Zariski closed algebras in varieties of universal algebra. (English) Zbl 1315.16020
The authors study identical relations of algebras in the general context of universal algebra. An algebra \(A\) is representable if it is a subalgebra of a finite dimensional algebra \(B\) over a larger base field. The purpose of this paper is to develop the theory of the Zariski closure of a representable algebra (not necessarily associative) over an arbitrary integral domain. As an application, the authors prove that the codimension sequence of any representable Zariski-closed PI-algebra over an arbitrary field is exponentially bounded.
Reviewer: Victor Petrogradsky (Brasilia)
MSC:
| 16R10 | \(T\)-ideals, identities, varieties of associative rings and algebras |
| 17A30 | Nonassociative algebras satisfying other identities |
| 08B99 | Varieties |
| 16R30 | Trace rings and invariant theory (associative rings and algebras) |
| 17A01 | General theory of nonassociative rings and algebras |
| 17B01 | Identities, free Lie (super)algebras |
| 17C05 | Identities and free Jordan structures |
Keywords:
Zariski closed algebras; polynomial identities; codimension sequences; T-ideals; affine algebras; representable algebras; universal algebrasReferences:
| [1] | Aljadeff, E., Giambruno, A., La Mattina, D.: Graded polynomial identities and exponential growth. J. Reine Angew. Math. 650, 83-100 (2011) · Zbl 1215.16014 |
| [2] | Amitsur, S.A.: The sequence of codimensions of a PI-algebra. Israel J. Math. 47, 1-22 (1984) · Zbl 0534.16012 · doi:10.1007/BF02760558 |
| [3] | Bahturin, Yu.A., Drensky, V.: Graded polynomial identities of matrices. Linear Algebra Appl. 357, 15-34 (2002) · Zbl 1019.16011 · doi:10.1016/S0024-3795(02)00356-7 |
| [4] | Belov, A., Rowen, L.H.: Computational aspects of polynomial identities. Research Notes in Mathematics, vol. 9. AK Peters (2005) · Zbl 1076.16018 |
| [5] | Belov, A., Rowen, L.H., Vishne, U.: Structure of Zariski closed algebras. Trans. Am. Math. Soc. 362, 4695-4734 (2010) · Zbl 1221.16021 · doi:10.1090/S0002-9947-10-04993-7 |
| [6] | Belov, A., Rowen, L.H., Vishne, U.: PI-varieties associated to full quivers of representations of algebras. Trans. Am. Math. Soc., to appear · Zbl 1302.16014 |
| [7] | Belov, A., Rowen, L.H., Vishne, U.: Specht’s problem for associative affine algebras over commutative Noetherian rings. Trans. Am. Math. Soc., to appear · Zbl 1332.16015 |
| [8] | Belov, A., Rowen, L.H., Vishne, U.: Spechtian Varieties. preprint (2014) · Zbl 1226.16017 |
| [9] | Berele, A.: Properties of hook Schur functions with applications to p.i. algebras. Adv. Appl. Math. 41(1), 52-75 (2008) · Zbl 1145.05052 · doi:10.1016/j.aam.2007.03.002 |
| [10] | Berele, A., Regev, A.: Asymptotic behaviour of codimensions of p.i. algebras satisfying Capelli identities. Trans. Am. Math. Soc. 360(10), 5155-5172 (2008) · Zbl 1161.16016 · doi:10.1090/S0002-9947-08-04500-5 |
| [11] | Birkhoff, G.: Lattice theory. Am. Math. Soc. Colloq. Pub. 25 (1967) · Zbl 0153.02501 |
| [12] | Cohn, P.M.: Universal algebra. Reidel, Amsterdam (1981) · Zbl 0461.08001 · doi:10.1007/978-94-009-8399-1 |
| [13] | Drensky, V.S., Regev, A.: Exact asymptotic behaviour of the codimensions of some P.I. algebras. Israel J. Math. 96(part A), 231-242 (1996) · Zbl 0884.16014 · doi:10.1007/BF02785540 |
| [14] | Giambruno, A., La Mattina, D.: Graded polynomial identities and codimensions: computing the exponential growth. Adv. Math. 225(2), 859-881 (2010) · Zbl 1203.16021 · doi:10.1016/j.aim.2010.03.013 |
| [15] | Giambruno, A., Zaicev, M.: Exponential codimension growth of P.I. algebras: an exact estimate. Adv. Math. 142, 221-243 (1999) · Zbl 0920.16013 · doi:10.1006/aima.1998.1790 |
| [16] | Giambruno, A., Zaicev, M.: Polynomial identities and asymptotic methods. Mathematical Surveys and Monographs, vol. 122. AMS, Providence (2005) · Zbl 1105.16001 |
| [17] | Giambruno, A., Zaicev, M.: Proper identities, Lie identities and exponential codimension growth. J. Algebra 320(5), 1933-1962 (2008) · Zbl 1153.16018 · doi:10.1016/j.jalgebra.2008.06.009 |
| [18] | Giambruno, A., Zelmanov, E.: On growth of codimensions of Jordan algebras. In: Groups, algebras and applications. Contemp. Math., vol. 537, pp. 205-210 (2011) · Zbl 1243.17020 |
| [19] | Gordienko, A.S.: Codimensions of polynomial identities of representations of Lie algebras. Proc. Am. Math. Soc. 141(10), 3369-3382 (2013) · Zbl 1337.17008 · doi:10.1090/S0002-9939-2013-11622-9 |
| [20] | Higgins, P.J.: Algebras with a scheme of operators. Math. Nachr. 27, 115-132 (1963) · Zbl 0117.25903 · doi:10.1002/mana.19630270108 |
| [21] | Jacobson, N.: Basic Algebra. Freeman, San Francisco (1980) · Zbl 0441.16001 |
| [22] | Kemer, A.R.: Identities of finitely generated algebras over an infinite field. Math. USSR Izv. 37, 69-97 (1991) · Zbl 0784.16016 · doi:10.1070/IM1991v037n01ABEH002053 |
| [23] | Kemer, A.R.: Identities of associative algebras. Transl. Math. Monogr. Am. Math. Soc. 87 (1991) · Zbl 0736.16013 |
| [24] | Kurosh, A.G.: Free sums of multiple operator algebras (Russian). Sibirsk. Mat. Z. 1, 62-70 (1960). correction, 638 · Zbl 0096.25304 |
| [25] | Kurosh, A.G.: Higher algebra. Translated from the Russian by George Yankovsky. Reprint of the 1972 translation. “Mir”, Moscow, 428 pp. (1988) |
| [26] | Mishchenko, S.P.: Growth of varieties of Lie algebras. English. Russian originalRussian Math. Surv. 45(6), 27-52 (1990). translation from Uspekhi Mat. Nauk 45(1990), No. 6(276), 25-45 · Zbl 0718.17004 · doi:10.1070/RM1990v045n06ABEH002710 |
| [27] | Petrogradsky, V.M.: Growth of polynilpotent varieties of Lie algebras, and rapidly increasing entire functions. Mat. Sb. 188, 119-138 (1997). English translation: Sb. Math. 188 (1997), 913931. · Zbl 0890.17002 · doi:10.4213/sm232 |
| [28] | Petrogradsky, V.M.: Codimension growth of strong Lie nilpotent associative algebras. Commun. Algebra 39, 918-928 (2011) · Zbl 1226.16017 · doi:10.1080/00927870903527592 |
| [29] | Razmyslov, Yu.P.: Algebras satisfying identity relations of Capelli type (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 45, 143-166, 240 (1981) · Zbl 0465.16009 |
| [30] | Razmyslov, Yu.P.: Identities of algebras and their representations. Sovremennaya Algebra. [Modern Algebra] “Nauka”, Moscow, 432 pp. Translations of Mathematical Monographs, vol. 138. American Mathematical Society, Providence. xiv+318 pp. (1994) · Zbl 0827.17001 |
| [31] | Regev, A.: The representation of Sn and explicit identities of PI-algebra. J. Algebra 51, 25-40 (1978) · Zbl 0374.16009 · doi:10.1016/0021-8693(78)90133-3 |
| [32] | Rowen, L.H.: Polynomial identities in ring theory. Academic Press Pure and Applied Mathematics, vol. 84 (1980) · Zbl 0461.16001 |
| [33] | Stanley, RP.: Enumerative Combinatorics. Cambridge University Press, Cambridge (1999) · Zbl 0928.05001 · doi:10.1017/CBO9780511609589 |
| [34] | Volichenko, I. B.: Varieties of Lie algebras with identity [[X1, X2, X3], [X4, X5, X6]] = 0 over a field of characteristic zero (Russian). Sibirsk. Mat. Zh. 25(3), 40-54 (1984) · Zbl 0575.17006 |
| [35] | Zaicev, M.V.: Integrality of exponents of growth of identities of finite-dimensional Lie algebras (Russian). Izv. Ross. Akad. Nauk. Ser. Mat. 66, 23-48 (2002). translation in Izv. Math. 66 (2002), 463-487 · Zbl 1057.17003 · doi:10.4213/im386 |
| [36] | Zubrilin, K.A.: Algebras that satisfy the Capelli identities (Russian). Mat. Sb. 186(3), 53-64 (1995). translation in Sb. Math. 186 (1995), no. 3, 359-370 · Zbl 0854.17003 |
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