TWO NON-NILPOTENT LINEAR TRANSFORMATIONS THAT SATISFY THE CUBIC q-SERRE RELATIONS

By Tatsuro Ito, Paul M. Terwilliger. 2005

74 citations recorded by the source corpus.

Let 𝕂 denote an algebraically closed field with characteristic 0, and let q denote a nonzero scalar in 𝕂 that is not a root of unity. Let 𝔸q denote the unital associative 𝕂-algebra defined by generators x,y and relations \begin{eqnarray*} x^{3}y - [3]_{q} x^{2}yx + [3]_{q} xyx^{2} - yx^{3} &=& 0,\\ y^{3}x - [3]_{q} y^{2}xy + [3]_{q} yxy^{2} - xy^{3} &=& 0, \end{eqnarray*} where [3]q = (q3 - q-3)/(q - q-1). We classify up to isomorphism the finite-dimensional irreducible 𝔸q-modules on which neither of x,y is nilpotent. We discuss how these modules are related to tridiagonal pairs.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/12fcdb7d833d06fef8c237a393b0d46bb80607c1

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In Discrete Mathematics and Combinatorics

1. Tatsuro Ito
2. Paul M. Terwilliger
3. Raimundas Vidunas
4. B. Hartwig

In Geometry and Topology

1. Tatsuro Ito
2. Paul M. Terwilliger
3. Alexei S. Zhedanov
4. Luc Vinet
5. Plamen Iliev
6. 伊藤 達郎

In Mathematical Physics

1. Paul M. Terwilliger
2. Tatsuro Ito
3. Raimundas Vidunas

Citations in the corpus

• Finite-Dimensional Irreducible Modules for the Three-Point 𝔰𝔩2 Loop Algebra (2007, background): “ndegenerate symmetric bilinear form h,iσ on V such that hξ.u,viσ = −hu,σ(ξ).viσ ξ∈ ⊠, u,v∈ V. We remark on our motivation to investigate ⊠. This has to do with the tridiagonal pairs [19], [20], [21], [22] and the closely related Leonard pairs [27], [28], [29], [35], [36], [37], [39], [40]. A Leonard”
• Tridiagonal pairs of q-Racah type (2008, background): “It is an open problem to classify the sharp TD pairs up to isomorphism, but progress is being made [24,27,28,44,46,60].”
• The augmented tridiagonal algebra (2009, background): “This fact motivates our ongoing investigation of the tridiagonal pairs [3], [4], [5], [6], [7], [8].”
• Linear transformations that are tridiagonal with respect to both eigenbases of a Leonard pair (2006, unknown): “See [1], [2], [4], [5], [6], [7], [8], [10], [11], [20], [29] for related topics.”