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

Yang-Baxter equations


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category: algebra, physics

Last revised on September 20, 2022 at 18:17:44. See the history of this page for a list of all contributions to it.