Minimal varieties of algebras of exponential growth
Authors:
A. Giambruno and M. Zaicev
Journal:
Electron. Res. Announc. Amer. Math. Soc. 6 (2000), 40-44
MSC (2000):
Primary 16R10, 16P90
DOI:
https://doi.org/10.1090/S1079-6762-00-00078-0
Published electronically:
June 6, 2000
MathSciNet review:
1767635
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Abstract: The exponent of a variety of algebras over a field of characteristic zero has been recently proved to be an integer. Through this scale we can now classify all minimal varieties of a given exponent and of finite basic rank. As a consequence we describe the corresponding T-ideals of the free algebra, and we compute the asymptotics of the related codimension sequences. We then verify in this setting some known conjectures.
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br1 A. Berele and A. Regev, On the codimensions of the verbally prime P.I. algebras, Israel J. Math. 91 (1995), 239–247.
br2 A. Berele and A. Regev, Codimensions of products and intersections of verbally prime T-ideals, Israel J. Math. 103 (1998), 17–28.
dren1 V. Drensky, Extremal varieties of algebras I, Serdica 13 (1987), 320–332. (Russian)
dren2 V. Drensky, Extremal varieties of algebras II, Serdica 14 (1988), 20–27. (Russian)
dren3 V. Drensky, Gelfand-Kirillov dimension of PI-algebras, In: Methods in Ring Theory, Lect. Notes in Pure and Appl. Math., vol. 198, Marcel Dekker, New York, 1998, pp. 97–113.
giazai1 A. Giambruno and M. Zaicev On codimension growth of finitely generated associative algebras, Adv. Math. 140 (1998), 145–155.
giazai2 A. Giambruno and M. Zaicev, Exponential codimension growth of PI-algebras: an exact estimate, Adv. Math. 142 (1999), 221–243.
giazai3 A. Giambruno and M. Zaicev, A characterization of algebras with polynomial growth of the codimensions, Proc. Amer. Math. Soc. (to appear).
kem1 A. Kemer, T-ideals with power growth of the codimensions are Specht, Sibirskii Matematicheskii Zhurnal, 19 (1978), 54–69; English translation: Siberian Math. J. 19 (1978), 37–48.
kem2 A. Kemer, Ideals of identities of associative algebras, AMS Translations of Mathematical Monographs, Vol. 87, 1988.
lew J. Lewin, A matrix representation for associative algebras. I, Trans. Amer. Math. Soc. 188 (1974), 293–308.
pro C. Procesi, Non-commutative affine rings, Atti Acc. Naz. Lincei, Ser. VIII 8 (1967), 239–255.
reg A. Regev, Existence of identities in $A \otimes B$, Israel J. Math. 11 (1972), 131–152.
reg2 A. Regev, Codimensions and trace codimensions of matrices are asymptotically equal, Israel J. Math. 47 (1984), 246–250.
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Additional Information
A. Giambruno
Affiliation:
Dipartimento di Matematica ed Applicazioni, Università di Palermo, 90123 Palermo, Italy
MR Author ID:
73185
ORCID:
0000-0002-3422-2539
Email:
a.giambruno@unipa.it
M. Zaicev
Affiliation:
Department of Algebra, Faculty of Mathematics and Mechanics, Moscow State University, Moscow 119899, Russia
MR Author ID:
256798
Email:
zaicev@mech.math.msu.su
Keywords:
Varieties of algebras,
polynomial identities
Received by editor(s):
October 4, 1999
Published electronically:
June 6, 2000
Additional Notes:
The first author was partially supported by MURST of Italy; the second author was partially supported by the RFBR grants 99-01-00233 and 96-15-96050.
Communicated by:
Efim Zelmanov
Article copyright:
© Copyright 2000
American Mathematical Society