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
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
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.
- Allan Berele, Generic verbally prime PI-algebras and their GK-dimensions, Comm. Algebra 21 (1993), no. 5, 1487β1504. MR 1213968, DOI https://doi.org/10.1080/00927879308824632
- Allan Berele and Amitai Regev, On the codimensions of the verbally prime P.I. algebras, Israel J. Math. 91 (1995), no. 1-3, 239β247. MR 1348314, DOI https://doi.org/10.1007/BF02761648
- A. Berele and A. Regev, Codimensions of products and of intersections of verbally prime $T$-ideals, Israel J. Math. 103 (1998), 17β28. MR 1613536, DOI https://doi.org/10.1007/BF02762265
- Veselin S. Drenski, Extremal varieties of algebras. I, Serdica 13 (1987), no. 4, 320β332 (Russian). MR 929452
- Veselin S. Drenski, Extremal varieties of algebras. II, Serdica 14 (1988), no. 1, 20β27 (Russian). MR 944480
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. 97β113.
- A. Giambruno and M. Zaicev, On codimension growth of finitely generated associative algebras, Adv. Math. 140 (1998), no. 2, 145β155. MR 1658530, DOI https://doi.org/10.1006/aima.1998.1766
- A. Giambruno and M. Zaicev, Exponential codimension growth of PI algebras: an exact estimate, Adv. Math. 142 (1999), no. 2, 221β243. MR 1680198, DOI https://doi.org/10.1006/aima.1998.1790
giazai3 A. Giambruno and M. Zaicev, A characterization of algebras with polynomial growth of the codimensions, Proc. Amer. Math. Soc. (to appear).
- A. R. Kemer, The Spechtian nature of $T$-ideals whose condimensions have power growth, Sibirsk. Mat. Ε½. 19 (1978), no. 1, 54β69, 237 (Russian). MR 0466190
- Aleksandr Robertovich Kemer, Ideals of identities of associative algebras, Translations of Mathematical Monographs, vol. 87, American Mathematical Society, Providence, RI, 1991. Translated from the Russian by C. W. Kohls. MR 1108620
- Jacques Lewin, A matrix representation for associative algebras. I, II, Trans. Amer. Math. Soc. 188 (1974), 293β308; ibid. 188 (1974), 309β317. MR 338081, DOI https://doi.org/10.1090/S0002-9947-1974-0338081-5
- Claudio Procesi, Non-commutative affine rings, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. Ia (8) 8 (1967), 237β255 (English, with Italian summary). MR 224657
- Amitai Regev, Existence of identities in $A\otimes B$, Israel J. Math. 11 (1972), 131β152. MR 314893, DOI https://doi.org/10.1007/BF02762615
- Amitai Regev, Codimensions and trace codimensions of matrices are asymptotically equal, Israel J. Math. 47 (1984), no. 2-3, 246β250. MR 738172, DOI https://doi.org/10.1007/BF02760520
ber A. Berele, Generic verbally prime PI-algebras and their GK-dimension, Comm. Algebra 21 (1993), 1478β1504.
br1 A. Berele and A. Regev, On the codimensions of the verbally prime P.I. algebras, Israel J. Math. 91 (1995), 239β247.
br2 A. Berele and A. Regev, Codimensions of products and intersections of verbally prime T-ideals, Israel J. Math. 103 (1998), 17β28.
dren1 V. Drensky, Extremal varieties of algebras I, Serdica 13 (1987), 320β332. (Russian)
dren2 V. Drensky, Extremal varieties of algebras II, Serdica 14 (1988), 20β27. (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. 97β113.
giazai1 A. Giambruno and M. Zaicev On codimension growth of finitely generated associative algebras, Adv. Math. 140 (1998), 145β155.
giazai2 A. Giambruno and M. Zaicev, Exponential codimension growth of PI-algebras: an exact estimate, Adv. Math. 142 (1999), 221β243.
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), 54β69; English translation: Siberian Math. J. 19 (1978), 37β48.
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), 293β308.
pro C. Procesi, Non-commutative affine rings, Atti Acc. Naz. Lincei, Ser. VIII 8 (1967), 239β255.
reg A. Regev, Existence of identities in $A \otimes B$, Israel J. Math. 11 (1972), 131β152.
reg2 A. Regev, Codimensions and trace codimensions of matrices are asymptotically equal, Israel J. Math. 47 (1984), 246β250.
Similar Articles
Retrieve articles in Electronic Research Announcements of the American Mathematical Society
with MSC (2000):
16R10,
16P90
Retrieve articles in all journals
with MSC (2000):
16R10,
16P90
Additional Information
A. Giambruno
Affiliation:
Dipartimento di Matematica ed Applicazioni, UniversitΓ 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
