Solutions of associative Yang-Baxter equation and $D-$equation in low dimensions and associated Frobenius algebras and Connes cocycles

This work addresses some relevant characteristics of associative algebras in low dimensions. Especially, given 1 and 2 dimensional associative algebras, we explicitly solve associative Yang-Baxter equations and use skew-symmetric solutions to perform double constructions of Frobenius algebras. Besides, we determine related compatible dendriform algebras and solutions of their $D-$equations. Finally, using symmetric solutions of the latter equations, we proceed to double constructions of corresponding Connes cocycles.