一类商代数上的双-Frobenius代数结构

Bi-Frobenius Algebraic Structure on a Type of Quotient Algebras

设C[X]为复数域上的一元多项式代数,I为n+1次Dickson多项式E_(n+1)(X)生成的C[X]的理想,C[X]/I为商代数.证明了商代数C[X]/I既是Frobenius代数,又是Frobenius余代数.进一步,该商代数在恒等对极下还是双-Frobenius代数.

Let C[X]be a polynomial algebra with one variable over the field of complex numbers,I the ideal of C[X]generated by the n+1-th Dickson polynomial En+1(X)and C[X]/I the quotient algebra of C[X]modulo the ideal I.It is shown that the quotient algebra C[X]/I is both a Frobenius algebra and a Frobenius coalgebra.Moreover,it is a bi-Frobenius algebra whose antipode is given by the identity map.