Structure of Zariski-closed algebras

Belov-Kanel, Alexei; Rowen, Louis; Vishne, Uzi

doi:10.1090/S0002-9947-10-04993-7

Structure of Zariski-closed algebras
HTML articles powered by AMS MathViewer

by Alexei Belov-Kanel, Louis Rowen and Uzi Vishne PDF

Trans. Amer. Math. Soc. 362 (2010), 4695-4734 Request permission

Abstract:

The objective of this paper is to describe the structure of Zariski-closed algebras, which provide a useful generalization to finite dimensional algebras in the study of representable algebras over finite fields. Our results include a version of Wedderburn’s principal theorem as well as a more explicit description using representations, in terms of “gluing” in Wedderburn components. Finally, we construct “generic” Zariski-closed algebras, whose description is considerably more complicated than the description of generic algebra of finite dimensional algebras.

Special attention is given to infinite dimensional algebras over finite fields.

References

Belov, A. Ya., Algebras with polynomial identities: Representations and combinatorical methods, Doctor of Science Dissertation, Moscow (2002).
A. Ya. Belov, Counterexamples to the Specht problem, Mat. Sb. 191 (2000), no. 3, 13–24 (Russian, with Russian summary); English transl., Sb. Math. 191 (2000), no. 3-4, 329–340. MR 1773251, DOI 10.1070/SM2000v191n03ABEH000460
Alexei Kanel-Belov and Louis Halle Rowen, Computational aspects of polynomial identities, Research Notes in Mathematics, vol. 9, A K Peters, Ltd., Wellesley, MA, 2005. MR 2124127, DOI 10.1201/9781439863725
A. Giambruno and M. Zaicev, Minimal varieties of algebras of exponential growth, Adv. Math. 174 (2003), no. 2, 310–323. MR 1963697, DOI 10.1016/S0001-8708(02)00047-6
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
Tatsuji Kambayashi, Masayoshi Miyanishi, and Mitsuhiro Takeuchi, Unipotent algebraic groups, Lecture Notes in Mathematics, Vol. 414, Springer-Verlag, Berlin-New York, 1974. MR 0376696, DOI 10.1007/BFb0070517
Jacques Lewin, A matrix representation for associative algebras. I, II, Trans. Amer. Math. Soc. 188 (1974), 293–308; ibid. 188 (1974), 309–317. MR 338081, DOI 10.1090/S0002-9947-1974-0338081-5
Maxwell Rosenlicht, Questions of rationality for solvable algebraic groups over nonperfect fields, Ann. Mat. Pura Appl. (4) 61 (1963), 97–120 (English, with Italian summary). MR 158891, DOI 10.1007/BF02412850
Mohan S. Putcha, Linear algebraic monoids, London Mathematical Society Lecture Note Series, vol. 133, Cambridge University Press, Cambridge, 1988. MR 964690, DOI 10.1017/CBO9780511600661
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
Louis H. Rowen, Ring theory, Student edition, Academic Press, Inc., Boston, MA, 1991. MR 1095047
Louis Halle Rowen, Graduate algebra: noncommutative view, Graduate Studies in Mathematics, vol. 91, American Mathematical Society, Providence, RI, 2008. MR 2462400, DOI 10.1090/gsm/091
Tits, J., “Lectures on algebraic groups”, Dept. of Math., Yale Univ. – New Haven, 1966/67.