Báo cáo toán học: The Quantum Double of a Dual Andruskiewitsch-Schneider Algebra Is a Tame Algebra

Trong bài báo này, chúng ta nghiên cứu lý thuyết đại diện của tăng gấp đôi lượng tử D (Γn, d). Chúng tôi cung cấp cho các cấu trúc của mô-đun projective D (Γn, d) lần đầu tiên. Bằng cách này, chúng tôi cung cấp cho các Ext-run (quan hệ) của D (Γn, d) và hiển thị D (Γn, d) là một chế ngự...
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Báo cáo toán học: "The Quantum Double of a Dual Andruskiewitsch-Schneider Algebra Is a Tame Algebra"Vietnam Journal of Mathematics 34:2 (2006) 189–207 9LHWQD P -RXUQDO RI 0$ 7+ (0$ 7, &6 ‹ 9$67 The Quantum Double of a Dual Andruskiewitsch-Schneider Algebra Is a Tame Algebra* Meihua Shi Dept. of Math. Zhejiang Education Institute Hangzhou, Zhejiang 310012, China Received July 5, 2005 Revised December 22, 2005Abstract. In this paper, we study the representation theory of the quantum doubleD(Γn,d ). We give the structure of projective modules of D(Γn,d ) at first. By this,we give the Ext-quiver (with relations) of D(Γn,d ) and show that D(Γn,d ) is a tamealgebra.2000 Mathematics Subject Classification: 16W30Keywords: Representation Theory, Quantum double Tame Algebra1. IntroductionIn this paper, k is an algebraically closed field of characteristic 0 and an algebrais a finite dimensional associative k -algebra with identity element. Although the quantum doubles of finite dimension Hopf algebras are impor-tant, not very much is known about their representations in general. A completelist of simple modules of the quantum doubles of Taft algebras is given by Chenin [2]. He also gives all indecomposable modules for the quantum double ofa special Taft algebra in [3]. From this, we can deduce immediately that thequantum double of this special Taft algebra is tame. The authors of [7] study∗ Project(No.10371107) supported by the Natural Science Foundation of China.190 Meihua Shithe representation theory of the quantum doubles of the duals of the general-ized Taft algebras in detail. They describe all simple modules, indecomposablemodules, quivers with relations and AR-quivers of the quantum doubles of theduals of the generalized Taft algebras explicitly and show that these quantumdoubles are tame. The structures of basic Hopf algebras of finite representation type are gottenin [11]. In fact, the authors of [11] show that basic Hopf algebras of finiterepresentation type and monomial Hopf algebras (see [4]) are the same. But forthe structure of tame basic Hopf algebras, we know little. In [10], the authorgives the structure theorem for tame basic Hopf algebras in the graded case. Inorder to study tame basic Hopf algebras or generally tame Hopf algebras, weneed more examples of tame Hopf algebras. The Andruskiewitsch-Schneider algebra is a kind of generalization of gener-alized Taft algebra and of course Taft algebra. Therefore, it is natural to askthe following question: whether is the quantum double of dual Andruskiewitsch-Schneider algebra a tame algebra? In this paper, we give an affirmative answer.As a consequence, we give some new examples of tame Hopf algebra. Our method is direct. Explicitly, we firstly give the structure of projectivemodules of the Drinfel Double of a dual Andruskiewitsch-Schneider algebra bydirect computations. Then using this, we get its Ext-quiver with relations whichwill help us to get the desired conclusion.2. Main ResultsIn this section, we will study the Drinfeld Double D(Γn,d ), which is a general-ization of [7], of (A(n, d, μ, q ))∗cop . Our main result is to give the structure ofprojective modules of D(Γn,d ). By this, we give the Ext-quiver (with relations)of D(Γn,d ) and show that D(Γn,d ) is a tame algebra. This section relays heavilyon [7] and we refer the reader to this paper. The algebra Γn,d := kZn /J d with d|n is described by quiver and relations.The quiver is a cycle, mwith n vertices e0 , . . . , en−1 . We shall denote by γi the path of length m startingat the vertex ei . The relations are all paths of length d 2. We give the Hopf structure on Γn,d . We fix a primitive d-th root of unity qThe Quantum Double of a Dual Andruskiewitsch-Schneider 191and a μ ∈ k . ej ⊗ el + α0 − βt , 0 1 ej ⊗ γl1 + q l γj ⊗ el + α1 − βt 1 1Δ(et ) = Δ(γt ) = t ...