The crystal commutor and Drinfeld’s unitarized R-matrix

The crystal commutor and Drinfeld’s unitarized R-matrix

Drinfeld defined a unitarized R-matrix for any quantum group Uq(g). This gives a commutor for the category of Uq(g) representations, making it into a coboundary category. Henriques and Kamnitzer defined another commutor which also gives Uq(g) representations the structure of a coboundary category. We show that a particular case of Henriques and Kamnitzer’s construction agrees with Drinfeld’s co...

On the Combinatorics of Crystal Graphs, Ii. the Crystal Commutor

We present an explicit combinatorial realization of the commutor in the category of crystals which was first studied by Henriques and Kamnitzer. Our realization is based on certain local moves defined by van Leeuwen.

A Definition of the Crystal Commutor Using Kashiwara’s Involution

Henriques and Kamnitzer defined and studied a commutor for the category of crystals of a finite dimensional complex reductive Lie algebra. We show that the action of this commutor on highest weight elements can be expressed very simply using Kashiwara’s involution on the Verma crystal.

The partitioned R - matrix

The (R,S)-symmetric and (R,S)-skew symmetric solutions of the pair of matrix equations A1XB1 = C1 and A2XB2 = C2

Let $Rin textbf{C}^{mtimes m}$ and $Sin textbf{C}^{ntimes n}$ be nontrivial involution matrices; i.e., $R=R^{-1}neq pm~I$ and $S=S^{-1}neq pm~I$. An $mtimes n$ complex matrix $A$ is said to be an $(R, S)$-symmetric ($(R, S)$-skew symmetric) matrix if $RAS =A$ ($ RAS =-A$). The $(R, S)$-symmetric and $(R, S)$-skew symmetric matrices have a number of special properties and widely used in eng...