# University of Hertfordshire

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

Research output: Contribution to journalArticle

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In: ArXiv e-prints, 06.07.2018.

Research output: Contribution to journalArticle

### Bibtex

@article{e8d46a777ed342d391c61fbf827d00db,
title = "Quasitriangular coideal subalgebras of $U_q(\mathfrak{g})$ in terms of generalized Satake diagrams",
abstract = "Let $\mathfrak{g}$ be a finite-dimensional semisimple complex Lie algebra and $\theta$ an involutive automorphism of $\mathfrak{g}$. According to Letzter, Kolb and Balagovi\'c 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 universal K-matrix $\mathcal{K}$. The objects $\theta$, $\mathfrak{k}$, $B$ and $\mathcal{K}$ can all be described in terms of 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 naturally defines a semisimple automorphism $\theta$ of $\mathfrak{g}$ restricting to the standard Cartan subalgebra $\mathfrak{h}$ as an involution. It also defines a subalgebra $\mathfrak{k}\subset \mathfrak{g}$ satisfying $\mathfrak{k} \cap \mathfrak{h} = \mathfrak{h}^\theta$, but not necessarily a fixed-point subalgebra. The subalgebra $\mathfrak{k}$ can be quantized to a coideal subalgebra of $U_q(\mathfrak{g})$ endowed with a universal K-matrix in the sense of Kolb and Balagovi\'c. We conjecture that all such coideal subalgebras of $U_q(\mathfrak{g})$ arise from generalized Satake diagrams in this way.",
keywords = "math.QA, math.RT",
author = "Vidas Regelskis and Bart Vlaar",
note = "18 pages; v2: minor corrections, examples added",
year = "2018",
month = "7",
day = "6",
language = "English",
journal = "ArXiv e-prints",

}

### RIS

TY - JOUR

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

AU - Regelskis, Vidas

AU - Vlaar, Bart

N1 - 18 pages; v2: minor corrections, examples added

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 $\mathfrak{g}$. According to Letzter, Kolb and Balagovi\'c 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 universal K-matrix $\mathcal{K}$. The objects $\theta$, $\mathfrak{k}$, $B$ and $\mathcal{K}$ can all be described in terms of 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 naturally defines a semisimple automorphism $\theta$ of $\mathfrak{g}$ restricting to the standard Cartan subalgebra $\mathfrak{h}$ as an involution. It also defines a subalgebra $\mathfrak{k}\subset \mathfrak{g}$ satisfying $\mathfrak{k} \cap \mathfrak{h} = \mathfrak{h}^\theta$, but not necessarily a fixed-point subalgebra. The subalgebra $\mathfrak{k}$ can be quantized to a coideal subalgebra of $U_q(\mathfrak{g})$ endowed with a universal K-matrix in the sense of Kolb and Balagovi\'c. 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 $\mathfrak{g}$. According to Letzter, Kolb and Balagovi\'c 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 universal K-matrix $\mathcal{K}$. The objects $\theta$, $\mathfrak{k}$, $B$ and $\mathcal{K}$ can all be described in terms of 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 naturally defines a semisimple automorphism $\theta$ of $\mathfrak{g}$ restricting to the standard Cartan subalgebra $\mathfrak{h}$ as an involution. It also defines a subalgebra $\mathfrak{k}\subset \mathfrak{g}$ satisfying $\mathfrak{k} \cap \mathfrak{h} = \mathfrak{h}^\theta$, but not necessarily a fixed-point subalgebra. The subalgebra $\mathfrak{k}$ can be quantized to a coideal subalgebra of $U_q(\mathfrak{g})$ endowed with a universal K-matrix in the sense of Kolb and Balagovi\'c. 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 - Article

JO - ArXiv e-prints

JF - ArXiv e-prints

ER -