Projective class ring of a restricted quantum group $ \overline{U}_{q}(\mathfrak{sl}^{*}_2) $

Pengcheng Ji; Jialei Chen; Fengxia Gao; Pengcheng Ji; Jialei Chen; Fengxia Gao

doi:10.3934/math.20231016

AIMS Mathematics

2023, Volume 8, Issue 9: 19933-19949. doi: 10.3934/math.20231016

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Research article

Projective class ring of a restricted quantum group $ \overline{U}_{q}(\mathfrak{sl}^{*}_2) $

1.
Faculty of Science, Beijing University of Technology, Beijing 100124, China
2.
School of Science, Henan Institute of Technology, Henan 453003, China

Received: 25 March 2023 Revised: 29 May 2023 Accepted: 01 June 2023 Published: 15 June 2023
MSC : 17B37, 16G20, 16D70, 16T05

In this paper, we compute the projective class ring of the new type restricted quantum group $ \overline{U}_{q}(\mathfrak{sl}^{*}_2) $. First, we describe the principal indecomposable projective $ \overline{U}_{q}(\mathfrak{sl}^{*}_2) $-modules and study their radicals, composition series, Cartan matrix of $ \overline{U}_{q}(\mathfrak{sl}^{*}_2) $ and so on. Then, we deconstruct the tensor products between two simple modules, two indecomposable projective modules and a simple module and an indecomposable projective module, into direct sum of some indecomposable representations. At last, we characterize the projective class ring by generators and relations explicitly.
- new type restricted quantum group,
- projective class ring
Citation: Pengcheng Ji, Jialei Chen, Fengxia Gao. Projective class ring of a restricted quantum group $ \overline{U}_{q}(\mathfrak{sl}^{*}_2) $[J]. AIMS Mathematics, 2023, 8(9): 19933-19949. doi: 10.3934/math.20231016

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Abstract

In this paper, we compute the projective class ring of the new type restricted quantum group $ \overline{U}_{q}(\mathfrak{sl}^{*}_2) $. First, we describe the principal indecomposable projective $ \overline{U}_{q}(\mathfrak{sl}^{*}_2) $-modules and study their radicals, composition series, Cartan matrix of $ \overline{U}_{q}(\mathfrak{sl}^{*}_2) $ and so on. Then, we deconstruct the tensor products between two simple modules, two indecomposable projective modules and a simple module and an indecomposable projective module, into direct sum of some indecomposable representations. At last, we characterize the projective class ring by generators and relations explicitly.

References

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