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8
390
А. Белов-Канель, Л. Ровен, Цзе-Тай Юй
ЧЕБЫШЕВСКИИ СБОРНИК
Том 20. Выпуск 3.
УДК 512
DOI 10.22405/2226-8383-2019-20-3-390-393
Гипотеза Якобиана для свободной ассоциативной алгебры (произвольной характеристики)
Белов-Канель Алексей Яковлевич — доктор физико-математических наук, федеральный профессор математики, профессор, университет Бар-Илана (г. Рамат-Ган, Израиль), Колледж математики и статистики, Шэньчжэньский университет, Шэньчжэнь, 518061, Китай. e-mail: beloval@cs.biu.ac.il; kanelster@gmail.com,
Ровен Луи Хейл — факультет математики, университет Бар-Илан (Израиль). e-mail: rowen@math.biu.ac.il
Цзе-Тай Юй — профессор, МФТИ, факультет математики, университет Сенгэн (Китай). e-mail: jietai@hotmail.com
Целью данной работы является использование Р/-теории для упрощения результатов Дикса и Левина [4] об автоморфизмах свободной алгебры Р{X}, а именно: если якобиан обратим, тогда каждый эндоморфизм является эпиморфизмом. Результаты переносятся на широкий класс колец.
Ключевые слова: Автоморфизмы, полиномиальные алгебры, свободные ассоциативные алгебры.
Библиография: 9 названий. Для цитирования:
А. Белов-Канель, Л. Ровен, Цзе-Тай Юй. Гипотеза Якобиана для свободной ассоциативной алгебры (произвольной характеристики) // Чебышевский сборник, 2019, т. 20, вып. 3, с. 390-
А. Белов-Канель, Л. Ровен, Цзе-Тай Юй
Аннотация
393.
CHEBYSHEVSKII SBORNIK Vol. 20. No. 3.
UDC 512
DOI 10.22405/2226-8383-2019-20-3-390-393
The Jacobian Conjecture for the free associative algebra
(of arbitrary characteristic)
A. Belov-Kanel, L. Rowen and Jie-Tai Yu
Belov-Kanel Alexei Yakovlevich — doctor of physical and mathematical sciences, federal professor, professor, Bar-Ilan University (Ramat Gan, Israel), College of Mathematics and Statistics, Shenzhen University, Shenzhen, 518061, China. e-mail: heloval@cs.hiu.ac.il; kanelster@gmail.com,
Rowen Louis Haile — Department of Mathematics, Bar-Ilan University (Israel). e-mail: rowen@math.biu.ac.il
Jie-Tai Yu — professor, MIPT, Department of Mathematics, Sengeng University (China). e-mail: jietai@hotmail.com
Abstract
The object of this note is to use Pi-theory to simplify the results of Dicks and Lewin [4] on the automorphisms of the free algebra F[X}, namely that if the Jacobian is invertible, then every endomorphism is an epimorphism. We then show how the same proof applies to a somewhat wider class of rings.
Keywords: Automorphisms, polynomial algebras, free associative algebras.
Bibliography: 9 titles.
For citation:
A. Belov-Kanel, L. Rowen and Jie-Tai Yu, 2019, "The Jacobian Conjecture for the free associative algebra (of arbitrary characteristic)", Chebyshevskii sbornik, vol. 20, no. 3, pp. 390-393.
1. Introduction and main results
The object of this note is to use Pl-theorv to simplify the results of Dicks and Lewin [4] on the automorphisms of the free algebra F[X}, namely that if the Jacobian is invertible, then every endomorphism is an epimorphism. We then show how the same proof applies to a somewhat wider class of rings.
2. Hopfian rings
Definition 1. An algebra R is Hopfian if every epimorphism, (i.e., onto algebra homomor-phism) R ^ R is an isomorphism.
Dicks and Lewin [4, Proposition 3.1] proved that an endomorphism of the free associative algebra F[X} is an epimorphism iff its Jacobian matrix is invertible. In this way, they reduced the Jacobian conjecture for F[X} to the question of whether F[X} is Hopfian, and proved it for the free algebra in two variables. In fact, this had already been resolved for any finite set of variables by Orzech and Ribes [6], with a more direct proof given in [3]. Also see [9] for a treatment of the Jacobian conjecture over a free algebra, and fl] for an overview of Yagzev's method to attack the Jacobian conjecture.
In this section we give a quick proof of the fact that the free associative algebra F[X} is Hopfian, relying on considerations of growth, with a generalization obtained from the proof. . Recall that the Gelfand-Kirillov dimension GKdim(A) of an affine algebra A = F[a\,..., ai} is
GKdim(A) := lim logn dn, (1)
where An = ^ Fa¿1 ■ ■ ■ ain and dn = dim^ An.
The standard reference on Gelfand-Kirillov dimension is [5] Although the dn depend on the choice of the generating set a\,..., at, GKdim(A) is independent of the choice of the generating set. We can tighten this fact a bit: Suppose that A' = F[a[,..., a'e} and d'n = dim^ A'n. We say
392
A. BejiOB-Kanejib, il. Pobbh, U^e-Taft K)h
that the growth rate of the dn is less than or equal to the growth rate of the d'n if there are constants c, k such that d'n ^ cdkn- This defines an equivalence, and it is easy to see that the growth rate of A with respect to any two sets of generators is the same.
Lemma 1. Suppose R is an affine algebra in which the growth of R/I is less than the growth of R, for each ideal 1 of R. Then R is Hopfian.
In particular, if GKdim(R/I) < GKdim(E) for all ideals I of R, then R is Hopfian.
proof. For any epimorphism tp : R ^ R, one has <p(R) = R/kerp, but then <p(R) and R have the same growth rates, implying ker^> = 0. □
The hypothesis of Lemma 1 holds for prime Pi-algebras, cf. [2, Theorem 11.2.12], so we have:
Corollary 1. Any prime affine Pi-algebra is Hopfian.
Remark 1. R and R/I could have different growth rates even if GKdim(R/I) < GKdim(E). For example, let R be the subalgebra of the free associative algebra generated by all subwords of un for any n, where u1 = xyx and un+1 = x1 unx10 yx10 unx10 , a prime algebra, o/GKdim2, and I be the ideal generated by all words of degree 2 in y. Then GKdim(E//) = 2, although the growth rate of R/1 is less than that of R. This example is not a Pi-algebra.
A T-ideal of an ideal R is an ideal invariant under all ring endomorphisms.
Lemma 2. If X is a T-ideal of R, then any endomorphism p of R clearly induces an endomorphism of R/X.
Proof. Define <p : R/X ^ R/X bv <p(a + X) = tp(a) + X. This is well-defined since p(X) C X by □
Theorem 1 (f6j). When X is a finite set of noncommuting indeterminates, the free associative algebra F{X} is Hopfian.
proof. Let <p : F{X} ^ F{X} be an epimorphism, with some nonzero polynomial f £ ker(^). Let n = deg(/). Let Xra be the T-ideal of identities of the algebra of generic n x n matrices. Then p ^^^^^ an endomorphism of A : F{X}/Xn, whose kernel does not contain f, since the easy part of the Amitsur-Levitzki theorem says that the degree of any identity of n x n matrices is at least 2n > n. ^^^s ^te epimorphism induced by p has non-zero kernel, contradicting Lemma 1. □
The same idea of proof yields a stronger result. We say that R is T-residually Hopfian if the intersection of those T-ideals / of R fa which R/I is Hopfian is 0. Examples include almost
GKdim 2.
T
proof. Let <p : R ^ E be an epimorphism, with some nonzero polynomial f £ ker(^). Bv
hypothesis there is some ^^^al X not containing r, but Lemma 2 implies that R/X is not Hopfian, □
Corollary 2.3 belongs to Alexei Kanel-Belov, his work was supported by the Russian Science Foundation under grant 17-11-01377. Louis Rowen was supported by ISF grant N 1623/16. Bar-Ilan University, Mipt, Shengeng University
Conclusions. In the paper we show that some ideas from P/-theorv can be used for polynomial authomorphisms. Note that many specialists in P/-theorv got different results in this arrear.
REFERENCES
1. Belov, A., Bokut, L., Rowen, L., and Yu, J.-T., The Jacobian Conjecture, together with Specht and Burnside-type problems, Automorphisms in Birational and Affine Geometry (Bellavista Relax Hotel, Levico Terme -Trento, October 29th - November 3rd, 2012, Italy), Springer Proceedings in Mathematics k, Statistics, 79, Springer Verlag, 2014, 249-285, ISBN 978-3-31905681-4, http://link.springer.com/chapter/10.1007/978-3-319-05681-4^15, arXiv: 1308.0674
2. Belov, A. and Rowen, L.H. Computational Aspects of Polynomial Identities, Research Notes in Mathematics 9, AK Peters, 2005.
3. Cohn, P.M., Free Ideal Rings and Localization
4. Dicks, W. and Lewin, J., A jacobian conjecture for free associative algebras, Communications in Algebra 10:12 (1982) 1285-1306.
5. Krause, G.R., and Lenagan, T.H., Growth of Algebras and Gelfand-Kirillov Dimension, Amer. Math. Soc. Graduate Studies in Mathematics 22 (2000).
6. Orzech, M. and Ribes, L., Residual Finiteness and the Hopf Property in Rings, Journal of Algebra 15 (1970), 81-88.
7. Orzech, M., Onto endomorphisms are isomorphosms, Amer. Math. Monthly 78 (1971), 357-362.
8. Rowen, L.H. and Small, L., Representable rings and GK dimension (2015).
9. Schofield, A., Representations of rings over skew fields, LMS Lecture note series 92 1985, 223 pages.
Получено 16.10.2019 г. Принято в печать 12.11.2019 г.
