Normal basis and transitivity of crossed products for Hopf algebras
Normal basis and transitivity of crossed products for Hopf algebras
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DOI：
10.1016/0021-8693(92)90034-j
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发表时间：
1992-11
影响因子：
0.9
通讯作者：
H. Schneider
中科院分区：
文献类型：
--
作者：
H. Schneider
基本概念设k是一个交换环。所有代数和余代数都在k上定义，~ 3= Qk。设H是一个Hopf代数，而a（右）H-conzodzde代数，即a， a + a @ H是一个代数映射和一个右H-模结构（参见[lo]）。定义B:= AcoH:={ae4/A,(A)= A @ l)，协不变元的子代数。如果正则映射A aga -+ rl@ H， x@ y-+ c-u?？oQyl，其中x J 'E A， yl是A (y)的符号，是双射的。如果存在一个正确的H共线性* *可逆映射j: H—+ A，则扩展B c A称为H交叉积（或裂），然后j定义了一个同构B@ HZ A， B 0 11 H B j (H)，并且A作为代数被标识为一个破碎积b# (TH)，其中Q: H@ H—+ B是一个可逆的2-环。h交叉产物是具有正交基的h -伽罗瓦扩展，即。4 z BQ H为左b模和右H模（参见[21]，第8节；18、2.1;19、2.2;6，定理9和定理11；1、4.14;2、1.181）~ H交叉积的概念推广了交叉积的各种经典概念。商Hopf代数现在取一个满射Hopf代数映射p: H-+ i7。则（A@ p） A是一个r -模代数结构A。定义A ':= AcoR。如果k是一个域，a = b# d H, H ‘是p的（左）Hopf核，则a ’是子代数b# （i H '）。注意，B c A （H= R）和H 'c H （A= H）是特殊情况。本文的出发点是以下问题。假设A是
Basic Notions Let k be a commutative ring. All algebras and coalgebras will be defined over k, and~ 3= Qk. Let H be a Hopf algebra, and-4 a (right) H-conzodzde algebra, ie, A,: A+ A@ H is an algebra map and a righ H-comodule structure (cf.[lo]). Define B:= AcoH:={ae4/A,(a)= a@ l), the subalgebra of coinvariant elements. Then the ring extension B c A is called an H-Galois extension, if the canonical map A ag A-+ rl@ H, x@ y-+ c-u?? oQyl, where x, J’E A, and CyO@ yl is a notation for A,(y), is bijective. The extension B c A will be called an H-crossed product (or cleft) if there is a right H-colinear and*-invertible map j: H--+ A. Then j defines an isomorphism B@ HZ A, b 0 11 H b. j (h), and A as an algebra is identified with a smashed product B# (TH, where Q: H@ H-+ B is an invertible 2-cocycle. H-crossed products are H-Galois extensions with a rzorrnal basis, ie,. 4 z BQ H as left B-modules and right H-modules (cf.[21, Section 8; 18, 2.1; 19, 2.2; 6, Theorems 9 and 11; 1, 4.14; 2, 1.181)~ The notion of H-crossed products generalizes various classical concepts of crossed products.Quotient Hopf Algebras Now take a surjective Hopf algebra map p: H-+ i7. Then (A@ p) A, is an R-comodule algebra structure an A. Define A’:= AcoR. If k is a field, A= B# d H, and H’is the (left) Hopf kernel of p, then A’is the subalgebra B#(i H’. Note that B c A (H= R) and H’c H (A= H) are special cases. The starting point of this paper is the following question. Assume A is