Fixed subalgebra of a commutative Frobenius algebra

Abstract

Let B B be a finite-dimensional commutative algebra generated by a single element, and let A = B ⊗ B A = B \otimes B . We prove that the fixed subalgebra of A A under the involution b 1 ⊗ b 2 ↦ b 2 ⊗ b 1 {b_1} \otimes {b_2} \mapsto {b_2} \otimes {b_1} is Frobenius if and only if either the characteristic of B B is different from 2 or B B is separable.