nLab associative Yang-Baxter equation

Idea

The associative Yang-Baxter equation (AYBE) is an associative analogue of the classical Yang-Baxter equation.

Definition

If $A$ is a $k$-algebra then, in Leningrad notation, the AYBE for a matrix $r\in A\otimes A$ is the condition

$$r_{1 3} r_{1 2} - r_{1 2} r_{2 3} + r_{2 3} r_{1 3} = 0$$

If $r = \sum a_i\otimes b_i$, then the operator $P:A\to A$ given by

$$P(x) = \sum a_i x b_i$$

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

The skew-symmetric solutions ($r_{1 2} = - r_{2 1}$) 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

Related concepts

Yang-Baxter equations

• classical Yang-Baxter equation

• quantum Yang-Baxter equation

• dynamical quantum Yang-Baxter equation

• associative Yang-Baxter equation

• infinitesimal braid relation

• set theoretic Yang-Baxter equation

• quasitriangular bialgebra

References

• 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 $A_\infty$-constraint, Adv. Math. 168 (2002, 56-95) open access pdf

• Chengming Bai, Li Guo, Xiang Ni, $\mathcal{O}$-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, in: Advances in quantum computation (Tyler, TX, 2007) (Providence, RI) (K. Mahdavi and D. Koslover, eds.), Contemp. Math. 482 (2009) 91–106 arXiv:math.QA/0612493

• V. Sokolov, Classification of constant solutions for associative Yang-Baxter on $gl(3)$, arxiv/1212.6421

• A. Odesskii, V. Rubtsov, V. Sokolov, Double Poisson brackets on free associative algebras, in: Noncommutative Birational Geometry, Representations and Combinatorics, Contemp. Math. 592, Amer. Math. Soc. (2013) 225–239 doi arxiv/1208.2935

• A. Odesskii, V. Rubtsov, V. Sokolov, Parameter-dependent associative Yang–Baxter equations and Poisson brackets, Int. J. Geom. Meth. Mod. Phys. 11:09, 1460036 (2014) Proc. XXII IFWGP, Univ. of Évora, Portugal, 2013 doi

category: algebra, physics