Skip to main content
Log in

Functional Identities on Matrix Algebras

  • Published:
Algebras and Representation Theory Aims and scope Submit manuscript

Abstract

Complete solutions of functional identities \({\sum }_{k\in K}F_{k}\left (\overline {x}_{m}^{k}\right )x_{k} = {\sum }_{l\in L}x_{l}G_{l}\left (\overline {x}_{m}^{l}\right )\) on the matrix algebra M n (\(\mathbb {F}\)) are given. The nonstandard parts of these solutions turn out to follow from the Cayley-Hamilton identity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Beidar, K.I., Martindale, W.S., Mikhalevm, A.V.: Rings with generalized identities, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc. (1996)

  2. Brešar, M.: Commuting maps: A survey. Taiwan. J. Math. 8(2004), 361–397

  3. Brešar, M., Chebotar, M.A., Martindale, W.S.: Functional identities, 3rd. Birkhäuser Verlag (2007)

  4. Brešar, M., Procesi, C., Špenko, Š.: Quasi-identities and the Cayley-Hamilton polynomial. Adv. Math. 280, 439–471 (2015)

  5. Brešar, M., Špenko, Š.: Functional identities in one variable. J. Algebra 401, 234–244 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  6. Eisenbud, D.: Commutative algebra with a view toward algebraic geometry. Graduate Texts in Mathematics. Springer-Verlag (1995)

  7. Littlewood, D.E.: Identical relations satisfied in an algebra. Proc. London Math. Soc. 32, 312–320 (1931)

    Article  MathSciNet  Google Scholar 

  8. Onn, S.: Hilbert series of group representations and Gröbner bases for generic modules. J. Algebraic Combin. 3, 187–206 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  9. Procesi, C.: Lie groups: An approach through invariants and representations. Springer Universitext (2007)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Špela Špenko.

Additional information

Presented by Vlastimil Dlab.

Supported by ARRS Grant P1-0288.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Brešar, M., Špenko, Š. Functional Identities on Matrix Algebras. Algebr Represent Theor 18, 1337–1356 (2015). https://doi.org/10.1007/s10468-015-9543-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10468-015-9543-x

Keywords

Mathematics Subject Classification (2010)

Navigation