Differential-graded Hopf algebras and quantum group differential calculi
The paper studies differential calculi for an algebra \(A\) or a Hopf algebra \(H\) when working with the category \(\underline {DGV}\) of complexes of \(k\)-vector spaces (differential graded vector spaces, the gradation over \(\mathbb{N}\)), i.e., with objects \(M^\bullet= (M^\bullet,d^\bullet)\) formed by sequences of \(k\)-linear maps \(d^i: M^i\to M^{i+1}\) satisfying \(d^{i+1}d^i=0\). A morphism \(f^\bullet: M^\bullet\to N^\bullet\) is called a differentiable map as it obeys \(f^{i+1}d^i= d^if^i\). The subcategory with objects terminating at the \(n\)th term \((M^i=0\) for \(i>n)\) is denoted by \(\underline {DGV}^n\). Following the Introduction and Preliminaries, Section 3 is devoted to the notion of a differential calculus \(A^\bullet\) which is a differential graded algebra generated by \(A^0\) and \(d^0(A^0)\subset A^1\). In Section 4, a universal construction is presented for higher degree differential graded algebras from those of lower degree. It is obtained as a left adjoint to the cutoff functor \(\underline {DGV}^m\to\underline {DGV}^n\), with \(m>n\). In an analogous manner, differential graded coalgebras and bialgebras are studied in Section 5. A basic principle is that the comultiplication should be differentiable. This is in a nice agreement with some known results since the first order differential calculi over a Hopf algebra \(H\), for which the comultiplication is once differentiable, are precisely the bicovariant differential calculi in the sense of Woronowicz. In Section 6 one shows that the condition on differentiability of the antipode does not imply any additional restriction. Section 7 is a brief review of Woronowicz' structure theory of first order bicovariant differential calculi. The main results are contained in Section 8. The exterior algebra \(\wedge^\bullet(H^1)\) is constructed in an alternative way (originally due to Woronowicz) when starting from a bicovariant first order differential calculus \(H^{\leq 1}\). It is proven that the exterior algebra is a differential graded Hopf algebra.
scientific article; zbMATH DE number 869812