# nLab associative Yang-Baxter equation

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## 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

## 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, 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

category: algebra, physics

Last revised on November 26, 2019 at 07:32:26. See the history of this page for a list of all contributions to it.