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Braided tensor product and Lie algebra in a braided category. (Produit tensoriel tressé et algèbre de Lie dans une catégorie tressée.) (French) Zbl 0958.18004
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 A. Joyal and 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.
MSC:
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
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