Skip to main content
×
Home
    • Aa
    • Aa

Gröbner bases for quadratic algebras of skew type

  • Ferran Cedó (a1) and Jan Okniński (a2)
Abstract
Abstract

Non-degenerate monoids of skew type are considered. This is a class of monoids S defined by n generators and quadratic relations of certain type, which includes the class of monoids yielding set-theoretic solutions of the quantum Yang–Baxter equation, also called binomial monoids (or monoids of I-type with square-free defining relations). It is shown that under any degree-lexicographic order on the associated free monoid FMn. of rank n the set of normal forms of elements of S is a regular language in FMn. As one of the key ingredients of the proof, it is shown that an identity of the form xN yN = yN xN holds in S. The latter is derived via an investigation of the structure of S viewed as a semigroup of matrices over a field. It also follows that the semigroup algebra K[S] is a finite module over a finitely generated commutative subalgebra of the form K[A] for a submonoid A of S.

Copyright
References
Hide All
2. J. P. Bell and P. Colak , Primitivity of finitely presented monomial algebras, J. Pure Appl. Alg. 213 (2009), 12991305.

3. T. Gateva-Ivanova , Skew polynomial rings with binomial relations, J. Alg. 185 (1996), 710753.

5. T. Gateva-Ivanova and M. Van den Bergh , Semigroups of I-type, J. Alg. 206 (1998), 97112.

7. E. Jespers and J. Okniński , Noetherian semigroup algebras (Springer, 2007).

9. J. Månsson and P. Nordbeck , Regular Gröbner bases, J. Symb. Computat. 33 (2002), 163281.

12. J. Okniński , Semigroups of matrices (World Scientific, Singapore, 1998).

13. J. Okniński , Structure of prime finitely presented monomial algebras, J. Alg. 320 (2008), 31993205.

15. W. Rump , A decomposition theorem for square-free unitary solutions of the quantum Yang–Baxter equation, Adv. Math. 193 (2005), 4055.

16. V. A. Ufnarovskii , Combinatorial and asymptotic methods in algebra, in Encyclopedia of Mathematical Sciences, Volume 57, pp. 1196 (Springer, 1995).

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 5 *
Loading metrics...

Abstract views

Total abstract views: 81 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 17th October 2017. This data will be updated every 24 hours.