Complexity of multiplication in commutative group algebras over fields of characteristic 0

Let rk A denote the bilinear complexity (also known as rank) of a finite-dimension associative algebra A . Algebras of minimal rank are widely studied from the point of view of bilinear complexity. These are the algebras A for which the Alder-Strassen inequality is satisfied as an equality, i.e., rk A = 2dim A − t , where t is the number of maximum two-sided ideals in A . It is proved in this work that an arbitrary commutative group algebra over a field of characteristic 0 is an algebra of minimal rank. The structure and precise values of the bilinear complexity of commutative group algebras over a field of rational numbers are obtained.