Graded semisimple algebras are symmetric (Q2404943)

A finite-dimensional associative and unital algebra over a field is Frobenius when there is a balanced (or associative) nondegenerated bilinear form defined on it. If the bilinear form is symmetric, then the algebra is said to be symmetric. A classical result states that every semi-simple algebra, always of finite dimension, is symmetric. The aim of the paper is to state the analogue to this result in the realm of group-graded algebras, the grading group being finite. To this end, the notion of graded symmetric algebra considered fits to the notion of a Frobenius symmetric algebra in the tensor category of finite-dimensional graded vector spaces. The semi-simple algebras in this category enjoy a Wedderburn-type structure theorem [the third author and \textit{F. Van Oystaeyen}, Methods of graded rings. Berlin: Springer (2004; Zbl 1043.16017)], which allows to reduce many problems to the case of graded division algebras. This is the case of the paper under review, where it is proved that a graded semisimple algebra is symmetric (Theorem A). The proof thus requires, as a crucial step, to prove the statement for a graded division algebra. An interesting example shows that the center of a graded symmetric algebra needs not to by symmetric. However, it is proved that, if the characteristic of the ground field does not divide the order of the grading group, then the center of a graded division algebra is already symmetric (Theorem B).