102 research outputs found
Effective Detection of Nonsplit Module Extensions
Let n be a positive integer, and let R be a finitely presented (but not
necessarily finite dimensional) associative algebra over a computable field. We
examine algorithmic tests for deciding (1) if every n-dimensional
representation of R is semisimple, and (2) if there exist nonsplit extensions
of non-isomorphic irreducible R-modules whose dimensions sum to no greater than
n. Our basic strategy is to reduce each of the considered representation
theoretic decision problems to the problem of deciding whether a particular set
of commutative polynomials has a common zero. Standard methods of computational
algebraic geometry can then be applied (in principle).Comment: AMS-TeX; 13 pages; no figures. Revised version. To appear in Journal
of Pure and Applied Algebr
Constructing irreducible representations of finitely presented algebras
By combining well-known techniques from both noncommutative algebra and
computational commutative algebra, we observe that an algorithmic approach can
be applied to the study of irreducible representations of finitely presented
algebras. In slightly more detail: Assume that is a positive integer, that
is a computable field, that denotes the algebraic closure of ,
and that denotes the algebra of matrices with
entries in . Let be a finitely presented -algebra. Calculating
over , the procedure will (a) decide whether an irreducible representation
exists, and (b) explicitly construct an irreducible
representation if at least one exists. (For (b), it is
necessary to assume that is equipped with a factoring algorithm.) An
elementary example is worked through.Comment: 9 pages. Final version. To appear in J. Symbolic Computatio
Module Extensions Over Classical Lie Superalgebras
We study certain filtrations of indecomposable injective modules over
classical Lie superalgebras, applying a general approach for noetherian rings
developed by Brown, Jategaonkar, Lenagan, and Warfield. To indicate the
consequences of our analysis, suppose that is a complex classical simple
Lie superalgebra and that is an indecomposable injective -module with
nonzero (and so necessarily simple) socle . (Recall that every essential
extension of , and in particular every nonsplit extension of by a simple
module, can be formed from -subfactors of .) A direct transposition of
the Lie algebra theory to this setting is impossible. However, we are able to
present a finite upper bound, easily calculated and dependent only on , for
the number of isomorphism classes of simple highest weight -modules
appearing as -subfactors of .Comment: 20 page
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