Quantized GIM Algebras and their Images in Quantized Kac-Moody Algebras
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Quantized GIM Algebras and their Images in Quantized Kac-Moody Algebras Yun Gao1 · Naihong Hu2 · Li-meng Xia3 Received: 25 November 2019 / Accepted: 5 March 2020 / © Springer Nature B.V. 2020
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 Sq 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. Keywords Generalized intersection matrices · Quantum algebras · Kac-Moody algebras · Involutory subalgebras · Lusztig symmetries Mathematics Subject Classification (2010) 17B37 · 17B35 · 17B65
1 Introduction Slodowy’s GIM Lie algebra theory is a generalization of Kac-Moody Lie algebras. As the analogy of Kac-Moody Lie algebras, a GIM Lie algebra can be defined by a generalized intersection matrix (GIM for short). For more information on GIMs and GIM Lie algebras, one is referred to [5, 6, 10, 15, 16] and [11], etc. In 1990s, the work of Berman and Moody, Benkart and Zelmanov, and Neher on the classification of root-graded Lie algebras [2, 5, 13] revealed that some families of Intersection Matrix (IM) algebras were the universal covering algebras of some well-known Lie algebras. Many other authors also paid attention to these new algebras. The first author studied the compact forms of IM Lie algebras arising from conjugation over the complex field [8].
Presented by: Peter Littelmann Li-meng Xia
[email protected] 1
Department of Mathematics and Statistics, York University, Toronto, ON, Canada
2
School of Mathematical Science, SKLPMMP, East China Normal University, Shanghai, China
3
Institute of Applied System Analysis, Jiangsu University, Zhenjiang, China
Y. Gao et al.
Peng found relations between IM Lie algebras and the representations of tilted algebras via Ringel-Hall algebras [14]. Berman, Jurisich, and Tan showed that the presentation of GIM Lie algebras could be put into a broader framework that incorporated the Borcherds algebras [4]. In [19], some new Lie algebras arising from some GIMs are classified. In [9], the classification for all finite dimensional irreducible modules is given. Many other references are referred (see [1], [7] and etc.). In quantized setting, Tan studied Uq (ggim ) of GIM Lie algebras with structural matrices of 2-affinizations of ADE-types: in [17], he introduced the Lusztig symmetries of Uq (ggim ); in [18], he defined the Drinfel’d-Jimbo coproduct of Uq (ggim ) over a quotient tensor space. In this paper, we focus on a class of simply-laced GIMs. Let M = M(mi,j )n×n be an indecomposable GIM and mi,i = 2, mi,j = 0, ±1 for all i = j . In particular, M is symmetric under this assumption.
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