associative Yang-Baxter equation

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

- 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*, 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*, arxiv/1208.2935

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