Homologies of Algebraic Structures via Braidings and Quantum Shuffles

By Victoria Lebed. 2012

58 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c23262b122cf5b38a89dce9fca3fc3e9aebfbc30

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In Discrete Mathematics and Combinatorics

1. Leandro Vendramin
2. Victoria Lebed

In Geometry and Topology

1. Victoria Lebed
2. Leandro Vendramin
3. Agata Smoktunowicz

Citations in the corpus

• Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation (2015, methodology): “In this section we recall and slightly extend the (co)homology theory for (X, σ), developed in [26].”