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## Hochschild cohomology of Fermat type polynomials with non–abelian symmetries

Journal of Geometry and Physics. 2022. Vol. 174. Article 104450.

Basalaev A., Ionov A.

For a polynomial f=x_1^n+…+x_N^n let Gf be the non–abelian maximal group of symmetries of f. This is a group generated by all g in GL(N,C), rescaling and permuting the variables, so that f(x)=f(g x). For any G subgroup in Gf we compute explicitly Hochschild cohomology of the category of *G*–equivariant matrix factorizations of *f*. We introduce the pairing on it showing that it is a Frobenius algebra

Publication based on the results of:

Kaledin D., Lowen W., Advances in Mathematics 2015 Vol. 272 P. 652-698

We use (non-)additive sheaves to introduce an (absolute) notion of Hochschild cohomology for exact categories as Ext's in a suitable bisheaf category. We compare our approach to various definitions present in the literature. ...

Added: February 9, 2015

Ionov A., / Cornell University. Series arXiv "math". 2016. No. 1611.03962.

We apply the technique of the paper "The abelian/nonabelian correspondence and Frobenius manifolds" by I. Ciocan-Fontanine, B. Kim, C. Sabbah to construct Saito primitive forms for Gepner singularities. ...

Added: November 16, 2016

Basalaev A., Buryak A., International Mathematics Research Notices 2021 Vol. 2021 No. 7 P. 5460-5491

A well-known construction of B. Dubrovin and K. Saito endows the parameter space of a universal unfolding of a simple singularity with a Frobenius manifold structure. In our paper, we present a generalization of this construction for the singularities of types A and D that gives a solution of the open WDVV equations. For the A-singularity, the resulting solution describes ...

Added: April 21, 2020

Nakatsu T., Kato A., Noumi M. et al., Physics Letters B 1994 Vol. 322 No. 3 P. 192-197

We study the relation between topological string theory and singularity theory using the partition function of A_N-1 topological string defined by matrix integral of Kontsevich type. Genus expansion of the free energy is considered, and the genus g=0 contribution is shown to be described by a special solution of N-reduced dispersionless KP system. We show ...

Added: August 14, 2014

Van H. D., Lowen W., Advances in Mathematics 2018 Vol. 330 P. 173-228

The aim of this work is to construct a complex which through its higher structure directly controlls deformations of general prestacks, building on the work of Gerstenhaber and Schack for presheaves of algebras. In defining a Gerstenhaber–Schack complex for an arbitrary prestack , we have to introduce a differential with an infinite sequence of components instead of ...

Added: September 13, 2018

Basalaev A., ASIAN JOURNAL OF MATHEMATICS 2023 Vol. 26 No. 1 P. 45-80

We give explicitly in the closed formulae the genus zero primary potentials of the three $6$-dimensional FJRW theories of the simple–elliptic singularity $\tilde{E}_7$ with the non–maximal symmetry groups. For each of these FJRW theories we establish the CY/LG correspondence to the Gromov–Witten theory of the elliptic orbifold $[\mathcal{E} / (\mathbb{Z}/2\mathbb{Z})]$ — the orbifold quotient of ...

Added: February 26, 2019

Basalaev A., Takahashi A., Werner E., Journal of Singularities 2023 Vol. 26 P. 92-127

An important invariant of a polynomial f is its Jacobian algebra defined by its partial derivatives. Let f be invariant with respect to the action of a finite group of diagonal symmetries G. We axiomatically define an orbifold Jacobian Z/2Z-graded algebra for the pair (f,G) and show its existence and uniqueness in the case, when ...

Added: February 26, 2019

Kuznetsov A., Journal fuer die reine und angewandte Mathematik 2015 Vol. 2015 No. 708 P. 213-243

We define the normal Hochschild cohomology of an admissible subcategory of the derived category of coherent sheaves on a smooth projective variety $X$ --- a graded vector space which controls the restriction morphism from the Hochschild cohomology of $X$ to the Hochschild cohomology of the orthogonal complement of this admissible subcategory. When the subcategory is ...

Added: December 22, 2013

Bokut L., Chen Y., Kalorkoti K. et al., World Scientific, 2020

The book is about (associative, Lie and other) algebras, groups, semigroups presented by generators and defining relations. They play a great role in modern mathematics. It is enough to mention the quantum groups and Hopf algebra theory, the Kac–Moody and Borcherds algebra theory, the braid groups and Hecke algebra theory, the Coxeter groups and semisimple ...

Added: September 27, 2021

Ionov A., Journal of Geometry and Physics 2019 Vol. 140 P. 125-130

We provide a construction of Saito primitive forms for Gepner singularity by studying the relation between Saito primitive forms for Gepner singularities and primitive forms for singularities of the form F_{k,n} = ∑^n_{i=1} x^k_i invariant under the natural S_n-action. ...

Added: November 8, 2019

Lopatkin V., Journal of Algebra and its Applications 2016 Vol. 15 No. 4 Article 1650082

In this paper, we calculate the cohomology ring and the Hochschild cohomology ring of the plactic monoid algebra via the Anick resolution using a Gröbner–Shirshov basis. ...

Added: October 29, 2021

Lopatkin V., Journal of Algebra 2019 Vol. 520 P. 59-89

In this paper, we describe the K-module HH1(LK(Γ)) of outer derivations of the Leavitt path algebra LK(Γ) of a row-finite graph Γ with coefficients in an associative commutative ring K with unit. We explicitly describe a set of generators of HH1(LK(Γ)) and relations among them. We also describe a Lie algebra structure of outer derivation algebra of the Toeplitz algebra. We prove that every derivation of a ...

Added: October 29, 2021

Mironov A., Morozov A., Natanzon S. M., Journal of Knot Theory and Its Ramifications 2014 Vol. 23 No. 6 P. 1-16

The classical Hurwitz numbers of degree n together with the Hurwitz numbers of the seamed surfaces of degree n give rise to the Klein topological field theory. We extend this construction to the Hurwitz numbers of all degrees at once. The corresponding Cardy-Frobenius algebra is induced by arbitrary Young diagrams and arbitrary bipartite graphs. It ...

Added: April 2, 2014

Basalaev A., Ionov A., Theoretical and Mathematical Physics 2021 Vol. 209 No. 2 P. 1491-1506

We study Landau-Ginzburg orbifolds (f,G) with f=xn1+…+xnN and G=S⋉Gd, where S⊆SN and Gd is either the maximal group of scalar symmetries of f or the intersection of the maximal diagonal symmetries of f with SLN(ℂ). We construct a mirror map between the corresponding phase spaces and prove that it is an isomorphism restricted to a ...

Added: November 23, 2021

Basalaev A., Takahashi A., International Mathematical Research Notices 2022 Vol. 2022 No. 19 P. 14865-14922

For any triple of positive integers A′=(a′1,a′2,a′3) and c∈C∗, cusp polynomial fA′=xa′11+xa′22+xa′33−c−1x1x2x3 is known to be mirror to Geigle–Lenzing orbifold projective line P1a′1,a′2,a′3. More precisely, with a suitable choice of a primitive form, the Frobenius manifold of a cusp polynomial fA′ turns out to be isomorphic to the Frobenius manifold of the Gromov–Witten theory of ...

Added: September 9, 2022

Ionov A., / Cornell University. Series arXiv:1504.07930 "math.arxiv". 2015.

Cardy-Frobenius algebra is the algebraic structure on the space of states in open-closed topological field theory. We prove that every semisimple super Cardy-Frobenius algebras is the direct sum of the super Cardy-Frobenius algebras of three simple types. We also apply our results to singularity theory via Landau-Ginzburg models and matrix factorizations. ...

Added: November 8, 2016

Kuznetsov A., / Cornell University. Series math "arxiv.org". 2012. No. 1211.4693.

We define the normal Hochschild cohomology of an admissible subcategory of the derived category of coherent sheaves on a smooth projective variety $X$ --- a graded vector space which controls the restriction morphism from the Hochschild cohomology of $X$ to the Hochschild cohomology of the orthogonal complement of this admissible subcategory. When the subcategory is ...

Added: October 4, 2013

Sirotin V., Arkhipova M., Dubrova T. A. et al., Bielsko-Biala : University of Bielsko-Biala Press, 2016

The main attributes of modern enterprises should be the flexibility and the ability of forecasting the future. Constant adaptation to the changing environment and the rapidity of undertaking certain actions which are conditioned by specific situations determine the rules for the future position of market competition. Effective and efficient adjustment of the company in line ...

Added: November 2, 2016

Pahomov F., Известия РАН. Серия математическая 2016 Т. 80 № 6 С. 173-216

Полимодальная логика доказуемости
GLP была введена Г. К. Джапаридзе в 1986 г. Она является логикой доказуемости для ряда цепочек предикатов доказуемости возрастающей силы. Всякой полимодальной логике соответствует многообразие полимодальных алгебр. Л. Д. Беклемишевым и А. Виссером был поставлен вопрос о разрешимости элементарной теории свободной GLP-алгебры, порожденной константами 0, 1 [1]. В этой статье для любого натурального n решается аналогичный вопрос для логик GLPn, являющихся ...

Added: December 4, 2017

Furmanov K. K., Nikol'skii I. M., Computational Mathematics and Modeling 2016 Vol. 27 No. 2 P. 247-253

Added: December 22, 2016

Buchstaber V., Limonchenko I., / Cornell University. Series math "arxiv.org". 2018. No. 1808.08851.

We introduce the notions of algebraic and geometric direct families of polytopes and develop a theory of such families. The theory is then applied to the problem of existence of nontrivial higher Massey products in cohomology of moment-angle-complexes. ...

Added: September 29, 2019

Min Namkung, Younghun K., Scientific Reports 2018 Vol. 8 No. 1 P. 16915-1-16915-18

Sequential state discrimination is a strategy for quantum state discrimination of a sender’s quantum
states when N receivers are separately located. In this report, we propose optical designs that can
perform sequential state discrimination of two coherent states. For this purpose, we consider not
only binary phase-shifting-key (BPSK) signals but also general coherent states, with arbitrary prior
probabilities. Since ...

Added: November 16, 2020

Shiryaev A., Zhitlukhin M., Ziemba W., / SSRN. Series Social Science Research Network "Social Science Research Network". 2013.

We study the land and stock markets in Japan circa 1990. While the Nikkei stock average in the late 1980s and its -48% crash in 1990 is generally recognized as a financial market bubble, a bigger bubble and crash was in the golf course membership index market. The crash in the Nikkei which started on ...

Added: March 9, 2014

Maslov V., Теоретическая и математическая физика 2019 Т. 201 № 1 С. 65-83

We study the process of a nucleon separating from an atomic nucleus from the mathematical standpoint
using experimental values of the binding energy for the nucleus of the given substance. A nucleon becomes
a boson at the instant of separating from a fermionic nucleus. We study the further transformations of
boson and fermion states of separation in a ...

Added: November 1, 2019