of unity. 0. Introduction For fixed positive integers m and n the quantized enveloping superalgebra U = Uq(gl(m; n)) of the general linear Lie superalgebra gl(m; n) is a deformation of the universal enveloping superalgebra of gl(m; n) depending on a parameter q in the defining field F. The superalgebra U, which is defined by R. B. Zhang in [6], is the quantum general linear superalgebra (the term \supergroup" in that paper is changed to \superalgebra" in later work). The superalgebra U has a subalgebra B+ having a quotient U0 isomorphic to the quantized enveloping algebra of the Lie algebra gl(m)⊕gl(n). We can regard a given finite-dimensional 0 + simple U -module L as a B -module and then form the induced U-module K(L) = U⊗B+ L, called a Kac module. Kac modules are of interest because every finite-dimensional simple U-module is a quotient of K(L) for some L. The aim of this paper is to provide necessary and sufficient conditions for K(L) itself to be simple. The study of the simplicity of K(L) was initiated by R. B. Zhang in [6]. Our setting is a bit more general than the setting in that paper. For instance, in [6] the field F is taken to be the field of complex numbers, while our assumptions allow for other fields as well. Also, see Remark 4.4 below. We recover some results in [6] using different methods and providing detailed proofs for some claims that are made in that paper without proofs or with only sketched proofs. 2010 Mathematics Subject Classification. 17B37, 17B50. Key words and phrases. Quantum supergroup, quantum enveloping superalgebra, general linear, repre- sentation, Kac module, simple module. 1Corresponding author: Randall R. Holmes, [email protected], 334.844.6571. The authors thank the referee for a careful reading of the paper and for several useful suggestions. 1 2 The paper is organized as follows. In Section1 the superalgebra U is defined in terms of generators and relations. Section2 gathers together several formulas involving elements of U that will be used throughout the paper. In Section3 some structural features of U are obtained, including a PBW-type basis. In Section4 the Kac module is defined and a general criterion is given for its simplicity. Section5 provides a more explicit simplicity criterion. The remaining sections specialize to the two cases where q is not a root of unity (Section6) and where q is a root of unity (Section7). In both cases, finite-dimensional simple U0-modules are constructed and then explicit criteria are given for simplicity of the corresponding Kac modules. 1. The superalgebra U Fix positive integers m and n. Put I0 = f(i; j)j1 ≤ i < j ≤ m or m < i < j ≤ m + ng; I1 = f(i; j)j1 ≤ i ≤ m < j ≤ m + ng; and I = I0 [I1. Let F be a commutative ring with identity and let q be a unit in F such that q − q−1 is also a −1 2 unit. Assume that there exists a ring automorphism Ω of F such that Ω(q) = q and Ω = 1F. (For instance, we could have F = C(q), q an indeterminate, and Ω : x 7! x (x 2 C); q 7! q−1. Or we could have F = C, take q 6= ±1 to be a root of unity, and take Ω to be complex conjugation.) From Section4 on we assume that F is a field, the more general situation being required only for the proof of Theorem 3.1. −1 Denote by X the set of symbols Ei;i+1;Fi;i+1;Kk;Kk with i 2 [1; m + n) and k 2 [1; m + n]. (Here and below we use interval notation to denote the set of integers in the indicated range.) Let U^ be the free F-algebra on the set X . This algebra has the structure of F-superalgebra (i.e., Z2-graded F-algebra) uniquely determined by assigning degree 1 (odd parity) to the elements Em;m+1 and Fm;m+1 and degree 0 (even parity) to the remaining elements of X . (See [3, Section 1.2] for basic notions regarding superalgebras and Lie superalgebras.) ηk For k 2 [1; m + n], put qk = q , where ηk is 1 or −1 according as k ≤ m or k > m. For a = (i; j) 2 I, define the elements Ea = Eij and Fa = Fij of U^ recursively by −1 Ea = EikEkj − qk EkjEik; Fa = FkjFik − qkFikFkj; for i < k < j. It is straightforward to check that these definitions are independent of the choice of k and that the elements Ea and Fa have degree 0 if a 2 I0 and degree 1 if a 2 I1. SIMPLICITY OF KAC MODULES 3 The supercommutator of homogeneous elements x and y of a superalgebra is given by [x; y] := xy − (−1)x¯y¯yx; wherex ¯ denotes the degree of x. Denote by U = UF;q the F-algebra with generators X and the relations R given below. (Thus, U is the quotient of U^ by the ideal generated by all x − y, with x = y a relation in R.) −1 (R1) KkKl = KlKk;KkKk = 1, −1 (δkj −δk;j+1) (R2) KkEj;j+1Kk = qk Ej;j+1; −1 −(δkj −δk;j+1) KkFj;j+1Kk = qk Fj;j+1, −1 −1 KiKi+1 − Ki Ki+1 (R3)[ Ei;i+1;Fj;j+1] = δij −1 , qi − qi 2 2 (R4) Em;m+1 = 0;Fm;m+1 = 0, (R5) Ei;i+1Ej;j+1 = Ej;j+1Ei;i+1;Fi;i+1Fj;j+1 = Fj;j+1Fi;i+1 (ji − jj > 1), 2 −1 2 (R6) Ei;i+1Ej;j+1 − (q + q )Ei;i+1Ej;j+1Ei;i+1 + Ej;j+1Ei;i+1 = 0 (ji − jj = 1; i 6= m), 2 −1 2 (R7) Fi;i+1Fj;j+1 − (q + q )Fi;i+1Fj;j+1Fi;i+1 + Fj;j+1Fi;i+1 = 0 (ji − jj = 1; i 6= m), (R8)[ Em−1;m+2;Em;m+1] = 0; [Fm−1;m+2;Fm;m+1] = 0, with i; j 2 [1; m+n) and k; l 2 [1; m+n], where δij is the Kronecker delta. (The last relation is omitted if either m = 1 or n = 1.) We use the same letters to denote the images of elements of the generating set X in U under the canonical map U^ ! U. The superalgebra structure on U^ induces a superalgebra structure on U. The superalgebra U is the quantum general linear superalgebra, denoted Uq(gl(m; n)) in [5],[6]. It is a quantum deformation of the universal enveloping superalgebra of the Lie superalgebra gl(m; n). A bijective Z-linear map f : U ! U is an antiautomorphism of U if f(xy) = f(y)f(x) for all x; y 2 U: Lemma 1.1 (cf. [6, Appendix A]). The automorphism Ω of F extends to an antiautomor- phism Ω of U with −1 Ω: Ei;i+1 7! Fi;i+1;Fi;i+1 7! Ei;i+1;Kj 7! Kj for all i 2 [1; m + n); j 2 [1; m + n]. We have Ω(Ea) = Fa for every a 2 I. Proof. Denote by U0 the opposite ring of U regarded as an F-module via cx := Ω(c)x (c 2 F, x 2 U). There is a uniquely determined F-algebra homomorphism U^ ! U0 that maps the 4 −1 elements of X as indicated in the statement and with Kj 7! Kj. The ideal generated by all x − y with x = y in the relation R is contained in the kernel of this homomorphism, so we get an induced F-algebra homomorphism Ω : U ! U0. Regarding Ω as a function 2 Ω: U ! U we have Ω = 1U, so Ω is bijective and it is an antiautomorphism extending the given automorphism of F. The final claim follows immediately from the definitions of Ea and Fa. In [6, Appendix A], R. B. Zhang defines certain automorphisms Ti, i 2 [1; m + n) n m, of U. He calls these generalized Lusztig automorphisms since they generalize the automorphisms defined by Lusztig in [4, Theorem 3.1]. We will require only the following properties of these automorphisms. (In the formulas, empty products of automorphisms are regarded as the identity automorphism of U.) Lemma 1.2 ([6, Appendix A]). Let a = (i; j) 2 I with j 6= i + 1. If a 2 I0 let i < k < j be arbitrary; if a 2 I1, put k = m. We have j−i+1 −1 −1 −1 (a) Ea = (−1) TiTi+1 ··· Tk−1Tj−1Tj−2 ··· Tk+1(Ek;k+1), j−i+1 −1 −1 −1 (b) Fa = (−1) TiTi+1 ··· Tk−1Tj−1Tj−2 ··· Tk+1(Fk;k+1). 2. Some formulas We collect here several formulas involving the elements of U that we will require. −1 For a = (i; j) 2 I, put Ka = KiKj . Lemma 2.1 (cf. [6, Lemma 2]). For every a = (i; j) 2 I, k 2 [1; m + n], and b 2 I1, −1 δki−δkj −1 −(δki−δkj ) (a) KkEaKk = qk Ea and KkFaKk = qk Fa, −1 Ka − Ka (b)[ Ea;Fa] = −1 , qi − qi 2 2 (c) Eb = 0 and Fb = 0.
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