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COMMON PART

Project Number23-21-00504

Project titleCohomology of Conformal Algebras

Project LeadLopatkin Viktor

AffiliationNational Research University Higher School of Economics,

Research area 01 - MATHEMATICS, INFORMATICS, AND SYSTEM SCIENCES, 01-102 - Algebra

Keywordscohomology, conformal algebras, algebraic discrete Morse theory, Anick resolution, Grobner-Shirshov bases.

PROJECT CONTENT

Annotation
Vertex algebras appeared as an algebraic formalization of the properties of the operator product expansion (OPE) in the 2-dimensional conformal field theory. The singular part of the OPE describes the commutator of two fields, and the corresponding algebraic structures are called conformal (Lie) algebras [V. Kac, 1996] or vertex Lie algebras (e.g., [I. Frenkel, D. Ben-Zvi, 2001]). An algebraic study of conformal algebras is an interesting mathematical problem with numerous ties to other areas. Structure theory of finite Lie conformal algebras was developed by [A.D'Andrea, V. Kac, 1998], simple and semisimple finite Lie conformal superalgebras were described by [S.-J. Cheng, V. Kac, 1997], [D. Fattori, V. Kac, 2002, 2004]. Representations and cohomologies of conformal algebras were studied in a series of papers started with [B. Bakalov, V.Kac, A.Voronov, 1999]. Cohomologies of conformal algebras have the same relations to derivations, extensions, deformations, and crossed modules as the cohomologies of ``ordinary algebras''. In particular, the study of Hochschild cohomologies of (semi)simple associative conformal algebras with finite faithful representation [P. Kolesnikov, R. Kozlov, 2019] allowed to find the ultimate solution of the radical splitting problem for associative algebras of conformal endomorphisms. A similar question for Lie conformal algebras remains to be an open problem. The Hochschild cohomologies of associative conformal algebras and of pre-Lie conformal algebras emerge in the theory of Maurer-Cartan equations for conformal algebras. However, there is an essential difference between the properties of associative and Lie conformal cohomologies. It is well known that the nth cohomology group of a Lie algebra L defined via its standard complex coincides with the nth Hochschild cohomology group of the universal enveloping Lie algebra U(L). For conformal algebras, it is not always true. For example, the (centerless) Virasoro conformal algebra Vir has a nonzero 2nd cohomology group corresponding to the non trivial central extension giving rise to the classical Virasoro Lie algebra. On the other hand, the (associative) conformal Weyl algebra, which is isomorphic to the universal associative envelope U(2) of Vir, has all trivial 2nd Hochschild cohomology groups. For the next universal envelope in the series, U(3), there still exists an irreducible module over Vir for which the cohomologies of Vir differ from those of U(3). Our aim is to study the entire series of envelopes U(n), to calculate their Hochschild cohomology groups and compare them to the known cohomologies of Vir. We will also study the deformations of simple associative conformal algebras with finite faithful representation.

Expected results
We plan to develop the Anick resolution technique in the case of conformal algebras theory. It is well known that in the case of associative alebras we can construct Anick resolution if a Grobner-Shirshov basis is known. However the classical version of Anick resolution is not useful in pratice because all its differentials are defined inductively. On the other hand the algebraic discrete Morse theory allows to define them in terms of directed weighted graph which is obtained by the chain complex. Thus we plan to make the corresponding technique for the case of conformal alebras. Since the Grobner-Shirshov basis theory for conformal algebars has been developed it then allows us to develop this theory in this cases. However we will take an interest not only on the Anick resolution approuach. The algebraic discrete Morse theory technique allows to make new complexes starting with some one, and moreover, all such complexes are homotopically equivalent the origin one. Our aim is to apply it for the case of conformal algebras. This approach is very interesting and useful for practice and theory development. Indeed, a calculations of conformal cohomologies is usually performed in an indirect way; they rest upon deep and nontrivial auxiliary construction. There is a natural question: whether one can arrive at these results in a more universal and natural fashion? The main aim of this project is to develop a modification of the algebraic discrete Morse theory machinery (we call it the Morse matching method) to calculate cohomology of conformal algebras. We believe that calculation of homology of conformal algebras should be obtained in a natural manner; they should be deduced from an intrinsic structure of a combinatorial presentation of algebras (i.e., presentation via generators and relations). In this case this machinery looks natural and powerful and we demonstrate it in some examples. We also take an interest on multiplicative structrue of cohomology rings of conformal algebras. In the literature there are no any results or attempts to research this structure because of complicated of calculation of cohomology of conformal algebras.Our approach allows to reach this aim. We are going also to research cohomology not only with coefficients in trivial modules but in any irreducible ones. All of the above problems and the expected results belong to relevant sections of mathematical physics. A number of strong research groups around the world are working in this area. Advances in the declared directions should lead to significant progress in the understanding of the structure of the mathematical objects under study, which in turn will contribute to solving applied problems and expanding the computational capabilities of modern mathematical science.

REPORTS