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Algebras and Representation Theory

, Volume 14, Issue 2, pp 317–339 | Cite as

Deep Matrix Modules

  • Christopher KennedyEmail author
Article
  • 42 Downloads

Abstract

A deep matrix algebra, \(\mathcal{DM}(X,\mathbb{K})\), is a unital associative algebra over a field \(\mathbb{K}\) with basis all deep matrix units, \(\mathfrak{e}(h,k)\), indexed by pairs of elements h and k taken from a free monoid generated by a set X. After briefly describing the construction of \(\mathcal{DM}(X,\mathbb{K})\), we determine necessary and sufficient conditions for constructing representations for \(\mathcal{DM}(X,\mathbb{K})\). With these conditions in place, we define null modules and give three canonical examples of such. A classification of general null modules is then given in terms of the canonical examples along with their submodules and quotients. In the final section, additional examples of natural actions for \(\mathcal{DM}(X,\mathbb{K})\) are given and their submodules determined depending on the cardinality of the set X.

Keywords

Deep matrix algebra Deep matrix module Irreducible modules 

Mathematics Subject Classification (2000)

16D10 

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References

  1. 1.
    Cuntz, J.: Simple C *-algebras generated by isometries. Commun. Math. Phys. 57, 173–185 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Kennedy, C.: An Exploration of Deep Matrix Algebras. Proquest, Ann Arbor (2004)Google Scholar
  3. 3.
    Kennedy, C.: Deep matrix algebras of finite type. Algebr. Represent. Theory 9, 525–537 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Kennedy, C.: Simple and nearly simple deep matrix algebras. Algebr. Represent. Theory. doi: 10.1007/s10468-008-9127-0 (2009)Google Scholar
  5. 5.
    McCrimmon, K.: Deep matrices and their frankenstein actions. In: Sabinnene, L., Sbitneva, L. Shestakov, I. (eds.) Non-Associative Algebra and Its Applications. Lecture Notes in Pure and Appl. Math., vol. 246. CRC, Boca Raton (2006)CrossRefGoogle Scholar
  6. 6.
    McCrimmon, K., Faulkner J.: Finitely deep matrices. Commun. Algebra 36, 3897–3911 (2008)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of MathematicsChristopher Newport UniversityNewport NewsUSA

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