Abstract
In this chapter we consider some basic algorithmic problems related to finite dimensional associative algebras. Our starting point is the structure theory of these algebras. This theory gives a description of the main structural ingredients of finite dimensional associative algebras, and specifies the way the algebra is constructed from these building blocks. We describe polynomial time algorithms to find the main components: the Jacobson radical and the simple direct summands of the radical-free part.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
E. H. Bareiss (1968): Sylvester’s identity and multistep integer-preserving Gaussian elimination, Mathematics of Computation 103, 565–578.
R.E. Beck, B. Kolman, and I.N. Stewart (1977): Computing the structure of a Lie algebra, Computers in nonassociative rings and algebras, Academic Press, New York, 167–188.
E. R. Berlekamp (1968): Algebraic Coding Theory, McGraw-Hill.
E. R. Berlekamp (1970): Factoring polynomials over large finite fields, Math, of Computation 24, 713–715.
A.M. Cohen, G. Ivanyos, and D. B. Wales (1997): Finding the radical of an algebra of linear transformations, J. of Pure and Applied Algebra 117 & 118, 177–193.
A. M. Cohen and L. Meertens (1995): The ACELA project: Aims and Plans, to appear in: Human Interaction for Symbolic Computation, ed. N. Kajler, Texts and Monographs in Symbolic Computation, Springer-Verlag, Berlin Heidelberg New York.
G. E. Collins, M. Mignotte, and F. Winkler (1983): Arithmetic in basic algebraic domains, in: Computer Algebra. Symbolic and Algebraic Computation, 2nd edn., Springer-Verlag, Berlin Heidelberg New York, 189–220.
T. H. Cormen, C. E. Leiserson, and R. L. Rivest (1990): Introduction to Algorithms, The MIT Press.
L.E. Dickson (1923): Algebras and Their Arithmetics, University of Chicago.
W. M. Eberly (1989): Computations for Algebras and Group Representations, Ph.D. thesis, Dept. of Computer Science, University of Toronto.
W. M. Eberly and M. Giesbrecht (1996): Efficient decomposition of associative algebras, Proc. of ISSAC’96, ACM Press, 170–178.
J. Edmonds (1967): System of distinct representatives and linear algebra, Journal of Research of the National Bureau of Standards 718, 241–245.
K. Friedl and L. Rónyai (1985): Polynomial time solution of some problems in computational algebra, Proc. 17th ACM STOC, 153–162.
W. A. de Graaf (1997): Algorithms for Finite-Dimensional Lie Algebras, Ph.D. Thesis, Technische Universiteit Eindhoven.
W. A. de Graaf, G. Ivanyos, and L. Rónyai (1996): Computing Cartan subalge-bras of Lie algebras, Applicable Algebra in Engineering, Communication and Computing 7, 339–349.
I.N. Herstein (1968): Noncommutative Rings, Math. Association of America.
J. E. Hopcroft and J. D. Ullman (1979): Introduction to Automata Theory, Languages and Computation, Addison-Wesley.
J.E. Humphreys (1980): Introduction to Lie Algebra and Representation Theory, Graduate Texts in Mathematics 9, Springer-Verlag, Berlin Heidelberg New York.
G. Ivanyos, L. Rónyai, and á. Szántó (1994): Decomposition of algebras over F q (Xi,..., Xm), Applicable Algebra in Engineering, Communication and Computing 5, 71–90.
N. Jacobson (1962): Lie Algebras, John Wiley.
R. Kannan and A. Bachern (1979): Polynomial algorithms for computing the Smith and Hermite normal forms of an integer matrix, SIAM J. on Computing 4, 499–507.
D.E. Knuth (1981): The art of computer programming, Vol. 2, Seminumerical algorithms, Addison-Wesley.
S. Lang (1965): Algebra, Addison-Wesley.
R. Lidl and H. Niederreiter (1983): Finite Fields, Addison-Wesley.
M. Mignotte (1992): Mathematics for Computer Algebra, Springer-Verlag, Berlin Heidelberg New York.
R. S. Pierce (1982): Associative Algebras, Springer-Verlag, Berlin Heidelberg New York.
L. Rónyai (1988): Zero divisors in quaternion algebras, Journal of Algorithms 9, 494–506.
L. Rónyai (1990): Computing the structure of finite algebras, J. of Symbolic Computation 9, 355–373.
L. Rónyai (1993): Computations in associative algebras, Groups and Computation, DIMACS Series 11, American Mathematical Society, 221–243.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Ivanyos, G., Rónyai, L. (1999). Computations in Associative and Lie Algebras. In: Cohen, A.M., Cuypers, H., Sterk, H. (eds) Some Tapas of Computer Algebra. Algorithms and Computation in Mathematics, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03891-8_5
Download citation
DOI: https://doi.org/10.1007/978-3-662-03891-8_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-08335-8
Online ISBN: 978-3-662-03891-8
eBook Packages: Springer Book Archive