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Anti-Frobenius algebras and associative Yang–Baxter equation
Matematičeskoe modelirovanie, Tome 26 (2014) no. 11, pp. 51-56 Cet article a éte moissonné depuis la source Math-Net.Ru

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Associative Yang–Baxter equation arises in different areas of algebra, e.g., when studying double quadratic Poisson brackets, non-abelian quadratic Poisson brackets, or associative algebras with cyclic 2-cocycle (anti-Frobenius algebras). Precisely, faithful representations of anti-Frobenius algebras (up to isomorphism) are in one-to-one correspondence with skew-symmetric solutions of associative Yang–Baxter equation (up to equivalence). Following the work of Odesskii, Rubtsov and Sokolov and using computer algebra system Sage, we found some constant skew-symmetric solutions of associative Yang–Baxter equation and construct corresponded non-abelian quadratic Poisson brackets.
Keywords: associative Yang–Baxter equation, anti-Frobenius algebras, non-abelian quadratic Poisson brackets. 
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A. I. Zobnin. Anti-Frobenius algebras and associative Yang–Baxter equation. Matematičeskoe modelirovanie, Tome 26 (2014) no. 11, pp. 51-56. http://geodesic.mathdoc.fr/item/MM_2014_26_11_a7/

[1] M. Aguiar, “On the associative analog of Lie bialgebras”, J. Alg., 244:2 (2001), 492–532 | DOI | MR | Zbl

[2] M. Van den Bergh, “Double Poisson algebras”, Trans. Amer. Math. Soc., 360:11 (2008), 5711–5769 | DOI | MR | Zbl

[3] A. Connes, “Non-commutative Algebraic Geometry”, Publ. Math. Inst. Haute Études Sci., 62 (1985), 257–360 | MR

[4] Functional Anal. Appl., 16:4 (1982), 326–328 | DOI | MR

[5] A. V. Odesskii, V. N. Rubtsov, V. V. Sokolov, “Bi-Hamiltonian ordinary differential equations with matrix variables”, Theoretical and Mathematical Physics, 171:1, April (2012), 442–447 | DOI | MR | Zbl

[6] A. V. Odesskii, V. N. Rubtsov, V. V. Sokolov, “Double Poisson brackets on free associative algebras”, Contemporary Mathematics, 592, 2013, 225–241 | DOI | MR

[7] A. Polischuk, “Classical Yang–Baxter Equation and the $A_\infty$-Constraint”, Advances in Mathematics, 168:1 (2002), 56–95 | DOI | MR

[8] V. Sokolov, “Classification of constant solutions of associative Yang–Baxter equation on $\mathfrak{gl}(3)$”, Theoretical and Mathematical Physics, 176:3 (2013), 1156–1162 | DOI | MR | Zbl