Nearly Frobenius dimension of Frobenius algebras.

Let K be a field. A Frobenius algebra (A,B) is an associative K-algebra A with a non-degenerate bilinear form B : A×A → K such that B(ab,c) = B(a,bc) for all a,b,c ∈ A. It essentially first appeared in the papers of G. Frobenius, and then named by Brauer and Nesbitt. As it is well known, Frobenius algebras have wide applications in many fields in mathematics and mathematical physics, such as representations of finite groups and Yang-Baxter equation, etc. Firstly, this paper shows that the Frobenius space associated to a Frobenius algebra is generated as left A-module over a filed and further in a commutative ring, respectively. It is well known that each Frobenius algebra over a commutative ring can determine a class of solutions of the quantum Yang-Baxter equation. Secondly, the solutions of the quantum Yang-Baxter equation starting from elements in the Frobenius space are given in this paper.