PI semigroup algebras of linear semigroups

Okniński, Jan; Putcha, Mohan S.

doi:10.1090/S0002-9939-1990-1013977-4

PI semigroup algebras of linear semigroups
HTML articles powered by AMS MathViewer

by Jan Okniński and Mohan S. Putcha PDF

Proc. Amer. Math. Soc. 109 (1990), 39-46 Request permission

Abstract:

It is well-known that if a semigroup algebra $K[S]$ over a field $K$ satisfies a polynomial identity then the semigroup $S$ has the permutation property. The converse is not true in general even when $S$ is a group. In this paper we consider linear semigroups $S \subseteq {\mathcal {M}_n}(F)$ having the permutation property. We show then that $K[S]$ has a polynomial identity of degree bounded by a fixed function of $n$ and the number of irreducible components of the Zariski closure of $S$.

References

Russell D. Blyth, Rewriting products of group elements. II, J. Algebra 119 (1988), no. 1, 246–259. MR 971358, DOI 10.1016/0021-8693(88)90088-9
A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. I, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1961. MR 0132791
Mario Curzio, Patrizia Longobardi, Mercede Maj, and Derek J. S. Robinson, A permutational property of groups, Arch. Math. (Basel) 44 (1985), no. 5, 385–389. MR 792360, DOI 10.1007/BF01229319
O. I. Domanov, Identities of semigroup algebras of completely $0$-simple semi-groups, Mat. Zametki 18 (1975), no. 2, 203–212 (Russian). MR 409705
O. I. Domanov, Identities of semigroup algebras of $0$-simple semigroups, Sibirsk. Mat. Ž. 17 (1976), no. 6, 1406–1407, 1440 (Russian). MR 0439968
Max Garzon and Yechezkel Zalcstein, On permutation properties in groups and semigroups, Semigroup Forum 35 (1987), no. 3, 337–351. MR 900108, DOI 10.1007/BF02573115
Max Garzon and Yechezkel Zalcstein, Linear semigroups with permutation properties, Semigroup Forum 35 (1987), no. 3, 369–371. MR 900110, DOI 10.1007/BF02573117
James E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975. MR 0396773, DOI 10.1007/978-1-4684-9443-3

Strongly

-regular matrix semigroups

93

Jan Okniński, On semigroup algebras satisfying polynomial identities, Semigroup Forum 33 (1986), no. 1, 87–102. MR 834552, DOI 10.1007/BF02573184
Jan Okniński, On cancellative semigroup rings, Comm. Algebra 15 (1987), no. 8, 1667–1677. MR 884768, DOI 10.1080/00927878708823495

A note on the PI-property of semigroup rings

Jan Okniński, Semigroup algebras, Monographs and Textbooks in Pure and Applied Mathematics, vol. 138, Marcel Dekker, Inc., New York, 1991. MR 1083356

Complex representations of matrix semigroups

Donald S. Passman, The algebraic structure of group rings, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. MR 0470211
Mohan S. Putcha, Green’s relations on a connected algebraic monoid, Linear and Multilinear Algebra 12 (1982/83), no. 3, 205–214. MR 678826, DOI 10.1080/03081088208817484
Mohan S. Putcha, Matrix semigroups, Proc. Amer. Math. Soc. 88 (1983), no. 3, 386–390. MR 699399, DOI 10.1090/S0002-9939-1983-0699399-0

Linear algebraic Monoids

Mohan S. Putcha and Adil Yaqub, Semigroups satisfying permutation identities, Semigroup Forum 3 (1971/72), no. 1, 68–73. MR 292969, DOI 10.1007/BF02572944
Antonio Restivo and Christophe Reutenauer, On the Burnside problem for semigroups, J. Algebra 89 (1984), no. 1, 102–104. MR 748230, DOI 10.1016/0021-8693(84)90237-0
Louis Halle Rowen, Polynomial identities in ring theory, Pure and Applied Mathematics, vol. 84, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 576061
I. R. Shafarevich, Basic algebraic geometry, Die Grundlehren der mathematischen Wissenschaften, Band 213, Springer-Verlag, New York-Heidelberg, 1974. Translated from the Russian by K. A. Hirsch. MR 0366917, DOI 10.1007/978-3-642-96200-4