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Homologies of Algebraic Structures via Braidings and Quantum Shuffles
Abstract
In this paper we construct ''structural'' pre-braidings characterizing different algebraic structures: a rack, an associative algebra, a Leibniz algebra and their representations. Some of these pre-braidings seem original. On the other hand, we propose a general homology theory for pre-braided vector spaces and braided modules, based on the quantum co-shuffle comultiplication. Applied to the structural pre-braidings above, it gives a generalization and a unification of many known homology theories. All the constructions are categorified, resulting in particular in their super- and co-versions. Loday's hyper-boundaries, as well as certain homology operations are efficiently treated using the ''shuffle'' tools- info:eu-repo/semantics/preprint
- Preprints, Working Papers, ...
- pre-braiding
- braided (co)algebra
- braided homology
- character
- braided module
- quantum shuffle algebra
- Koszul complex
- rack homology
- Hochschild homology
- Leibniz algebra
- pre-braided object
- MSC 2010: 18D10, 20F36, 18G60, 55N35, 16E40, 17A32, 17D99, 18G30, 05E99
- [MATH.MATH-CT]Mathematics [math]/Category Theory [math.CT]
- [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA]
- [MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM]
- [MATH.MATH-KT]Mathematics [math]/K-Theory and Homology [math.KT]