On the representation type of tensor product algebras
On the representation type of tensor product algebras
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DOI：
10.4064/fm-144-2-143-161
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发表时间：
1994
影响因子：
0.6
通讯作者：
Z. Leszczyński
中科院分区：
文献类型：
--
作者：
Z. Leszczyński
研究了有限维代数的张量积代数的表示类型。利用代数A，B的Gabriel箭图给出了代数A，B的特征，使得A ∈ B是驯服表示型.导论.本文中的代数是指固定代数闭域K上的有限维代数。所有的代数都假定是关于直积基本不可分解的。我们的目的是确定两个代数B和C的张量积代数B的表示类型，即用具有描述代数B和C的关系的箭图表示。我们研究的动机之一是引入一种统一的方法来研究几类重要的代数的表示类型，包括：（i）有限群G的群代数B[G]，其系数在代数B中（在[MS，S1]中研究）。(ii)下三角n× n矩阵代数（0.1）Tn（B）= n × n矩阵代数B 0 . . . 0 B B。. . 0... .. . . . B B . . . B是n ≥ 2且系数在代数B中的代数（在[AR，Br 2，L1，L2，LS，S2]中研究）。(iii)[HM]研究了Tn（B），n，r ≥ 2的因子代数Tn，r（B）：= Tn（B）/Jrn（B），其中Jn（B）是严格下三角n × n矩阵的理想. (iv)系数在代数B中的有界矩阵Q的路代数BQ。1991年数学科目分类：小学16 G60。
The representation type of tensor product algebras of finite-dimensional algebras is considered. The characterization of algebras A, B such that A⊗B is of tame representation type is given in terms of the Gabriel quivers of the algebras A, B. Introduction. In this paper by an algebra we mean a finite-dimensional algebra over a fixed algebraically closed field K. All algebras are assumed to be basic indecomposable with respect to the direct product. Our aim is to determine the representation type of the tensor product algebra B⊗K C of two algebras B and C in terms of the quivers with relations describing the algebras B and C. One of the motivations for our study is to introduce a unified approach to the investigation of the representation type of several important classes of algebras including: (i) The group algebras B[G] of a finite group G with coefficients in an algebra B (studied in [MS, S1]). (ii) The lower triangular n× n matrix algebras (0.1) Tn(B) = B 0 . . . 0 B B . . . 0 .. .. . . . B B . . . B with n ≥ 2 and with coefficients in an algebra B (studied in [AR, Br2, L1, L2, LS, S2]). (iii) The factor algebras Tn,r(B) := Tn(B)/J r n(B) of Tn(B), n, r ≥ 2, studied in [HM], where Jn(B) the ideal of strictly lower triangular n × n matrices. (iv) The path algebra BQ of a bound quiver Q with coefficients in an algebra B. 1991 Mathematics Subject Classification: Primary 16G60.