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Representations and almost split sequences for Hopf algebras. (English) Zbl 0855.16034

Bautista, Raymundo (ed.) et al., Representation theory of algebras. Seventh international conference, August 22-26, 1994, Cocoyoc, Mexico. Providence, RI: American Mathematical Society. CMS Conf. Proc. 18, 237-245 (1996).
Many results for group rings are generalized to finite dimensional Hopf algebras.
In Section 1, the authors obtain some properties of homomorphism groups and tensor products for finite dimensional Hopf algebras which generalize from the group ring situation. In Section 2, it is shown that a Hopf algebra is semisimple if and only if the trivial module is projective, or if and only if the trivial module is injective if the Hopf algebra is finite dimensional. Moreover, a finite dimensional Hopf algebra \(H\) over a field \(K\) with an involutive antipode is semisimple if and only if there is a projective \(H\)-module \(P\) such that \(\dim_K(P)\) is invertible in \(K\). In Section 3, the authors generalize some of the results of M. Auslander and J. F. Carlson [J. Algebra 103, 122-140 (1986; Zbl 0594.20005)] on group rings to a general finite dimensional Hopf algebra \(H\) over a field \(K\) with an involutive antipode. The main result is: for \(M\) a splitting trace module, if \(0\to\tau(\mathbf{1}_H)\to E @>\beta>>\text\textbf{1}_H\to 0\) is an almost split sequence, then \(\eta:0\to M\otimes_K\tau(\mathbf{1}_H)\to M\otimes_K E @>{\text{id}_M\otimes\beta}>> M\to 0\) is an almost split sequence modulo injectives.
For the entire collection see [Zbl 0837.00015].
Reviewer: Li Fang (Nanjing)

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
16P10 Finite rings and finite-dimensional associative algebras
16S34 Group rings
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras

Citations:

Zbl 0594.20005
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