Generalized Smash
Products |
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Authors: | Email author" target="_blank">Zhi?Xiang?WuEmail author |
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Institution: | (1) Department of Mathematics, Zhejiang University, Hangzhou, 310027, P. R. China |
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Abstract: | Abstract
In this paper, we study the ring #(D,B) and obtain two very interesting
results. First we prove in Theorem 3 that the category of
rational left BU-modules is
equivalent to both the category of #-rational left modules and
the category of all (B,D)-Hopf modules
D
. Cai and Chen have
proved this result in the case B = D = A. Secondly they have proved that if
A has a nonzero left integral
then A#A
*rat is a
dense subring of End
k
(A). We prove that #(A,A) is a dense subring of
End
k
(Q), where Q is a certain subspace of #(A,A) under the condition that the
antipode is bijective (see Theorem 18). This condition is weaker
than the condition that A has
a nonzero integral. It is well known the antipode is bijective
in case A has a nonzero
integral. Furthermore if A
has nonzero left integral, Q
can be chosen to be A (see
Corollary 19) and #(A,A) is
both left and right primitive. Thus A#A
*rat ⊆
#(A,A) ≃
End
k
(A). Moreover we prove that the left
singular ideal of the ring #(A,A) is zero. A corollary of this is a
criterion for A with nonzero
left integral to be finite-dimensional, namely the ring
#(A,A) has a finite uniform
dimension. |
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Keywords: | Generalized Smash Product #-rational module Uniform dimension |
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