Quasitriangular coideal subalgebras of Uq(g) in terms of generalized Satake diagrams
Vidas Regelskis, Bart Vlaar
Abstract
Let g be a finite-dimensional semisimple complex Lie algebra and an involutive automorphism of g. According to Letzter, Kolb and Balagovi the fixed-point subalgebra k = g has a quantum counterpart B, a coideal subalgebra of the Drinfeld-Jimbo quantum group Uq(g) possessing a universal K-matrix K. The objects , k, B and K can all be described in terms of Satake diagrams. In the present work, we extend this construction to generalized Satake diagrams, combinatorial data first considered by Heck. A generalized Satake diagram naturally defines a semisimple automorphism of g restricting to the standard Cartan subalgebra h as an involution. It also defines a subalgebra k g satisfying k h = h , but not necessarily a fixed-point subalgebra. The subalgebra k can be quantized to a coideal subalgebra of Uq(g) endowed with a universal K-matrix in the sense of Kolb and Balagovi. We conjecture that all such coideal subalgebras of Uq(g) arise from generalized Satake diagrams in this way.