A Family of Commutative Endomorphism Algebras

LetK(X) be the quotient field of the polynomial ringK[X] over an algebraically closed fieldK. Given an integern ≥ 2, a finite sequencem = (m2,…,mn) ofn − 1 positive integers, a sequenceα = (α2,…,αn) ofK-linear functionals onK(X), and a height functionh, there is attached a Kronecker moduleE = E(m, h, α). In this paper we show that whenEhas no finite-dimensional direct summand then End Eis a commutativeK-subalgebra of the algebra ofn × nmatrices overK(X). Consequently, when End Eis an integral domain its transcendence degree overKis at most one. We give sufficient conditions for End Eto be isomorphic toKand also show thatKis the only field that occurs as End E. Examples whereK[X],K[X, X − 1], and some subalgebras ofK(X)noccur as End Eare given. The determination of all the commutativeK-subalgebras ofMn(K(X)) realizable as End Eremains open. Yet surprisingly it can be shown that the coordinate ring of a cubic equation in Weierstrass normal formY2 = X3 + aX2 + bX + cis realizable.