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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 21. Выпуск 1.
УДК 512.7 DOI 10.22405/2226-8383-2020-21-1-51-61
Некоммутативная теорема Бялыницкого — Бирули1
А. Я. Белов-Канель, А. М. Елишев, Ф. Разавиниа, Ю Джи-Тай, Венчао Жэнг
Белов-Канель Алексей Яковлевич — доктор физико-математических наук, профессор, Колледж математики и статистики, Шэньчжэньский университет (г. Шэньчжэнь). e-mail: kanel@mccme.ru
Елишев Андрей Михайлович — кандидат физико-математических наук, ассистент кафедры дискретной математики и лаборатории продвинутой комбинаторики и сетевых приложений, Московский физико-технический институт (Национальный исследовательский университет) (г. Москва). e-mail: elishev@phystech.edu
Разавиниа Фаррох — факультет математики, Университет Порту, Московский физико-технический институт (Национальный исследовательский университет) (г. Порту). e-mail: f.razavinia@phystech.edu,
К) Джи-Тай — Ph.D., профессор, Колледж математики и статистики, Шэньчжэньский университет (г. Шэньчжэнь). e-mail: jietaiyu@szu.edu.cn
Венчао Жэнг — факультет математики, Университет Бар-План (г. Рамат-Ган). e-mail: whzecomjm@gmail.com
Аннотация
Изучение действий алгебраических групп на алгебраических многообразиях и их координатных алгебрах является важной областью исследований в алгебраической геометрии и теории колец. Эта область связана с теорией полиномиальных отображений, ручных и диких автоморфизмов, проблемой якобиана, теорией бесконечномерных многообразий по Шафаревичу, проблемой сокращения (вместе с другими подобными вопросами), теорией локально нильпотентных дифференцирований. Одной из центральных задач теории действий алгебраических групп является проблема линеаризации, изученная в работе Т. Кам-баяши и П. Расселла, утверждающая, что всякое действие тора на аффинном пространстве линейно в некоторой системе координат. Гипотеза о линеаризации была основана на хорошо известной классической теореме А. Бялыницкого — Бирули, которая гласит, что всякое эффективное регулярное действие тора максимальной размерности на аффинном пространстве над алгебраически замкнутым полем допускает линеаризацию.
Несмотря на то что гипотеза о линеаризации оказалась отрицательной в ее общем вн-де — контрпримеры в положительной характеристике были построены Т. Асанума — теорема Бялыницкого — Бирули остается важным результатом теории благодаря ее связи с теорией полиномиальных автоморфизмов. Недавние продвижения в последней мотивировали поиск различных некоммутативных разновидностей теоремы Бялыницкого — Бирули. В данной статье мы приведем доказательство теоремы о линеаризации эффективного действия максимального тора автоморфизмами свободной ассоциативной алгебры, являющейся таким образом свободным аналогом теоремы Бялыницкого — Бирули.
Ключевые слова: действия тора, задача линеаризации, полиномиальные автоморфизмы.
Библиография: 16 названий.
1 Исследование выполнено за счет гранта Российского научного фонда (проект 17-11-01377).
Для цитирования:
А. Я. Белов-Канель, А. М. Елишев, Ф. Разавиниа, Ю Джи-Тай, Венчао Жэнг Некоммутативная теорема Бялыницкого — Бирули // Чебышевский сборник, 2020, т. 21, вып. 1, с. 51-61.
CHEBYSHEVSKII SBORNIK Vol. 21. No. 1.
UDC 512.7 DOI 10.22405/2226-8383-2020-21-1-51-61
Noncommutative Bialynicki^Birula Theorem
A. Ya. Belov-Kanel, A. M. Elishev, F. Razavinia, Yu Jie-Tai, Wenchao Zhang
Belov-Kanel Alexei Yakovlevich — College of Mathematics and Statistics, Shenzhen University (Shenzhen).
e-mail: kanel@mccme.ru
Elishev Andrey Mikhailovich — Department of Discrete Mathematics, Moscow Institute of Physics and Technology (Moscow). e-mail: elishev@phystech.edu
Razavinia Farrokh — Department of Mathematics, University of Porto, Praga de Gomes Teixeira (Porto).
e-mail: f.razavinia@phystech.edu,
Yu Jie-Tai — College of Mathematics and Statistics, Shenzhen University (Shenzhen). e-mail: jietaiyu@szu.edu.cn
Wenchao Zhang — Mathematics Department, Bar-Ilan University (Ramat-Gan). e-mail: whzecomjm@gmail.com
Abstract
The study of algebraic group actions on varieties and coordinate algebras is a major area of research in algebraic geometry and ring theory. The subject has its connections with the theory of polynomial mappings, tame and wild automorphisms, the Jacobian conjecture of O.-H. Keller, infinite-dimensional varieties according to Shafarevich, the cancellation problem (together with various cancellation-type problems), the theory of locally nilpotent derivations, among other topics. One of the central problems in the theory of algebraic group actions has been the linearization problem, formulated and studied in the work of T. Kambayashi and P. Russell, which states that any algebraic torus action on an affine space is always linear with respect to some coordinate system. The linearization conjecture was inspired by the classical and well known result of A. Bialynicki-Birula, which states that every effective regular torus action of maximal dimension on the affine space over algebaically closed field is linearizable.
Although the linearization conjecture has turned out negative in its full generality, according to, among other results, the positive-characteristic counterexamples of T. Asanuma, the Bialynicki-Birula has remained an important milestone of the theory thanks to its connection to the theory of polynomial automorphisms. Recent progress in the latter area has stimulated the search for various noncommutative analogues of the Bialynicki-Birula theorem. In this paper, we give the proof of the linearization theorem for effective maximal torus actions by automorphisms of the free associative algebra, which is the free analogue of the Bialynicki-Birula theorem.
Keywords: torus actions, linearization problem, polynomial automorphisms.
Bibliography: 16 titles.
For citation:
A. Ya. Belov-Kanel, A. M. Elishev, F. Razavinia, Yu Jie-Tai, Wenchao Zhang, 2020, "Noncommutative Bialvnicki-Birula Theorem" , Chebyshevskii sbornik, vol. 21, no. 1, pp. 51-61.
1. Introduction
In this paper we consider algebraic torus actions on the affine space, according to Bialvnicki-Birula, and formulate certain noncommutative generalizations.
We begin by recalling a few basic definitions. Let K be an algebraically closed field.
Definition 1. An algebraic group is a variety G equipped with the structure of a group, such that the multiplication map m : G x G ^ G : (gi,g2) ^ g\g2 and the inverse map t : G ^ G : g ^ g~l are morphisms of varieties.
Definition 2. A G-variety is a variety equipped with an action of the algebraic group G,
a : G x X ^ X : (g,x) ^ g ■ x, which is also a morphism of varieties. We then say that a is an algebraic G-action.
Let K be our ground field, which is assumed to be algebraically closed. Let Z = {z\, z2,...} = = {zi : i £ I} be a finite or a countable set of variables (where I = {1,2,...} is an index set), and let Z* denote the free semigroup generated by Z, Z+ = Z*\{1}. Moreover let Fj(K) = K (Z) be the free associative K-algebra and Fj(K) = K ((Z)) be the algebra of formal power series in free variables.
Denote by W = (Z) the free monoid of words over the alphabet Z (with 1 as the empty word) such that |W| ^ ^^r |W| ^te length of the word W £ Z+.
For an alphabet Z, the free associative ^^^^^ra on Z is
K (Z) := ®wez*KW,
where the multiplication is K-bilinear extension of the concatenation on words, Z* denotes the free monoid on Z, and KW the free K-module on one element, the word W. Any element
of K (Z) can thus be written uniquely in the form
£
E ■
k=0 ii,...,i^El
where the coefficients a^,^,...,^ are elements of the field K and all but finitely many of these elements are zero.
In our context, the alphabet Z is the same as the set of algebra generators, therefore the terms "monomial"and "word"will be used interchangeably.
In the sequel, we employ a (slightly ambiguous) short-hand notation for a free algebra monomial. For an element z, its powers are defined as usual. Any monomial Zi1 Zi2 ... Zik can then be written in a reduced form with subwords zz ... z replaced by powers. We then write
~l = ~h Jk
* = 2 ...zjk
where by I we mean an assignment of ik to jk in the word z1. Sometimes we refer to I as a multiindex, although the term is not entirely accurate. If I is such a multi-index, its abosulte value |/1 is defined as the sum i\ + ■ ■ ■ + ik-
For a field ^t Kx = K\{0} denote the multiplicative group of its non-zero elements viewed K
Definition 3. An n-dimensional algebraic K-torus is a group
Tn ~ (Kx)ra
il ,%2 ,...,ik ^il ^2 . . . ^ik
(with obvious multiplication).
Denote by An the affine space of dimension n over K. Definition 4. A (left) torus action is a morphism
a : Tra x An ^ An. that fulfills the usual axioms (identity and compatibility):
a(1,x) = x, a(t1,a(t2,x)) = a(t1t2,x). The action a is effective if for every t = 1 there is an element x <E An such that a(t,x) = x. In [3], Bialvnicki-Birula proved the following two theorems. Theorem 1. Any regular action ofTnon An has a fixed point.
Th eorem 2. Any effective and regular action ofTn on An is a representation in some coordinate system,.
The term "regular"is to be understood here as in the algebro-geometric context of regular ll
effective regular maximal torus action on the affine space is conjugate to a linear action, or, as it is sometimes called, linearizable.
An algebraic group action on An is the same as an action by automorphisms on the algebra
K[«i, ...,xn]
of global sections of the structure sheaf. In other words, it is a homomorphism
a : Tn ^ AutK[x1,... ,xn}.
An action is effective iff Ker a = {1}.
The polynomial algebra is a quotient of the free associative algebra
Fn = K(zi, ...,zn)
by the commutator ideal I (it is the two-sided ideal generated by all elements of the form fg — gf). From the standpoint of Noncommutative geometry, the algebra r(X, Ox) of global sections (along with the category of f.g. projective modules) contains all the relevant topological data of X, and various non-commutative algebras (Pi-algebras) may be thought of as global function algebras over
ll
is a subject of legitimate interest.
ll
latter is formulated as follows.
Theorem 3 (Main Theorem). Suppose given a regular action a of the algebraic n-torus Tn on the free algebra Fn. If a is effective, then it is linearizable.
The linearization problem, as it has become known since Kambavashi, asks whether all (effective, regular) actions of a given type of algebraic groups on the affine space of given dimension are conjugate to representations. According to Theorem 3, the linearization problem extends to the noncommutative category. Several known results concerning the (commutative) linearization problem are summarized below.
1. Any effective regular torus action on A2 is linearizable (Gutwirth [8]).
2. Any effective regular torus action on Ara has a fixed point (Bialvnicki-Birula [3]).
3. Any effective regular action of Tra-1 on Ara is linearizable (Bialvnicki-Birula [4]).
4. Any (effective, regular) one-dimensional torus action (i.e., action of Kx) on A3 is linearizable (Koras and Russell [14]).
5. If the ground field is not algebraically closed, then a torus action on An need not be linearizable. In fl], Asanuma proved that over any field K, if there exists a non-rectifiable closed embedding from Am into An, then there exist non-linearizable effective actions of (Kx )r on A1+ra+m for 1 < r < 1 + m.
6. When K is infinite and has positive characteristic, there are examples of non-linearizable torus actions on An (Asanuma fl]).
Remark 1. A closed embedding l : Am ^ An is said to be rectifiable if it is conjugate to a linear embedding by an automorphism of An.
As can be inferred from the review above, the context of the linearization problem is rather broad, even in the case of torus actions. The regulating parameters are the dimensions of the torus and the afline space. This situation is due to the fact that the general form of the linearization conjecture (i.e., the conjecture that states that any effective regular torus action on any afline space is linearizable) has a negative answer.
Transition to the noncommutative geometry presents the inquirer with an even broader context: one now may vary the dimensions as well as impose restrictions on the action in the form of preservation of the Pi-identities. Caution is well advised. Some of the results are generalized in a straightforward manner — the main theorem of this paper being the typical example, others require more subtlety and effort (cf. 2 and the discussion at the end of the note). Of some note to us, given our ongoing work in deformation quantization (see, for instance, [12]) is the following instance of the linearization problem, which we formulate as a conjecture.
Conjecture 1. Forn > 1,letPn denote the commutative Poisson algebra, i.e. the polynomial algebra
K[21, . . .,Z2n] equipped with the Poisson bracket defined by
{Zi,Zj } — .
Then any effective regular action ofTnbyautomorphisms of Pn is linearizable.
It is interesting to note that the context of Conjecture 1 admits a vague analogy in the real transcendental category (with Pn replaced by an appropriate algebra of smooth functions, cf. for instance the work of Zung [16]). Although the instances of the linearization problem we consider in
ll
nature, it may be worthwhile to search for analytic analogues of the real transcendental linearization (however whether this will give a feasible approach to Conjecture 1 is unclear, the hurdles being significant and fairly obvious).
Acknowledgments
We are grateful to I. V. Arzhantsev, V. L. Dolnikov, R.N. Karasev, V. O. Manturov, A. M. Raigo-rodskii, G. B. Shabat and N. A. Vavilov for stimulating discussions.
The main result of this note was conceived in the prior work [13] of A. K.-B., J.-T. Y. and A. E.. Theorem 3 is due to A. E. and A. K.-B.; Lemma 1 and the review of known results for the linearization problem is due to F. R., J.-T. Y. and W. Z..
F. R. is also is responsible for the investigation of possible transcendental analogies.
A. E. and A. K.-B. are supported by the Russian Science Foundation grant No. 17-11-01377.
F. R. is supported by the FCT (Foundation for Science and Technology of Portugal) scholarship with reference number PD/BD/142959/2018.
2. Proof of the Main Theorem
The proof proceeds along the lines of the original commutative case proof of Bialvnicki-Birula. If a is the effective action of Theorem 3, then for each t £ Tn the automorphism
a(t) : Fn ^ Fn
is given by the n-tuple of images of the generators zi,..., zn of the free algebra:
(fi (t , Zl, ..., zn
Each of the f1,..., fn is a polynomial in the free variables.
Lemma 1. There is a translation of the free generators
(zi ,...,zn) ^ (zi — ci,...,zn — cn), (a £ K)
such that (for all t £ Tn) the polynomials fi(t, zi — ci,..., zn — on) have zero free part,.
PROOF. This is a direct corollary of Theorem 1. Indeed, any action a on the free algebra induces, by taking the canonical projection with respect to the commutator ideal /, an action a on the commutative algebra K[^,... If a is regular, then so is a. By Theorem 1, a (or rather, its geometric counterpart) has a fixed point, therefore the images of commutative generators Xi under a(t) (for everv t) will be polynomials with trivial degree-zero part. Consequently, the same will hold for a. □
We may then suppose, without loss of generality, that the polynomials fi have the form
n n N
fi(t,zi,...,zn) = j2aij(t)z3 + aiji(t)z3zi + ai'j(f)zJ
3=i j,l=i k=3 J,\J\=k
where by zJ we denote, as in the introduction, a particular monomial
¿ii ¿i2 ...
(a word in the alphabet [zi,..., zn} in the reduced notation; J is the multi-index in the sense described above); also, N is the degree of the automorphism (which is finite) and aij,a,iji,... are polynomials in ti,... ,tn.
As at is an automorphism, the matrix [aij] that determines the linear part is non-singular. Therefore, without loss of generality we may assume it to be diagonal (just as in the commutative case [3]) of the form
di&g^11 ...t%1n ,...,tr^"1 ...t™""). Now, just as in [3], we have the following
Lemma 2. The power m,atrix [rriij] is non-singular.
PROOF. Consider a linear action t defined by
r (t) : (zi,...,zn) (t?11 . ..t™1" Zl,...,tr1 . ..t™™ Zn), ...,tn) e Tn.
If T1 C Tn is any one-dimensional torus, the restriction of r to T1 is non-trivial. Indeed, were it to happen that for some T1;
T(t)z = z, t e Ti, (z = (zi,..., Zn))
then our initial action a, whose linear part is represented by r, would be identity modulo terms of degree > 1:
a(t)(zi) = Zi + aiji(t)zjzi +----.
3,1
Now, equality a(t2)(z) = a(t)(a(t)(z)) implies
a(t)(a(t)(zi)) = a(t) I Zi + ^ a^i(t)zjz¡ + ■
ji
= Zi + (liil (f)Z3 Zi + aijl(t)(Z3 + ajkm(t)zk zm +----)
jl jl km
(Zl + alk'm' (t)Zk'Zm, +----)+----
k'm'
= Zi + Y aifl (t2)ziZl +----
jl
which means that
2aiji(t) = ai:ji (t2)
and therefore aiji(t) = 0. The coefficients of the higher-degree terms are processed by induction (on the total degree of the monomial). Thus
a(t)(z) = z, t e Ti
which is a contradiction since a is effective. Finally, if [mij] were singular, then one would easily find a one-dimensional torus such that the restriction of t were trivial. □ Consider the action
tp(t) = t(t-1) o a(t).
The images under (p(t) are
(gi(z,t),.. .,gn(z,t)), (t = (ti,.. .,tn))
with
gi(z,t) = ^ gi,mi...m„ (z)t™1 ...Cn, mi,..., mn e Z. Define Gi(z) = gi,o...o(z) and consider the map P : Fn ^ Fn,
p :(zi,..., zn)^ (Gi(z),...,Gn (z)).
Lemma 3. P e AutFn and
P = r(t-i) op o a(t).
PROOF. This lemma mirrors the final part in the proof in [3]. The conjugation is straightforward, since for every s,t £ Tn one has
f(st) = t(t-is-r) o a(st) = t(t-r) o t(s-t) o a(s) o a(t) = t(t-r) o ip(s) o a(t).
Denote bv Fn the power series completion of the free algebra Fn, and 1 et a, f Mid /3 denote the endomorphisms of the power series algebra induced by corresponding morphisms of Fn. The endomorphisms f, f, ¡3 come from (polynomial) automorphisms and therefore are invertible.
Let
f-i(Zl) = Bi(z)= £ bitJ zJ j
(just as before, is the monomial with multi-index J). Then
f o f(t) o f-i(zi) = B^t™11 ... C1" Gi(z),..., t™-1 ... C™ Gn(z)). Now, from the conjugation property we must have
ff = f(t-i) o ff o f(t),
therefore 3(t) = f o f(t) o j3-'1- and
3r(t)(zi) = Y, hj (t?11 ... C1n )n ... (t?n1... tnnn )in G(Z)J; j
here the notation G(z)J stands for a wo rd in Gi (z) with multi-in dex J, while the exponents ji,..., jn count how many times a given index appears in J (or, equivalentlv, how many times a given generator Zi appears in the word zJ).
Therefore, the coefficient of a(t)(zi) at zJ has the form
bi,j(t™11 ... C1n )j1... (t?n1... C™ )jn + s
with S a finite sum of monomials of the form
(1^11 )h (t™"1 tm"" )l"
with (ji,... ,jn) = (li,..., Zn). Since the power matrix [m,ij] is non-singular, if bi, J = 0, we can find ai £ Tn such that the coefficient is not zero. Since a is an algebraic action, the degree
sup deg(<3) t
is a finite integer N. With the previous statement, this implies that
bi,j = 0, whenever | J| > N. Therefore, Bi(z) are polynomials in the free variables. WTiat remains is to notice that
= Bl(Gi(z),...,Gn(z)).
Thus ft is an automorphism. □ From Lemma 3 it follows that
t(t) = ft- o a(t) o ft which is the linearization of a. Theorem 3 is proved.
3. Discussion
The noncommutative torus action linearization theorem that we have proved has several useful applications. In the work [13] (cf. also [7]), it is used to investigate the properties of the group Aut Fn of automorphisms of the free algebra. As a corollary of Theorem 3, one gets
Corollary 1. Let 9 denote the standard action ofTnon К[x\,... ,xn] — i-e., the action
9t : (xi,.. .,xn) ^ (t 1x1,... , tnxn J .
Let в denote its lifting to an action on the free associative algebra Fn. Then 9 is also given by the standard torus action.
This statement plays a part, along with a number of results concerning the induced formal power series topology on AutFra, in the establishment of the following proposition (cf. [13]).
Proposition 1. When n > 3, any Ind-scheme automorphism <p of Aut(K(x1,...,xn)) is inner.
One could try and generalize the free algebra version of the Bialvnicki-Birula's theorem to other noncommutative situations. Another way of generalization lies in changing the dimension of the torus. In a complete analogy with further work of Bialvnicki-Birula [4], we expect the following to hold.
Conjecture 2. Any effective action ofTn-ion Fn is linearizable.
On the other hand, there is little reason to expect this statement to hold with further lowering of the torus dimension. In fact, even in the commutative case the conjecture that any effective toric action is linearizable, in spite of considerable effort (see [9]), proved negative (counterexamples in positive characteristic due to Asanuma, [1]).
Another direction would be to replace T bv an arbitrary reductive algebraic group, however the commutative case also does not hold even in characteristic zero (cf. [15]).
It is also a problem of legitimate interest to obtain the proof of Conjecture 1 — i.e. to resolve the linearization problem of the regular action of the n-dimensional torus on the group Sympl(k2ra) of polynomial svmplectomorphisms of the 2n-dimensional affine space (k is a field of characteristic zero). One could hope to utilize the latter result in order to obtain a description of the space of Ind-scheme automorphisms of Sympl( k2n) along the lines of [13]. This space plays a prominent role in the study of quantization of svmplectomorphisms, initiated by Kanel-Belov and Kontsevich [2],
n
and Wevl algebras has been posed as the main conjecture (Kontsevich conjecture). Recently, the first, the second and the fourth named authors have proposed a proof of this conjecture [10, 11].
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Получено 17.01.2019 г.
Принято в печать 20.03.2020 г.
