33 research outputs found
Homologies of Algebraic Structures via Braidings and Quantum Shuffles
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
Applications of self-distributivity to Yang-Baxter operators and their cohomology
Self-distributive (SD) structures form an important class of solutions to the
Yang--Baxter equation, which underlie spectacular knot-theoretic applications
of self-distributivity. It is less known that one go the other way round, and
construct an SD structure out of any left non-degenerate (LND) set-theoretic
YBE solution. This structure captures important properties of the solution:
invertibility, involutivity, biquandle-ness, the associated braid group
actions. Surprisingly, the tools used to study these associated SD structures
also apply to the cohomology of LND solutions, which generalizes SD cohomology.
Namely, they yield an explicit isomorphism between two cohomology theories for
these solutions, which until recently were studied independently. The whole
story leaves numerous open questions. One of them is the relation between the
cohomologies of a YBE solution and its associated SD structure. These and
related questions are covered in the present survey
Cohomology of idempotent braidings, with applications to factorizable monoids
We develop new methods for computing the Hochschild (co)homology of monoids
which can be presented as the structure monoids of idempotent set-theoretic
solutions to the Yang--Baxter equation. These include free and symmetric
monoids; factorizable monoids, for which we find a generalization of the
K{\"u}nneth formula for direct products; and plactic monoids. Our key result is
an identification of the (co)homologies in question with those of the
underlying YBE solutions, via the explicit quantum symmetrizer map. This
partially answers questions of Farinati--Garc{\'i}a-Galofre and Dilian Yang. We
also obtain new structural results on the (co)homology of general YBE
solutions
Cohomology of finite monogenic self-distributive structures
A shelf is a set with a binary operation~\op satisfying a \op (b \op c) =
(a \op b) \op (a \op c). Racks are shelves with invertible translations b
\mapsto a \op b; many of their aspects, including cohomological, are better
understood than those of general shelves. Finite monogenic shelves (FMS), of
which Laver tables and cyclic racks are the most famous examples, form a
remarkably rich family of structures and play an important role in set theory.
We compute the cohomology of FMS with arbitrary coefficients. On the way we
develop general tools for studying the cohomology of shelves. Moreover, inside
any finite shelf we identify a sub-rack which inherits its major
characteristics, including the cohomology. For FMS, these sub-racks are all
cyclic
Two- and three-cocycles for Laver tables
We determine all 2- and 3-cocycles for Laver tables, an infinite sequence of
finite structures obeying the left-selfdistributivity law; in particular, we
describe simple explicit bases. This provides a number of new positive braid
invariants and paves the way for further potential topological applications.
Incidentally, we establish and study a partial ordering on Laver tables given
by the right-divisibility relation