Homologies of Algebraic Structures via Braidings and Quantum Shuffles

Details Homologies of Algebraic Structures via Braidings and Quantum Shuffles Homologies of Algebraic Structures via Braidings and Quantum Shuffles

2012-04-15

arxiv.org/abs/1204.3312v1

Mathematics

Category Theory

Scientific paper

In this paper we construct braidings characterizing different algebraic structures: a rack, an associative algebra, a Leibniz algebra and their representations. Some of these braidings seem original. This produces, via braided space (co)homology coming from quantum (co)shuffle (co)multiplication, a family of (co)chain complexes for each of these structures. One recovers Koszul, rack, bar, Hochschild and Leibniz complexes in these families. All the constructions are categorified, resulting in particular in their super- and co-versions. Loday's hyper-boundaries are efficiently treated using the "shuffle" tools. A notion of modules over braided spaces, encompassing algebra, Lie and rack modules, is introduced.

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