Báo cáo toán học: On the Representation Categories of Matrix Quantum Groups of Type A

Một nhóm lượng tử của các loại, được định nghĩa trong điều khoản của một đối xứng Hecke. Chúng tôi hiển thị trong bài báo này rằng loại đại diện của một nhóm như vậy lượng tử là duy nhất được xác định như một loại abelian monoidal bện bằng cấp bậc hai đối xứng Hecke.
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Báo cáo toán học: " On the Representation Categories of Matrix Quantum Groups of Type A" 9LHWQDP -RXUQDOVietnam Journal of Mathematics 33:3 (2005) 357–367 RI 0$7+(0$7,&6 ‹ 9$67 On the Representation Categories of Matrix Quantum Groups of Type A* ` o’ Ph` ng Hˆ Hai uInstitute of Mathematics, 18 Hoang Quoc Viet Road, 10307, Hanoi, Vietnam; Dept. of Math., Univ. of Duisburg-Essen, 45117 Essen, Germany Dedicated to Professor Yu. I. Manin Received January 22, 2005 Revised March 3, 2005Abstract. A quantum groups of type A is defined in terms of a Hecke symmetry.We show in this paper that the representation category of such a quantum group isuniquely determined as an abelian braided monoidal category by the bi-rank of theHecke symmetry.1. IntroductionA matrix quantum group of type A is defined as the “spectrum” of the Hopfalgebra associated to a closed solution of the (quantized) Yang-Baxter equationand the Hecke equation (called a Hecke symmetry). Explicitly, let V be a vectorspace (over a field) of finite dimension d. An invertible operator R : V ⊗ V −→V ⊗ V is called a Hecke symmetry if it satisfies the equations R1 R2 R1 = R2 R1 R2 , (1)where R1 := R ⊗ idV , R2 := idV ⊗ R (the Yang-Baxter equation), (R + 1)(R − q ) = 0, q = 0; −1, (2)∗ Thiswork was supported in part by the Nat. Program for Basic Sciences Research of Vietnamand the “DFG-Schwerpunkt Komplexe Mannigfaltigkeiten”. ` o’358 Ph`ng Hˆ Hai u(the Hecke equation) and is closed in the sense that the half dual operator R : V ∗ ⊗ V −→ V ⊗ V ∗ , R (ξ ⊗ v ), w = ξ , R(v ⊗ w) ,is invertible. Given such a Hecke symmetry one constructs a Hopf algebra H as follows. ijFix a basis {xi ; 1 i d} of V and let Rkl be the matrix of R with respect tothis basis. As an algebra H is generated by two sets of generators {zj , ti ; 1 i ji d}, subject to the following relations (we will always adopt the conventionof summing over the indices that appear in both upper and lower places): ij p q i j mn Rpq zk zl = zm zn Rkl , z k tk = ti z j = δ j . i k i j kIn case R is the usual symmetry operator: R(v ⊗ w) = w ⊗ v (thus q = 1), H isisomorphic to the function algebra on the algebraic group GL(V ). The most well-known Hecke symmetry is the Drinfeld–Jimbo solutions of √series A to the Yang–Baxter equation (fix a square root q of q ) ⎡ qx ⊗ xi if i = j √i Rq (xi ⊗ xj ) = ⎣ qx ⊗ xi d if i > j (3) √j qxj ⊗ xi − (q − 1)xi ⊗ xj if i < j.In the “classical” limit q → 1, Rq reduces to the usual symmetry operator. There dis also a super version of these solutions due to Manin [12]. Let V be a vectorsuperspace of super-dimension (r|s), r + s = d, and let {xi } be a homogeneous r |sbasis of V , the parity of xi is denoted by ˆ. The Hecke symmetry Rq is given iby ⎡ ˆ (−1)i qxi ⊗ xi if i = j ij √ Rq |s (xi ⊗xj )b = ⎣ (−1)ˆˆ qxj ⊗ xi r ...