nLab
associative Yang-Baxter equation

The associative Yang-Baxter equation (AYBE) is an associative analogue of the classical Yang-Baxter equation. If AA is a kk-algebra then, in Leningrad notation, the AYBE for a matrix rAAr\in A\otimes A is the condition

r 13r 12r 12r 23+r 23r 13=0 r_{1 3} r_{1 2} - r_{1 2} r_{2 3} + r_{2 3} r_{1 3} = 0

If r=a ib ir = \sum a_i\otimes b_i, then the operator P:AAP:A\to A given by

P(x)=a ixb i P(x) = \sum a_i x b_i

is a Rota-Baxter operator of weight 00, i.e. (A,P)(A,P) is a Rota-Baxter algebra of weight λ=0\lambda = 0.

The skew-symmetric solutions (r 12=r 21r_{1 2} = - r_{2 1}) of AYBE give rise to

  • an algebra with the trace quadratic Poisson bracket
  • double Poisson structures on a free associative algebra
  • an anti-Frobenius associative subalgebra of a matrix algebra

Literature

  • V.N. Zhelyabin, Jordan bialgebras of symmetric elements and Lie bialgebras, Siberian Mathematical Journal 39 (1998, 261-276, open access pdf in Russian)
  • Marcelo Aguiar, Infinitesimal Hopf algebras, Contemporary Mathematics, 267 (2000) 1-29; Pre-Poisson algebras, Lett. Math. Phys. 54 (2000) 263-277, doi; On the associative analog of Lie bialgebras, Journal of Algebra 244 (2001, 492-532, open access pdf
  • A. Polishchuk, Classical Yang-Baxter equation and the A A_\infty-constraint, Adv. Math. 168 (2002, 56-95) open access pdf
  • Chengming Bai, Li Guo, Xiang Ni, 𝒪\mathcal{O}-operators on associative algebras and associative Yang-Baxter equations, Pacific J. Math. 256 (2012) 257-289, arxiv/0910.3261
  • Travis Schedler, Poisson algebras and Yang-Baxter equations, math.QA/0612493
  • V. Sokolov, Classification of constant solutions for associative Yang-Baxter on gl(3)gl(3), arxiv/1212.6421
  • A. Odesskii, V. Rubtsov, V. Sokolov, Double Poisson brackets on free associative algebras, arxiv/1208.2935
category: algebraphysics

Last revised on September 7, 2017 at 14:20:52. See the history of this page for a list of all contributions to it.