Semisimple algebras of almost minimal rank over the reals

Authors:
Markus Bläser;Andreas Meyer de Voltaire
Affiliations:
Computer Science, Saarland University, Postfach 151150, D-66041 Saarbrücken, Germany;Chair of Information Technology and Education, ETH Zürich, CH-8092 Zürich, Switzerland
Venue:
Theoretical Computer Science
Year:
2009

Citing 5
Cited 0

On associative algebras of minimal rank

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Algebraic complexity theory

Handbook of theoretical computer science (vol. A)
Lower bounds for the bilinear complexity of associative algebras

Computational Complexity
On the complexity of the multiplication of matrices of small formats

Journal of Complexity
A Complete Characterization of the Algebras of Minimal Bilinear Complexity

SIAM Journal on Computing

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Abstract

A famous lower bound for the bilinear complexity of the multiplication in associative algebras is the Alder-Strassen bound. Algebras for which this bound is tight are called algebras of minimal rank. After 25 years of research, these algebras are now well understood. Here we start the investigation of the algebras for which the Alder-Strassen bound is off by one. As a first result, we completely characterize the semisimple algebras over R whose bilinear complexity is by one larger than the Alder-Strassen bound. Furthermore, we characterize all algebras A (with radical) of minimal rank plus one over R for which A/radA has minimal rank plus one. The other possibility is that A/radA has minimal rank. For this case, we only present a partial result.