Research output: Working paper

**Quasitriangular coideal subalgebras of $U_q(\mathfrak{g})$ in terms of generalized Satake diagrams.** / Regelskis, Vidas; Vlaar, Bart.

Research output: Working paper

Regelskis, V & Vlaar, B 2018 'Quasitriangular coideal subalgebras of $U_q(\mathfrak{g})$ in terms of generalized Satake diagrams'. <https://arxiv.org/abs/1807.02388>

Regelskis, V., & Vlaar, B. (2018). *Quasitriangular coideal subalgebras of $U_q(\mathfrak{g})$ in terms of generalized Satake diagrams*. https://arxiv.org/abs/1807.02388

Regelskis V, Vlaar B. Quasitriangular coideal subalgebras of $U_q(\mathfrak{g})$ in terms of generalized Satake diagrams. 2018 Jul 6.

@techreport{266886cf20c2424997f02215117608bf,

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 $\mathfrak{g}$. 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.",

keywords = "math.QA, math.RT",

author = "Vidas Regelskis and Bart Vlaar",

note = "16 pages",

year = "2018",

month = jul,

day = "6",

language = "Undefined/Unknown",

type = "WorkingPaper",

}

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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 - Working paper

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

ER -