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Algebras of Minimal Rank over Arbitrary Fields

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STACS 2003 (STACS 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2607))

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Abstract

Let R(A) denote the rank (also called bilinear complexity) of a finite dimensional associative algebra A. A fundamental lower bound for R(A) is the so-called Alder-Strassen bound R(A) ≥ 2 dim A-t, where t is the number of maximal twosided ideals of A. The class of algebras for which the Alder-Strassen bound is sharp, the so-called algebras of minimal rank, has received a wide attention in algebraic complexity theory.

As the main contribution of this work, we characterize all algebras of minimal rank over arbitrary fields. This finally solves an open problem in algebraic complexity theory, see for instance [12], Sect. 12, Problem 4] or [6], Problem 17.5].

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References

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Author information

Authors and Affiliations

  1. Institut für Theoretische Informatik, Universität zu Lübeck, Wallstr. 40, 23560, Lübeck, Germany

    Markus Bläser

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  1. Markus Bläser
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Editors and Affiliations

  1. Institut für Informatik, Freie Universität Berlin, Takusstr. 9, 14195, Berlin, Germany

    Helmut Alt

  2. LIRMM, 161, Rue Ada, 34392, Montpellier Cedex 5, France

    Michel Habib

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© 2003 Springer-Verlag Berlin Heidelberg

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Bläser, M. (2003). Algebras of Minimal Rank over Arbitrary Fields. In: Alt, H., Habib, M. (eds) STACS 2003. STACS 2003. Lecture Notes in Computer Science, vol 2607. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36494-3_36

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  • DOI: https://doi.org/10.1007/3-540-36494-3_36

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-00623-7

  • Online ISBN: 978-3-540-36494-8

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