Polynomial functions on upper triangular matrix algebras

Sophie Frisch

doi:10.1007/s00605-016-1013-y

Abstract

There are two kinds of polynomial functions on matrix algebras over commutative rings: those induced by polynomials with coefficients in the algebra itself and those induced by polynomials with scalar coefficients. In the case of algebras of upper triangular matrices over a commutative ring, we characterize the former in terms of the latter (which are easier to handle because of substitution homomorphism). We conclude that the set of integer-valued polynomials with matrix coefficients on an algebra of upper triangular matrices is a ring, and that the set of null-polynomials with matrix coefficients on an algebra of upper triangular matrices is an ideal.

Highlights

Polynomial functions on non-commutative algebras over commutative rings come in two flavors: one, induced by polynomials with scalar coefficients, the other, by polynomials with coefficients in the non-commutative algebra itself [1,4,5,6,7,8,9,11,12]
For the algebra of upper triangular matrices over a commutative ring, we show how polynomial functions with matrix coefficients can be described in terms of polynomial functions with scalar coefficients
Our results have a bearing on several open questions in the subject of polynomial functions on non-commutative algebras

Summary

Introduction

Polynomial functions on non-commutative algebras over commutative rings come in two flavors: one, induced by polynomials with scalar coefficients, the other, by polynomials with coefficients in the non-commutative algebra itself [1,4,5,6,7,8,9,11,12]. For the algebra of upper triangular matrices over a commutative ring, we show how polynomial functions with matrix coefficients can be described in terms of polynomial functions with scalar coefficients. Our results have a bearing on several open questions in the subject of polynomial functions on non-commutative algebras They allow us to answer in the affirmative, in the case of algebras of upper triangular matrices, two questions Peruginelli and Werner [7] have characterized the algebras for which this relationship holds We take this investigation one step further and show for algebras of upper triangular matrices, where A = Tn(D) and B = Tn(K ), that IntB( A) can be described quite nicely in terms of IntK(A) (cf Remark 1.6, Theorem 4.2), but that the connection is not as simple as merely tensoring with A. Remark 1.6 If we identify IntTn(R)(Tn(S), Tn(I )) and Int Tn(R)(Tn(S), Tn(I ))—a priori subsets of (Tn(R))[x]—with their respective images in Tn(R[x]) under the ring isomorphism φ : (Tn(R))[x] → Tn(R[x]),

Path polynomials and polynomials with scalar coefficients

Lemmata for polynomials with matrix coefficients

Results for polynomials with matrix coefficients

Applications to null-polynomials and integer-valued polynomials