Computation of Hochschild cohomology via Morse matching тема диссертации и автореферата по ВАК РФ 01.01.06, кандидат наук Алхуссейн Хасан

• Алхуссейн Хасан
• кандидат наук
• 2022

Оглавление диссертации кандидат наук Алхуссейн Хасан

Contents.

Introduction

1. Main definitions and Statements

§1.1 Composition-Diamond Lemma for Associate Algebras

§1.2 Morse matching

2. The Hochschild cohomology of the group G2

§2.1 The Anick complex for G2

§2.2 Hochschild cohomologies of G2

3. Anick complex, Hochschild cohomology, Hilbert and Poincare series of the Manturov (3,4)-group

§3.1 The Anick complex for G4

§3.2 The Hochschild cohomology groups of G3

§3.3 Hilbert and Poincare series of the Manturov (3,4)-group

4. The Hochschild cohomology of the Chinese monoid algebra

§4.1 The Anick complex for the Chinese algebra

§4.2 Application of Anick resolution on k[Cn]

5. Grobner-Shirshov basis and Hochschild cohomology of the Manturov group rf

§5.1 Grobner-Shirshov basis for

§5.2 The Anick complex for

§5.3 Application of Anick resolution on

6. On the Hochschild cohomology of universal enveloping associative conformal algebras

§6.1 Preliminaries

§6.2 Cohomology of universal associative conformai envelopes of the

Virasoro conformai algebra U(3)

§6.3 Cohomologies of the Weyl conformal conformal algebra U(2) . . . 100 §6.4 Morse matching method for conformal cohomologies

References

Publication on the topic of the dissertation

Введение диссертации (часть автореферата) на тему «Computation of Hochschild cohomology via Morse matching»

Introduction

It is well known that the first cohomology group of an algebra is related to its derivations, and the elements of the second one describe its null extensions. Some believe that, in the theory of groups or algebras, we just need the first and the second groups of (co)homology, but it was shown that the elements of the third cohomology group can be applied to describe the obstacles to the construction of extensions. Homological algebra also introduces important numerical invariants in the group theory, e.g., (co)homological dimension and Euler characteristic. So we can say that homological methods allow us to get important information about the structure of an algebra.

The most important cohomology theory for associative algebras is the Hochschild cohomology. Given an associative algebra A over a field k and an A-bimodule W, the set of Hochschild cochains Cm(A,W) consists of all linear maps A®m ^ W, m > 0, and the mth Hochschild cohomology group is defined as Hm(A,W) = KerAm/IrnAm_i, where Am : Cm(A,W) ^ Cm+i(A,W) is the Hochschild differential

(Amf )(n,... ,rm+i) = rif (f2, ... ,rm+i)

m

+ ^(_1)7 (ri,.. .,riri+i,.. .,rm+i) + (_l)m+1/(ri,.. .,rm)rm+i, (1)

i=1

One of the most important features of the Hochschild cohomology groups is that the first one, Hi(A,W), describes outer derivations from A to W. The elements of H2(A, W) are in one-to-one correspondence with the equivalence classes of null extensions

0 ^ W ^ E ^ A ^ 0, W2 = 0.

It is often a difficult problem to find the groups Hm(A, W) for a given algebra A an A-bimodule W. The first important step of solution is to find a long exact sequence

of A-bimodules starting from A, a resolution of A. Next, one needs to calculate the derived functor to Hom(-,^) to find the cohomology groups. The most natural resolution known as the bar-resolution. It is easy to define but it is too bulky for the calculation of cohomologies. A smaller resolution was proposed by David J. Anick in 1986 [1]: he found a way to construct a free resolution for an associative algebra which is homotopy equivalent to the bar-resolution. The Anick resolution found numerous applications in combinatorial algebra, see [3]. This long exact sequence is more convenient for the calculation of cohomology groups, but the computation of differentials according to the original Anick algorithm described in [1] is extremely hard. In order to visualize the computation of differentials it is possible to use the discrete algebraic Morse theory based on the concept of a Morse matching (see, e.g., [2, 7, 8, 9, 10]). In [12]-[16], V.O.Manturov introduced a family of gruops Gkn depending on two positive integers n > k, and formulated the following principle:// dynamical systems describing a motion of n particles possess a nice codimension one property governed by k particles, then they have a topological invariants valued in groups G^, which generalize classical braid groups, virtual braid groups and other gruops in a very broad sense. In the paper [14], the author construct maps from fundamental gruops of configuration spaces to the groups Gkn. In [16], the authors construct a group corresponding to the motion of points in R3 from the point of view of Delaunay triangulations. They study homomorphisms from pure braids on n strands to the product of copies of r^, and also study the group of pure braids in R3, which is described by a fundamental group of the restricted configuration space of R3, and define the group homomorphism from the group of pure braids in R3 to T^, and give some comments about relations between the restricted configuration space of R3 and triangulations of the 3-dimensional ball and Pachner moves, therefore, it is useful to study this groups as finding Grobner—Shirshov basis and cohomology. The Chinese monoid Cn appeared in the classification of monoids with the growth function coinciding with that of the plactic monoid [20]. One of the motivations

for a study of the Chinese monoid is based on the expectation that it might play a similar role as the plactic monoid in several aspects of representation theory, quantum algebras, and in algebraic combinatorics. If n = 2 then the Chinese and the plactic monoids coincide and both constructions are strongly related to Young tableaux. Therefore, Chinese monoids are important in various aspects of representation theory and algebraic combinatorics. It also plays an important role in the area of classical Lie algebras [18, 19, 20, 21].The classical theory of finite-dimensional Lie algebras often needs universal constructions like free algebras and universal enveloping associative algebras. The study of such universal structures for conformal algebras was initiated in [48]. In particular, the emergence of associative conformal algebras is mainly motivated by the study of finite representations of Lie conformal algebras. In order to operate with conformal algebras defined by generators and relations, one needs combinatorial methods like Grobner-Shirshov bases. It was a motivation to study combinatorial issues in the theory of (associative) conformal algebras [33, 34].

It is well known (see, e.g., [37, Chapter XIII]) that the nth cohomology group of a Lie algebra g defined via its standard complex coincides with the nth Hochschild cohomology group of the universal enveloping Lie algebra U(g). For conformal algebras, it is not always true.

For example, consider the (centerless) Virasoro conformal algebra Vir generated as a C[5]-module by a single element v. It was proved in [31] that

|l, n = 2,3, dim Hm(Vir, C) = I

I 0, otherwise.

However, for the Weyl associative conformal algebra W which is exactly the universal associative envelope of Vir relative to the locality function N(v,v) = 2 we have %2(W, M) = 0 for every conformal W-bimodule M [47]. In the present work, we apply the Morse matching theory to find the Anick

resolution and calculate the groups of Hochschild m-cohomologies of G2 , rj and k[Cn] for m > 1 and some n > 2 with coefficients in various bimodules over a field k of characteristic zero. The main results of the work are the following:

1) We construct the Anick resolution and compute Hochschild cohomology groups for the group algebra of the Manturov group G2;

2) We find the dimensions of Hochschild cohomology groups for the group algebra of the group Gj and calculate its Hilbert and Poincare series.

3) We calculate the Gröbner-Shirshov basis of the Manturov group r5 relative to the tower order on monomials and find its second Hochschild cohomology group.

4) We find the second Hochschild cohomology group for the universal enveloping associative conformal algebra with locality level 3 of the Virasoro conformal algebra, and we prove that all Hochschild cohomologies of the Weyl conformal algebra are trivial.

In order to find explicitly the differentials of Anick resolution we apply the Morse matching theory. As an application, we derive explicit expressions for the Hilbert and Poincare series of Gl and k[Cn]. In case r5 we needed to find Grobner—Shirshov basis. For conformal algebras, we show that the Hochschild cohomology group V,m(W, C) is trivial for every n > 1. The second Hochschild cohomology group of universal enveloping associative conformal algebra U(3) = U(Vir, N = 3) of Vir relative to the locality function N(v, v) = 3 is 1-dimensional, but for higher Hochschild cohomologies the direct computation becomes too complicated since U(3) is of quadratic growth (Gelfand-Kirillov dimension = 2).

We developed a modification of the Morse matching method for calculation of Hochschild cohomologies of associative conformal algebras. As an application, we find higher Hochschild cohomologies of U(3) with coefficients in k.

Throughout the text below, the ground field k is of characteristic zero, Z+ is the set of non-negative integers.

Список литературы диссертационного исследования кандидат наук Алхуссейн Хасан, 2022 год

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Publication on the topic of the dissertation

[30] Х. Алхуссейн, П. С. Колесников, Комплекс Аника и когомологии Хох-шильда (2, 3)-группы Мантурова, Сиб. мат. журнал 61, no 1 (2020), 17-28.

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[33] H. AlHussein, Grobner-Shirshov basis and Hochschild cohomology of the group Г4, Siberian Electronic Mathematical Reports, 2022, Vol. 19, No. 1, pp. 211-236.

[34] H. AlHussein, P. S. Kolesnikov, On the Hochschild cohomology of universal enveloping associative conformal algebras, J. Math. Phys. 62, 121701 (2021).

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[36] Х. Алхуссейн, П.С. Колесников. О когомологиях Хохшильда универсальных ассоциативных обертывающих конформных алгебр, Труды Математического центра имени Н.И. Лобачевского. Том 60 // Материалы Международной конференции по алгебре, анализу и геометрии 2021 - Казань: Изд-во Академии наук РТ, 2021, стр. 17-19.

[37] H. AlHussein, P. S. Kolesnikov, Hochschild Cohomology via Morse Mathching and Anick Resolution // Int. Conf. "Topological Methods in Dynamics and Related Topics. Shilnikov Workshop Book of Abstracts, 9-13 December, 2019, Nizhny Novgorod, стр. 29-30.

[38] H. AlHussein, P. S. Kolesnikov, Hochschild Cohomology via Morse Mathching // Int. Conf. "MAL'TSEV MEETING Sobolev Institute of Mathematics, Novosibirsk State University, November 19-22, 2018, стр. 168.

[39] H. AlHussein, P. S. Kolesnikov, Hochschild Cohomology via Morse Mathching // Сборник тезисов международной конференции, посвящён-ной 90-летию кафедры высшей алгебры механико-математического факультета МГУ, Москва 2019, стр. 81.

[40] H. AlHussein, The Hochschild cohomology of the Chinese monoid algebra // The Eighth School-Conference on Lie Algebras, Algebraic Groups and Invariant Theory, Moscow, Russia, MCCME, January 27 - February 1, 2020, Abstracts, pp. 70-72.

[41] H. AlHussein, Hochschild cohomology of U(Vir,N=2,3) // Материалы Международного молодежного научного форума «Л0М0Н0С0В-2021», Москва МГУ 2021.