Tetrahedron and 3D reflection equation from PBW bases of the nilpotent subalgebra of quantum superalgebras. (English) Zbl 1490.16082
The paper under review studies transition matrices of PBW bases of the nilpotent sub-algebra of quantum superalgebras and examines relationships with three-dimensional (3D) integrability. The main motivation of the paper is to understand the cause behind the similar behaviour of 3D R and 3D L despite their completely different origins. Let \(F\) and \(V\) be the bosonic and Fermionic Fock spaces, respectively. The matrices \(\mathcal{R} \in \operatorname{End}(F \otimes F \otimes F)\) and \(\mathcal{L} \in \operatorname{End}(V \otimes V \otimes F)\) solutions of tetrahedron equations \[\mathcal{R}_{123}\mathcal{R}_{145}\mathcal{R}_{246}\mathcal{R}_{356} = \mathcal{R}_{356}\mathcal{R}_{246}\mathcal{R}_{145}\mathcal{R}_{123},\] \[\mathcal{L}_{123}\mathcal{L}_{145}\mathcal{L}_{246}\mathcal{R}_{356} = \mathcal{R}_{356}\mathcal{L}_{246}\mathcal{L}_{145}\mathcal{L}_{123},\] where the indices represent the tensor components on which each matrix acts non-trivially, are called 3D R and 3D L, respectively.
The paper starts with a review of all necessary basic facts about finite-dimensional Lie superalgebras \(\mathfrak{sl}(m|n)\) and \(\mathfrak{osp}(2m + 1|2n)\) (\(m\), \(n\) are non-negative integers and \(m + n \geq 2\)). Through the paper, \(\mathfrak{sl}(m|n)\) is simply called type A and \(\mathfrak{osp}(2m + 1|2n)\) type B, respectively. Then, quantum superalgebras and their PBW theorem are introduced according to [H. Yamane, Publ. Res. Inst. Math. Sci. 30, No. 1, 15–87 (1994; Zbl 0821.17005)]. Section 3 summarises the operators which give solutions to the tetrahedron and reflection equations. Sections 4 and 5 are the core parts of the paper. Section 4 (section 5, respectively) focus on transition matrices of PBW bases of the nilpotent subalgebra of quantum superalgebras of type A (type B, respectively) in the case of rank \(2\) and \(3\), and the author obtains several solutions to the tetrahedron equation (3D reflection equation, respectively). The paper concludes with an examination of the crystal limit of transition matrices.
The paper starts with a review of all necessary basic facts about finite-dimensional Lie superalgebras \(\mathfrak{sl}(m|n)\) and \(\mathfrak{osp}(2m + 1|2n)\) (\(m\), \(n\) are non-negative integers and \(m + n \geq 2\)). Through the paper, \(\mathfrak{sl}(m|n)\) is simply called type A and \(\mathfrak{osp}(2m + 1|2n)\) type B, respectively. Then, quantum superalgebras and their PBW theorem are introduced according to [H. Yamane, Publ. Res. Inst. Math. Sci. 30, No. 1, 15–87 (1994; Zbl 0821.17005)]. Section 3 summarises the operators which give solutions to the tetrahedron and reflection equations. Sections 4 and 5 are the core parts of the paper. Section 4 (section 5, respectively) focus on transition matrices of PBW bases of the nilpotent subalgebra of quantum superalgebras of type A (type B, respectively) in the case of rank \(2\) and \(3\), and the author obtains several solutions to the tetrahedron equation (3D reflection equation, respectively). The paper concludes with an examination of the crystal limit of transition matrices.
Reviewer: Ilaria Colazzo (Exeter)
MSC:
| 16T25 | Yang-Baxter equations |
| 17B38 | Yang-Baxter equations and Rota-Baxter operators |
| 81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |
Citations:
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