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- W2068059537 abstract "Given a field K, a Kronecker module V is a pair of K-linear spaces (U,V) together with a K-bilinear map K2×U→V. In finite dimensions this is also the notion of pencils of matrices. Every K[X]-module can be construed as a Kronecker module. In particular the K[X]-submodules of K(X) give rise to the Kronecker modules Rh where h is a height function, i.e. a function h:K∪{∞}→{∞,0,1,2,…}. The K[X]-module K[X] itself gives the Kronecker module P that goes with the height function which is ∞ at ∞ and 0 on K. The modules Rh that are infinite-dimensional come up precisely when h attains the value ∞ or when h is stictly positive on an infinite subset of K∪{∞}. The endomorphism algebra of Rh is called a pole algebra. Those Kronecker modules V that are extensions of finite-dimensional submodules of P by infinite-dimensional Rh lead to some engaging problems with matrix algebras. This is because the endomorphisms of such V constitute a K-subalgebra of n×n matrices over K(X), which is commutative if the extension is indecomposable. Among the algebras that are known to arise in the 2 × 2 case, when the extension is by P, are the coordinate rings of all elliptic curves. In this paper we replace P by an arbitrary infinite-dimensional Rh. The following new algebras are realized: infinite-dimensional pole algebras End Rh where h=0 on an infinite subset of K; maximal subalgberas of A×B for some pole algebras A,B; the quasi local ring K∝S where S is a K-vector space of dimension at most card K. In the process we identify those height functions h that will tolerate an indecomposable extension V having non-trivial endomorphisms." @default.
- W2068059537 created "2016-06-24" @default.
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- W2068059537 date "2003-11-01" @default.
- W2068059537 modified "2023-10-18" @default.
- W2068059537 title "Commutative algebras of rational function matrices as endomorphisms of Kronecker modules I" @default.
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- W2068059537 doi "https://doi.org/10.1016/s0024-3795(03)00539-1" @default.
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