Quasitriangular coideal subalgebras of $U_q(\mathfrak{g})$ in terms of generalized Satake diagrams

TY - UNPB

T1 - Quasitriangular coideal subalgebras of $U_q(\mathfrak{g})$ in terms of generalized Satake diagrams

AU - Regelskis, Vidas

AU - Vlaar, Bart

N1 - 16 pages

PY - 2018/7/6

Y1 - 2018/7/6

N2 - Let $\mathfrak{g}$ be a finite-dimensional semisimple complex Lie algebra and $\theta$ an involutive automorphism of $\mathfrak{g}$. It is well-known from works of Letzter, Kolb and Balagovi\'c that the fixed-point subalgebra $\mathfrak{k} = \mathfrak{g}^\theta$ has a quantum counterpart $B$, a coideal subalgebra of the Drinfeld-Jimbo quantum group $U_q(\mathfrak{g})$ possessing a cylinder-twisted universal K-matrix $\mathcal{K}$. The objects $\theta$, $\mathfrak{k}$, $B$ and $\mathcal{K}$ can all be described in terms of a combinatorial datum, a Satake diagram. In the present work we extend this construction to generalized Satake diagrams, objects first considered by Heck. A generalized Satake diagram defines a semisimple automorphism of $\mathfrak{g}$ restricting to the standard Cartan subalgebra $\mathfrak{h}$ as an involution. We show that it naturally leads to a subalgebra $\mathfrak{k}\subset \mathfrak{g}$, not necessarily a fixed-point subalgebra, but still satisfying $\mathfrak{k} \cap \mathfrak{h} = \mathfrak{h}^\theta$. Such a subalgebra $\mathfrak{k}$ can be quantized to a coideal subalgebra of $U_q(\mathfrak{g})$ endowed with a cylinder-twisted universal K-matrix. We conjecture that all such coideal subalgebras of $U_q(\mathfrak{g})$ arise from generalized Satake diagrams in this way.

AB - Let $\mathfrak{g}$ be a finite-dimensional semisimple complex Lie algebra and $\theta$ an involutive automorphism of $\mathfrak{g}$. It is well-known from works of Letzter, Kolb and Balagovi\'c that the fixed-point subalgebra $\mathfrak{k} = \mathfrak{g}^\theta$ has a quantum counterpart $B$, a coideal subalgebra of the Drinfeld-Jimbo quantum group $U_q(\mathfrak{g})$ possessing a cylinder-twisted universal K-matrix $\mathcal{K}$. The objects $\theta$, $\mathfrak{k}$, $B$ and $\mathcal{K}$ can all be described in terms of a combinatorial datum, a Satake diagram. In the present work we extend this construction to generalized Satake diagrams, objects first considered by Heck. A generalized Satake diagram defines a semisimple automorphism of $\mathfrak{g}$ restricting to the standard Cartan subalgebra $\mathfrak{h}$ as an involution. We show that it naturally leads to a subalgebra $\mathfrak{k}\subset \mathfrak{g}$, not necessarily a fixed-point subalgebra, but still satisfying $\mathfrak{k} \cap \mathfrak{h} = \mathfrak{h}^\theta$. Such a subalgebra $\mathfrak{k}$ can be quantized to a coideal subalgebra of $U_q(\mathfrak{g})$ endowed with a cylinder-twisted universal K-matrix. We conjecture that all such coideal subalgebras of $U_q(\mathfrak{g})$ arise from generalized Satake diagrams in this way.

KW - math.QA

KW - math.RT

M3 - Preprint

BT - Quasitriangular coideal subalgebras of $U_q(\mathfrak{g})$ in terms of generalized Satake diagrams

ER -