Quasitriangular coideal subalgebras of Uq(g) in terms of generalized Satake diagrams

Abstract

Let (Formula presented.) be a finite-dimensional semisimple complex Lie algebra and (Formula presented.) an involutive automorphism of (Formula presented.). According to Letzter, Kolb and Balagović the fixed-point subalgebra (Formula presented.) has a quantum counterpart (Formula presented.), a coideal subalgebra of the Drinfeld–Jimbo quantum group (Formula presented.) possessing a universal (Formula presented.) -matrix (Formula presented.). The objects (Formula presented.), (Formula presented.), (Formula presented.) and (Formula presented.) 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 (Formula presented.) of (Formula presented.) restricting to the standard Cartan subalgebra (Formula presented.) as an involution. It also defines a subalgebra (Formula presented.) satisfying (Formula presented.), but not necessarily a fixed-point subalgebra. The subalgebra (Formula presented.) can be quantized to a coideal subalgebra of (Formula presented.) endowed with a universal (Formula presented.) -matrix in the sense of Kolb and Balagović. We conjecture that all such coideal subalgebras of (Formula presented.) arise from generalized Satake diagrams in this way.