Quantized GIM Algebras and their Images in Quantized Kac-Moody Algebras

Yun Gao; Naihong Hu; Li meng Xia

doi:10.1007/s10468-020-09960-2

Quantized GIM Algebras and their Images in Quantized Kac-Moody Algebras

Yun Gao
, Naihong Hu
, Li meng Xia^*

^*Corresponding author for this work

School of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

For any simply-laced GIM Lie algebra L, we present the definition of quantum universal enveloping algebra U_q(L) , and prove that there is a quantum universal enveloping algebra U_q(A) of an associated Kac-Moody algebra A, together with an involution (ℚ-linear) σ, such that U_q(L) is isomorphic to the ℚ(q) -extension S~ _q of the σ-involutory subalgebra S_q of U_q(A). This result gives a quantum version of Berman’s work (Berman Comm. Algebra 17, 3165–3185, 1989) in the simply-laced cases. Finally, we describe an automorphism group of U_q(L) consisting of Lusztig symmetries as a braid group.

Original language	English
Pages (from-to)	565-584
Number of pages	20
Journal	Algebras and Representation Theory
Volume	24
Issue number	3
DOIs	https://doi.org/10.1007/s10468-020-09960-2
State	Published - Jun 2021

Keywords

Generalized intersection matrices
Involutory subalgebras
Kac-Moody algebras
Lusztig symmetries
Quantum algebras

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10.1007/s10468-020-09960-2

Cite this

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title = "Quantized GIM Algebras and their Images in Quantized Kac-Moody Algebras",

abstract = "For any simply-laced GIM Lie algebra L, we present the definition of quantum universal enveloping algebra Uq(L) , and prove that there is a quantum universal enveloping algebra Uq(A) of an associated Kac-Moody algebra A, together with an involution (ℚ-linear) σ, such that Uq(L) is isomorphic to the ℚ(q) -extension S\textasciitilde{} q of the σ-involutory subalgebra Sq of Uq(A). This result gives a quantum version of Berman{\textquoteright}s work (Berman Comm. Algebra 17, 3165–3185, 1989) in the simply-laced cases. Finally, we describe an automorphism group of Uq(L) consisting of Lusztig symmetries as a braid group.",

keywords = "Generalized intersection matrices, Involutory subalgebras, Kac-Moody algebras, Lusztig symmetries, Quantum algebras",

author = "Yun Gao and Naihong Hu and Xia, \{Li meng\}",

note = "Publisher Copyright: {\textcopyright} 2020, Springer Nature B.V.",

year = "2021",

month = jun,

doi = "10.1007/s10468-020-09960-2",

language = "英语",

volume = "24",

pages = "565--584",

journal = "Algebras and Representation Theory",

issn = "1386-923X",

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TY - JOUR

T1 - Quantized GIM Algebras and their Images in Quantized Kac-Moody Algebras

AU - Gao, Yun

AU - Hu, Naihong

AU - Xia, Li meng

PY - 2021/6

Y1 - 2021/6

N2 - For any simply-laced GIM Lie algebra L, we present the definition of quantum universal enveloping algebra Uq(L) , and prove that there is a quantum universal enveloping algebra Uq(A) of an associated Kac-Moody algebra A, together with an involution (ℚ-linear) σ, such that Uq(L) is isomorphic to the ℚ(q) -extension S~ q of the σ-involutory subalgebra Sq of Uq(A). This result gives a quantum version of Berman’s work (Berman Comm. Algebra 17, 3165–3185, 1989) in the simply-laced cases. Finally, we describe an automorphism group of Uq(L) consisting of Lusztig symmetries as a braid group.

AB - For any simply-laced GIM Lie algebra L, we present the definition of quantum universal enveloping algebra Uq(L) , and prove that there is a quantum universal enveloping algebra Uq(A) of an associated Kac-Moody algebra A, together with an involution (ℚ-linear) σ, such that Uq(L) is isomorphic to the ℚ(q) -extension S~ q of the σ-involutory subalgebra Sq of Uq(A). This result gives a quantum version of Berman’s work (Berman Comm. Algebra 17, 3165–3185, 1989) in the simply-laced cases. Finally, we describe an automorphism group of Uq(L) consisting of Lusztig symmetries as a braid group.

KW - Generalized intersection matrices

KW - Involutory subalgebras

KW - Kac-Moody algebras

KW - Lusztig symmetries

KW - Quantum algebras

UR - https://www.scopus.com/pages/publications/85091001569

U2 - 10.1007/s10468-020-09960-2

DO - 10.1007/s10468-020-09960-2

M3 - 文章

AN - SCOPUS:85091001569

SN - 1386-923X

VL - 24

SP - 565

EP - 584

JO - Algebras and Representation Theory

JF - Algebras and Representation Theory

IS - 3

ER -

Quantized GIM Algebras and their Images in Quantized Kac-Moody Algebras

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Keywords

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