arXiv research

This paper constructs cohomological Hall algebras for 3-Calabi-Yau categories.

In this paper, we study both the continuous model and the discrete model of the Quantum Hall Effect (QHE) on the hyperbolic plane. The Hall conductivity is identified as a geometric invariant associated to an imprimitivity algebra of observables. We define a twisted analogue of the Kasparov map, which enables us to use…

Constructs cohomology decompositions for symmetric stacks.

The elliptic Hall algebra governs torus link homology.

The trace of the affine Hecke category is compared with the elliptic Hall algebra.

Proves a pentagon relation in skein theory.

The BPS decomposition theorem splits cohomology of symmetric stacks into invariant parts.

We study both the continuous model and the discrete model of the integer quantum Hall effect on the hyperbolic plane in the presence of disorder, extending the results of an earlier paper [CHMM]. Here we model impurities, that is we consider the effect of a random or almost periodic potential as opposed to just periodi…

Given a finitely-generated group G, and a finite group Γ, Philip Hall defined δ_Γto be the number of factor groups of G that are isomorphic to Γ. We show how to compute the Hall invariants by cohomological and combinatorial methods, when G is finitely-presented, and Γbelongs to a certain class of metabelian groups. Key…

Geometrically proves Zabrodin-Wiegmann conjecture for integer QH states.

We further develop the asymptotic analytic approach to the study of scattering diagrams. We do so by analyzing the asymptotic behavior of Maurer-Cartan elements of a differential graded Lie algebra constructed from a (not-necessarily tropical) monoid-graded Lie algebra. In this framework, we give alternative differenti…

Symmetric function $L_{M,N}$ lifts torus link homology.

The article examines twisted cohomologies on algebraic and analytic varieties.

Study cohomology of hemistrict Lie 2-algebras, proving isomorphic results.

This study introduces a unified cohomology theory for braided algebras.

Town hall discusses AI's impact on statistics, culture, and training.

We introduce a bicomplex which computes the triple cohomology of Lie--Rinehart algebras. We prove that the triple cohomology is isomorphic to the Rinehart cohomology \cite{Ri} provided the Lie--Rinehart algebra is projective over the corresponding commutative algebra. As an application we construct a canonical class in…

Analyses cohomology relations for moving frames and coframes.

New algebraic tools solve Poisson and Lie bialgebra problems.

Proves integrability of strict Lie 2-algebras using cohomological methods.

In this article, we introduce a new cohomology theory associated to a Lie 2-algebras. This cohomology theory is shown to extend the classical cohomology theory of Lie algebras; in particular, we show that the second cohomology group classifies an appropriate type of extensions.

A celebrated theorem of Marshall Hall Jr. implies that finitely generated free groups are subgroup separable and that all of their finitely generated subgroups are retracts of finite-index subgroups. We use topological techniques inspired by the work of Stallings to prove that all limit groups share these two propertie…

We study the open string integrality invariants (LMOV invariants) for toric Calabi-Yau 3-folds with Aganagic-Vafa brane (AV-brane). In this paper, we focus on the case of the resolved conifold with one out AV-brane in any integer framing $τ$, which is the large $N$ duality of the Chern-Simons theory for a framed unknot…

Study on deformation cohomology for braided commutative structures.

The first cohomology of Poisson algebras is described and conditions for its vanishing are established.

Cohomology of Lie group quotient equals Lie algebra cohomology.

Cohomology of 'book' Lie algebra Poisson structure computed.

Introduces Lie-Yamaguti algebra bundles and their cohomology.

In this thesis, we introduce a new cohomology theory associated to a Lie 2-algebras and a new cohomology theory associated to a Lie 2-group. These cohomology theories are shown to extend the classical cohomology theories of Lie algebras and Lie groups in that their second groups classify extensions. We use this fact to…

In arXiv:1106.4305 extended superpolynomials were introduced for the torus links T[m,mk+r], which are functions on the entire space of time variables and, at expense of reducing the topological invariance, possess additional algebraic properties, resembling those of the matrix model partition functions and the KP/Toda …

Bayesian inference calibrates Hall thruster model uncertainty at varying pressures.

In this note we define a notion of Courant pair as a Courant algebra over the Lie algebra of linear derivations on an associative algebra. We study formal deformations of Courant pairs by constructing a cohomology bicomplex with coefficients in a module from the cochain complexes defining Hochschild cohomology and Leib…

This work is devoted to an intrinsic cohomology theory of Koszul-Vinberg algebras and their modules. Our results may be regarded as improvements of the attempt by Albert Nijenhuis in [NA]. The relationships between the cohomology theory developed here and some classical problems are pointed out, e.g. extensions of alge…

The paper studies deformations of Lie ideals in Lie algebras.

Study cohomology of Lie algebroids over algebraic spaces using derived functors and Čech cohomology.

We construct a new equivariant cohomology theory for a certain class of differential vertex algebras, which we call the chiral equivariant cohomology. A principal example of a differential vertex algebra in this class is the chiral de Rham complex of Malikov-Schechtman-Vaintrob of a manifold with a group action. The ma…

In the paper we consider the following conjecture: if a finite group $G$ possesses a solvable $π$-Hall subgroup $H$, then there exist elements $x,y,z,t\in G$ such that the identity $H\cap H^x\cap H^y\cap H^z\cap H^t=O_π(G)$ holds. The minimal counter example is shown to be an almost simple group of Lie type.

Novel cohomology theories for operadic algebras and spaces.

After a self-contained introduction to Lie algebra cohomology, we present some recent applications in mathematics and in physics. Contents: 1. Preliminaries: L_X, i_X, d 2. Elementary differential geometry on Lie groups 3. Lie algebra cohomology: a brief introduction 4. Symmetric polynomials and higher order cocycles 5…

In this paper, we study a refined L2 version of the semiclassical approximation of projectively invariant elliptic operators with invariant Morse type potentials on covering spaces of compact manifolds. We work on the level of spectral projections (and not just their traces) and obtain an information about classes of t…

Researchers compute Hochschild cohomology of Grassmannians.

Researchers compute cohomology of Lie groups using Lie algebras.

Develops a spectral sequence for Lie group actions on manifolds.

This research classifies deformations of Yang-Baxter operators using cohomology of $n$-Lie algebras.

Paper analyzes origami slope gaps and their distribution, finding a unique pattern.

In this paper we study the variability and rigidity of secondary characteristic classes which arise from flat connections on a manifold. Considering the connection as a Lie-algebra valued one-form, we study the characteristic map from Lie algebra cohomology to de Rham cohomology of the manifold, and prove that if the L…

The equivariant cohomology of a space with a group action is not only a ring but also an algebra over the cohomology ring of the classifying space of the acting group. We prove that toric manifolds (i.e. compact smooth toric varieties) are isomorphic as varieties if and only if their equivariant cohomology algebras are…

This is the second in a series of papers on a new equivariant cohomology that takes values in a vertex algebra. In an earlier paper, the first two authors gave a construction of the cohomology functor on the category of O(sg) algebras. The new cohomology theory can be viewed as a kind of "chiralization'' of the classic…