The trace function and Hopf algebras
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)].
scientific article; zbMATH DE number 554138
scientific article