2010/12/23 by Leonid Chekhov, Chekhov, Leonid, Marta Mazzocco +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Bilinear form #Braid group #Group (periodic table) #Homotopy and Cohomology in Algebraic Topology #Lie algebra #Lie algebroid #Mathematics #Physics #Poisson algebra #Poisson bracket #Poisson manifold #Pure mathematics #Triangular matrix #Universal enveloping algebra #Yangian #math-ph #math.MP #math.SG #msc:16T30
paper · pdf · doi:10.48550/arxiv.1012.5251
22 pages, 1 figure, 2nd version substantially elaborated: added: introduction, references, algebroid integrability condition, quantum braid-group action; 3rd version: added Lemma 4.3 on skew-symmetricity of bilinear form
openalex publication_date 2010/12/23 · arxiv created 2011/11/18 · arxiv updated 2011/11/21 · openalex created_date 2022/10/04 · openalex updated_date 2026/08/05
In this paper we study a quadratic Poisson algebra structure on the space of\nbilinear forms on CN with the property that for any n,m\∈ N such that\nn m =N, the restriction of the Poisson algebra to the space of bilinear forms\nwith block-upper-triangular matrix composed from blocks of size m\× m is\nPoisson. We classify all central elements and characterise the Lie algebroid\nstructure compatible with the Poisson algebra. We integrate this algebroid\nobtaining the corresponding groupoid of morphisms of block-upper-triangular\nbilinear forms. The groupoid elements automatically preserve the Poisson\nalgebra. We then obtain the braid group action on the Poisson algebra as\nelementary generators within the groupoid. We discuss the affinisation and\nquantisation of this Poisson algebra, showing that in the case m=1 the\nquantum affine algebra is the twisted q-Yangian for on and for m=2 is\nthe twisted q-Yangian for sp2n. We describe the quantum braid group\naction in these two examples and conjecture the form of this action for any\nm>2.\n