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Representations of Finite-Dimensional Hopf Algebras

✍ Scribed by Martin Lorenz

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
310 KB
Volume
188
Category
Article
ISSN
0021-8693

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✦ Synopsis

Let H denote a finite-dimensional Hopf algebra with antipode S over a field ‫މ‬ -.

w We give a new proof of the fact, due to Oberst and Schneider Manuscripta Math. 8 Ž .

x 1973 , 217᎐241 , that H is a symmetric algebra if and only if H is unimodular and S 2 is inner. If H is involutory and not semisimple, then the dimensions of all projective H-modules are shown to be divisible by char ‫މ‬ -. In the case where ‫މ‬ -is a splitting field for H, we give a formula for the rank of the Cartan matrix of H, reduced mod char ‫މ‬ -, in terms of an integral for H. Explicit computations of the Ž . Cartan matrix, the ring structure of G H , and the structure of the principal 0 indecomposable modules are carried out for certain specific Hopf algebras, in particular for the restricted enveloping algebras of completely solvable p-Lie Ž . algebras and of sl 2, ‫މ‬ -.

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