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 Mat. Zametki, 2000, Volume 68, Issue 6, Pages 887–897 (Mi mz1012)

On the Complexity Functions for $T$-Ideals of Associative Algebras

Ulyanovsk State University

Abstract: Let $c_n(\mathbf V)$ be the sequence of codimension growth for a variety $\mathbf V$ of associative algebras. We study the complexity function $\mathscr C(\mathbf V,z)=\sum_{n=0}^\infty c_n(\mathbf V)z^n/n!$, which is the exponential generating function for the sequence of codimensions. Earlier, the complexity functions were used to study varieties of Lie algebras. The objective of the note is to start the systematic investigation of complexity functions in the associative case. These functions turn out to be a useful tool to study the growth of varieties over a field of arbitrary characteristic. In the present note, the Schreier formula for the complexity functions of one-sided ideals of a free associative algebra is found. This formula is applied to the study of products of $T$-ideals. An exact formula is obtained for the complexity function of the variety $\mathbf U_c$ of associative algebras generated by the algebra of upper triangular matrices, and it is proved that the function $c_n(\mathbf U_c)$ is a quasi-polynomial. The complexity functions for proper identities are investigated. The results for the complexity functions are applied to study the asymptotics of codimension growth. Analogies between the complexity functions of varieties and the Hilbert–Poincaré series of finitely generated algebras are traced.

DOI: https://doi.org/10.4213/mzm1012

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English version:
Mathematical Notes, 2000, 68:6, 751–759

Bibliographic databases:

UDC: 512.55

Citation: V. M. Petrogradsky, “On the Complexity Functions for $T$-Ideals of Associative Algebras”, Mat. Zametki, 68:6 (2000), 887–897; Math. Notes, 68:6 (2000), 751–759

Citation in format AMSBIB
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This publication is cited in the following articles:
1. Drensky V., “Polynomial Identity Rings - Part a - Combinatorial Aspects in Pi-Rings”, Polynomial Identity Rings, Advanced Courses in Mathematics CRM Barcelona, Birkhauser Verlag Ag, 2004, 1+
2. Petrogradsky, VM, “Enumeration of algebras close to absolutely free algebras and binary trees”, Journal of Algebra, 290:2 (2005), 337
3. Giambruno, A, “Matrix algebras of polynomial codimension growth”, Israel Journal of Mathematics, 158:1 (2007), 367
4. Mishchenko, SP, “Poisson PI algebras”, Transactions of the American Mathematical Society, 359:10 (2007), 4669
5. Giambruno, A, “Proper identities, Lie identities and exponential codimension growth”, Journal of Algebra, 320:5 (2008), 1933
6. Giambruno A., La Mattina D., “Graded Polynomial Identities and Codimensions: Computing the Exponential Growth”, Adv. Math., 225:2 (2010), 859–881
7. S. M. Ratseev, “Identities in the varieties generated by the algebras of upper triangular matrices”, Siberian Math. J., 52:2 (2011), 329–339
8. Aljadeff E., Giambruno A., La Mattina D., “Graded Polynomial Identities and Exponential Growth”, J. Reine Angew. Math., 650 (2011), 83–100
9. Petrogradsky V.M., “Codimension Growth of Strong Lie Nilpotent Associative Algebras”, Commun. Algebr., 39:3 (2011), 918–928
10. Boumova S., Drensky V., “Algebraic Properties of Codimension Series of Pi-Algebras”, Isr. J. Math., 195:2 (2013), 593–611
11. Giambruno A., Souza Manuela da Silva, “Minimal Varieties of Graded Lie Algebras of Exponential Growth and the Special Lie Algebra $sl_2$”, J. Pure Appl. Algebr., 218:8 (2014), 1517–1527
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