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

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