R-matrices and transitivity

In my talk I connect two seemingly unrelated theories: solutions of classical (CYBE) and quantum (QYBE) Yang-Baster equations with applications to Lie bialgebras and quantum groups, on the one hand, and the combinatorial theory of C-transitivity recently developed in joint work with Ron Adin, Jacob Greenstein, Jianrong Li, Avichai Marmor, Yuval Roichman, on the other hand.

Namely, if a space V on whose tensor square we want to solve CYBE, is direct sum of n subspaces V_1,...,V_n and the solution of CYBE is the sum of r^{ij} acting on V_i \otimes V_j, then the r^{ij} must satisfy local CYBE which becomes a transitive CYBE when these n^2 of r^{ij} are replaced by an appropriate family r^{(c)}, c \in C, and the latter set does not depend on n. If C=\{-1,1\}, then the "default" solution of the transitive CYBE is when r^{(-1)} is the negative transpose of r^{(1)}, and the latter one is a solution of the ordinary CYBE. The link between QYBE and transitivity is nearly identical: replace r^{ij} with R^{ij}, the sum with the product, and the local QYBE with the transitive QYBE for a family R^{(c)}, c \in C.

In the second part of my talk, I will focus on a general theory of C-transitivity, which works for directed and undirected graphs (e.g., complete ones for CYBE/QYBE), matroids etc., and explain the Polynomiality Phenomenon for the number of C-transitive colorings of graphs, matroids and more general combinatorial structures. I will also give a commutative algebra interpretation of these colorings and introduce their q-analogs via the Hilbert series of the respective algebras.

If time permits, I will explain how the ordinary chromatic polynomials of graphs fit into the C-transitivity theory and how they can be "Hilbert-quantized."