Braided tensor product and Lie algebra in a braided category. (Q1567461)

The authors study to what extent the notion of a Lie algebra makes sense in an arbitrary additive braided monoidal category \(\mathcal C\). Braided monoidal categories have been introduced by \textit{A. Joyal} and \textit{R. Street} [Adv. Math., Vol. 102, No. 1, 20-78 (1993; Zbl 0817.18007)]. The authors give several equivalent forms of (a braid analog of) Jacobi's identity. They construct for each bialgebra \(B\) of \(\mathcal C\) a primitive part object which is shown to be a Lie algebra if and only if the braiding acts as a symmetry on \(B\). If \(\mathcal C\) is moreover abelian, they construct for each algebra \(A\) of \(\mathcal C\) an object of derivations of \(A\), which is shown to be a Lie algebra.