Victoria Lebed

Homology of left non-degenerate set-theoretic solutions to the Yang-Baxter equation

Abstract This paper deals with left non-degenerate set-theoretic solutions to the Yang–Baxter equation (= LND solutions), a vast class of algebraic structures encompassing groups, racks, and cycle sets. To each such solution there is associated a shelf (i.e., a self-distributive structure) which captures its major properties. We consider two (co)homology theories for LND solutions, one of which was previously known, in a reduced form, for biracks only. An explicit isomorphism between these theories is described. For groups and racks we recover their classical (co)homology, whereas for cycle sets we get new constructions. For a certain type of LND solutions, including quandles and non-degenerate cycle sets, the (co)homologies split into the degenerate and the normalized parts. We express 2-cocycles of our theories in terms of group cohomology, and, in the case of cycle sets, establish connexions with extensions. This leads to a construction of cycle sets with interesting properties.

Cohomology and extensions of braces

Braces and linear cycle sets are algebraic structures playing a major role in the classification of involutive set-theoretic solutions to the Yang-Baxter equation. This paper introduces two versions of their (co)homology theories. These theories mix the Harrison (co)homology for the abelian group structure and the (co)homology theory for general cycle sets, developed earlier by the authors. Different classes of brace extensions are completely classified in terms of second cohomology groups.

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.

R-Matrices, Yetter-Drinfel'd Modules and Yang-Baxter Equation

In the first part we recall two famous sources of solutions to the Yang-Baxter equation—R-matrices and Yetter-Drinfel0d (=YD) modules—and an interpretation of the former as a particular case of the latter. We show that this result holds true in the more general case of weak R-matrices, introduced here. In the second part we continue exploring the “braided” aspects of YD module structure, exhibiting a braided system encoding all the axioms from the definition of YD modules. The functoriality and several generalizations of this construction are studied using the original machinery of YD systems. As consequences, we get a conceptual interpretation of the tensor product structures for YD modules, and a generalization of the deformation cohomology of YD modules. This homology theory is thus included into the unifying framework of braided homologies, which contains among others Hochschild, Chevalley-Eilenberg, Gerstenhaber-Schack and quandle homologies.

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 Kunneth 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–Garcia-Galofre and Dilian Yang. We also obtain new structural results on the (co)homology of general YBE solutions.

REPRESENTATIONS OF CROSSED MODULES AND OTHER GENERALIZED YETTER-DRINFEL'D MODULES

The Yang-Baxter equation plays a fundamental role in various areas of mathematics. Its solutions, called braidings, are built, among others, from Yetter-Drinfel’d modules over a Hopf algebra, from self-distributive structures, and from crossed modules of groups. In the present paper these three sources of solutions are unified inside the framework of Yetter-Drinfel’d modules over a braided system. A systematic construction of braiding structures on such modules is provided. Some general categorical methods of obtaining such generalized Yetter-Drinfel’d (=GYD) modules are described. Among the braidings recovered using these constructions are the Woronowicz and the Hennings braidings on a Hopf algebra. We also introduce the notions of crossed modules of shelves / Leibniz algebras, and interpret them as GYD modules. This yields new sources of braidings. We discuss whether these braidings stem from a braided monoidal category, and discover several non-strict pre-tensor categories with interesting associators.

The shadow nature of positive and twisted quandle invariants of knots

Quandle cocycle invariants form a powerful and well-developed tool in knot theory. This paper treats their variations — namely, positive and twisted quandle cocycle invariants, and shadow invariant...