×

Algebras with intermediate growth of the codimensions. (English) Zbl 1111.16022

Let \(F\) be a field of characteristic zero and let \(A\) be a (not necessarily associative) \(F\)-algebra. By analogy with associative PI-algebras, one introduces the sequence \(c_n(A)\) of codimensions of the multilinear polynomial identities of \(A\), the \(S_n\)-cocharacter \(\chi_n(A)=\sum_{\lambda\vdash n}m_\lambda\chi_\lambda\), and the colength \(l_n(A)=\sum_{\lambda\vdash n}m_\lambda\). A valuable information on the combinatorial and structural properties of \(A\) can be obtained from the asymptotic behaviour of \(c_n(A)\), \(\chi_n(A)\), \(l_n(A)\). The authors show that for finite dimensional nonassociative algebras the colength sequence of \(A\) is polynomially bounded and the codimension sequence cannot have intermediate growth. Then the authors construct a series of examples of nonassociative algebras with intermediate growth of the codimensions. They show that for any real number \(0<\beta<1\), there exists an algebra \(A\) whose sequence of codimensions grows like \(n^{n^\beta}\).

MSC:

16R10 \(T\)-ideals, identities, varieties of associative rings and algebras
17A30 Nonassociative algebras satisfying other identities
17B01 Identities, free Lie (super)algebras
16P90 Growth rate, Gelfand-Kirillov dimension
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Bahturin, Y.; Drensky, V., Graded polynomial identities of matrices, Linear algebra appl., 357, 15-34, (2002) · Zbl 1019.16011
[2] Berele, A.; Regev, A., Applications of hook Young diagrams to PI algebras, J. algebra, 82, 2, 559-567, (1983) · Zbl 0517.16013
[3] Giambruno, A.; Mishchenko, S., On star-varieties with almost polynomial growth, Algebra colloq., 8, 1, 33-42, (2001) · Zbl 1007.16016
[4] Giambruno, A.; Mishchenko, S.; Zaicev, M., Polynomial identities on superalgebras and almost polynomial growth, Special issue dedicated to alexei ivanovich kostrikin, Comm. algebra, 29, 9, 3787-3800, (2001) · Zbl 1006.16024
[5] A. Giambruno, S. Mishchenko, M. Zaicev, Codimensions of algebras and growth functions, Preprint No. 264, Dipt. Mat. Appl., Università di Palermo (2004) 1-19 · Zbl 1133.17001
[6] Giambruno, A.; Regev, A.; Zaicev, M., On the codimension growth of finite-dimensional Lie algebras, J. algebra, 220, 2, 466-474, (1999) · Zbl 0938.17004
[7] Giambruno, A.; Regev, A.; Zaicev, M., Simple and semisimple Lie algebras and codimension growth, Trans. amer. math. soc., 352, 4, 1935-1946, (2000) · Zbl 0970.17004
[8] Giambruno, A.; Zaicev, M., On codimension growth of finitely generated associative algebras, Adv. math., 140, 145-155, (1998) · Zbl 0920.16012
[9] Giambruno, A.; Zaicev, M., Exponential codimension growth of PI-algebras: an exact estimate, Adv. math., 142, 221-243, (1999) · Zbl 0920.16013
[10] Giambruno, A.; Zaicev, M., Polynomial identities and asymptotic methods, Math. surveys monogr., vol. 122, (2005), Amer. Math. Soc. Providence, RI · Zbl 1105.16001
[11] James, G.; Kerber, A., The representation theory of the symmetric group, Encyclopedia math. appl., vol. 16, (1981), Addison-Wesley London
[12] Kemer, A., T-ideals with power growth of the codimensions are Specht, Sibirsk. mat. zh., Siberian math. J., 19, 37-48, (1978), (in Russian), translation in · Zbl 0411.16014
[13] Mishchenko, S., On varieties of Lie algebras of intermediate growth, Vestsī akad. navuk BSSR ser. Fīz.-mat. navuk, 126, 2, 42-45, (1987), (in Russian) · Zbl 0633.17013
[14] Mishchenko, S., Lower bounds on the dimensions of irreducible representations of symmetric groups and of the exponents of the exponential of varieties of Lie algebras, Mat. sb., Sb. math., 187, 81-92, (1996), (in Russian), translation in · Zbl 0873.17007
[15] Mishchenko, S.; Zaicev, M., Asymptotic behaviour of colength of varieties of Lie algebras, Serdica math. J., 26, 2, 145-154, (2000) · Zbl 0954.17004
[16] Petrogradsky, V., Growth of polynilpotent varieties of Lie algebras, and rapidly increasing entire functions, Mat. sb., Sb. math., 188, 6, 913-931, (1997), (in Russian), translation in · Zbl 0890.17002
[17] Procesi, C., Rings with polynomial identities, Pure appl. math., vol. 17, (1973), Dekker New York · Zbl 0262.16018
[18] Regev, A., Existence of identities in \(A \otimes B\), Israel J. math., 11, 131-152, (1972) · Zbl 0249.16007
[19] Robbins, H., A remark on Stirling’s formula, Amer. math. monthly, 62, 26-29, (1955) · Zbl 0068.05404
[20] Volichenko, I.B., Varieties of Lie algebras with identity \([[X_1, X_2, X_3], [X_4, X_5, X_6]] = 0\) over a field of characteristic zero, Sibirsk. mat. zh., 25, 3, 40-54, (1984), (in Russian)
[21] Zaicev, M.; Mishchenko, S., A criterion for the polynomial growth of varieties of Lie superalgebras, Izv. ross. akad. nauk ser. mat., Izv. math., 62, 5, 953-967, (1998), (in Russian), translation in · Zbl 0915.17004
[22] Zaicev, M.; Mishchenko, S., An example of a variety of Lie algebras with a fractional exponent, J. math. sci. (NY), 93, 6, 977-982, (1999), Algebra, vol. 11 · Zbl 0933.17004
[23] Zaicev, M., Integrality of exponents of growth of identities of finite-dimensional Lie algebras, Izv. ross. akad. nauk ser. mat., Izv. math., 66, 463-487, (2002), (in Russian), translation in · Zbl 1057.17003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.