The trace function and Hopf algebras
type: Item
instance of: scholarly article
title: The trace function and Hopf algebras
zbMATH Open document ID: 0801.16039
author: David E. Radford
published in: Journal of Algebra
publication date: 1994-12-01T00:00:00Z
review text: The trace function Tr on \(\text{End}_ k(A)\) plays an important role in the study of \(A\) a semisimple Hopf algebra over a field \(k\) of characteristic 0 [the author and \textit{R. G. Larson}, Am. J. Math. 110, No. 1, 187-195 (1988; Zbl 0637.16006); the author, ibid. 112, No. 2, 331- 357 (1990; Zbl 0699.16008)]. The aim of this paper is to give some trace formulas in some cases, discuss many properties of the trace function on \(\text{End}_ k(A)\) for \(A\) a finite dimensional Hopf algebra and apply them to unimodular and semisimple Hopf algebras. The author computes the trace in terms of integrals in several ways and gives trace derivations of two basic formulas. Then, \(\chi_ a \in A^*\) defined by \(\chi_ a(b) = \text{Tr}(l(b) \circ r(a))\) for \(b\in A\) is called cocommutative. A non-zero right integral \(\lambda\) for \(A^*\) is cocommutative iff \(A\) is unimodular and \(s^ 2 = 1\) where \(s\) is the antipode of \(A\). \(\chi_ a\) is a nonzero right integral for some \(a\in A\) iff \(A\) is semisimple and \(s^ 2 = 1\). As a application, it is shown that when \((\dim A)1 \neq 0\), \(A\) is unimodular iff \(p(\Lambda \triangleright 1) \neq 0\), \(A^*\) is unimodular iff \(p(1\triangleleft \Lambda) \neq 0\); \(A\) is semisimple iff \(1 < \Lambda \neq 0\); when \(A\) is semisimple and \(k\) is algebraically closed and \(\text{char}(k) = 0\) or \(=p\geq \sqrt {\dim A}\) then \(\text{Tr}_ A\) is a right integral for \(A\) iff \(\text{Tr}_ A = \chi\), iff \(A\) is commutative. In the end, define \(E_ A: \text{End}_ k(A) \to \text{End}_ k(A)\) by \(p(E_ A(f)(a)) = \text{Tr}(r(a) \circ f\circ r(p))\), \(f\in \text{End}_ k(A)\), \(p\in A^*\), \(a \in A\). Then \(E_ A\) is bijective. A method for computing \(\text{Tr}(E_ A(f))\) is given and some properties of \(E_ A\) are derived, which are used to study the endomorphism \(f_ A = E_ A(I)\) [the author and \textit{R. G. Larson}, J. Algebra 117, No. 2, 267-289 (1988; Zbl 0649.16005)].
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16S50
zbMATH DE Number: 554138
zbMATH Keywords: unimodular Hopf algebras
zbMATH Keywords: trace function
zbMATH Keywords: semisimple Hopf algebra
zbMATH Keywords: trace formulas
zbMATH Keywords: finite dimensional Hopf algebra
zbMATH Keywords: right integral
zbMATH Keywords: antipode
zbMATH Keywords: endomorphism
reviewed by: Fang Li
MaRDI profile type: Publication
full work available at URL: jabr.1994.1033
OpenAlex ID: W2061178501
DOI: 10.1006/JABR.1994.1033
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description: scientific article; zbMATH DE number 554138