Homologies of Algebraic Structures via Braidings and Quantum Shuffles

2012/04/15 by Victoria Lebed, Lebed, Victoria · 4 citations
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.1204.3312

openalex publication_date 2012/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• A classifying invariant of knots, the knot quandle

Cited by

• Subracks and second homology of the conjugacy classes of finite projective special linear groups of degree two
• Quantum Cocycle Invariants of Knots from Yang-Baxter Cohomology
• In Search of Homology for Quasigroups of Bol-Moufang Type
• From n-Leibniz algebras and linear n-racks to the solutions of the (higher analogue of) Yang-Baxter equation

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