Codimension growth of two-dimensional non-associative algebras
Authors:
A. Giambruno, S. Mishchenko and M. Zaicev
Journal:
Proc. Amer. Math. Soc. 135 (2007), 3405-3415
MSC (2000):
Primary 17A50, 16R10; Secondary 16P90
DOI:
https://doi.org/10.1090/S0002-9939-07-08673-X
Published electronically:
August 2, 2007
MathSciNet review:
2336552
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Let be a field of characteristic zero and let
be a two- dimensional non-associative algebra over
. We prove that the sequence
of codimensions of
is either bounded by
or grows exponentially as
. We also construct a family of two-dimensional algebras indexed by rational numbers with distinct T-ideals of polynomial identities and whose codimension sequence is
,
.
- 1. Yuri Bahturin and Vesselin Drensky, Graded polynomial identities of matrices, Linear Algebra Appl. 357 (2002), 15–34. MR 1935223, https://doi.org/10.1016/S0024-3795(02)00356-7
- 2. A. Giambruno, S. Mishchenko and M. Zaicev, Codimensions of algebras and growth functions, (preprint) U. of Palermo, Dipartimento di Matematica ed Applicazioni, Preprint N. 274 (2005).
- 3. A. Giambruno, S. Mishchenko, and M. Zaicev, Algebras with intermediate growth of the codimensions, Adv. in Appl. Math. 37 (2006), no. 3, 360–377. MR 2261178, https://doi.org/10.1016/j.aam.2005.02.005
- 4. A. Giambruno and M. Zaicev, On codimension growth of finitely generated associative algebras, Adv. Math. 140 (1998), no. 2, 145–155. MR 1658530, https://doi.org/10.1006/aima.1998.1766
- 5. A. Giambruno and M. Zaicev, Exponential codimension growth of PI algebras: an exact estimate, Adv. Math. 142 (1999), no. 2, 221–243. MR 1680198, https://doi.org/10.1006/aima.1998.1790
- 6. Antonio Giambruno and Mikhail Zaicev, Polynomial identities and asymptotic methods, Mathematical Surveys and Monographs, vol. 122, American Mathematical Society, Providence, RI, 2005. MR 2176105
- 7. Gordon James and Adalbert Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley Publishing Co., Reading, Mass., 1981. With a foreword by P. M. Cohn; With an introduction by Gilbert de B. Robinson. MR 644144
- 8. A. R. Kemer, The Spechtian nature of 𝑇-ideals whose condimensions have power growth, Sibirsk. Mat. Ž. 19 (1978), no. 1, 54–69, 237 (Russian). MR 0466190
- 9. S. P. Mishchenko, On varieties of Lie algebras of intermediate growth, Vestsī Akad. Navuk BSSR Ser. Fīz.-Mat. Navuk 2 (1987), 42–45, 126 (Russian, with English summary). MR 898426
- 10. S. P. Mishchenko, 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. 187 (1996), no. 1, 83–94 (Russian, with Russian summary); English transl., Sb. Math. 187 (1996), no. 1, 81–92. MR 1380205, https://doi.org/10.1070/SM1996v187n01ABEH000101
- 11. V. M. Petrogradskiĭ, Growth of polynilpotent varieties of Lie algebras, and rapidly increasing entire functions, Mat. Sb. 188 (1997), no. 6, 119–138 (Russian, with Russian summary); English transl., Sb. Math. 188 (1997), no. 6, 913–931. MR 1479133, https://doi.org/10.1070/SM1997v188n06ABEH000232
- 12. Amitai Regev, Existence of identities in 𝐴⊗𝐵, Israel J. Math. 11 (1972), 131–152. MR 314893, https://doi.org/10.1007/BF02762615
- 13. M. V. Zaicev and S. P. Mishchenko, An example of a variety of Lie algebras with a fractional exponent, J. Math. Sci. (New York) 93 (1999), no. 6, 977–982. Algebra, 11. MR 1698766, https://doi.org/10.1007/BF02366352
- 14. M. V. Zaĭtsev, Integrality of exponents of growth of identities of finite-dimensional Lie algebras, Izv. Ross. Akad. Nauk Ser. Mat. 66 (2002), no. 3, 23–48 (Russian, with Russian summary); English transl., Izv. Math. 66 (2002), no. 3, 463–487. MR 1921808, https://doi.org/10.1070/IM2002v066n03ABEH000386
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 17A50, 16R10, 16P90
Retrieve articles in all journals with MSC (2000): 17A50, 16R10, 16P90
Additional Information
A. Giambruno
Affiliation:
Dipartimento di Matematica e Applicazioni, Via Archirafi 34, 90123 Palermo, Italia
Email:
agiambr@unipa.it
S. Mishchenko
Affiliation:
Department of Algebra and Geometric Computations, Faculty of Mathematics and Mechanics, Ulyanovsk State University, Ulyanovsk 432700, Russia
Email:
mishchenkosp@.ulsu.ru
M. Zaicev
Affiliation:
Department of Algebra, Faculty of Mathematics and Mechanics, Moscow State University, Moscow, 119992 Russia
Email:
zaicev@mech.math.msu.su
DOI:
https://doi.org/10.1090/S0002-9939-07-08673-X
Keywords:
Polynomial identity,
codimension growth
Received by editor(s):
August 27, 2005
Received by editor(s) in revised form:
February 9, 2006
Published electronically:
August 2, 2007
Additional Notes:
The first author was partially supported by MIUR of Italy; the second author was partially supported by RFFI, grant 01-01-00739 and UR 04.01.036; the third author was partially supported by SSH, grant 1910.2003.1.
Communicated by:
Martin Lorenz
Article copyright:
© Copyright 2007
American Mathematical Society