arXiv research

Koszul duality for manifold modules proven.

Advances in Koszul modules and syzygies of algebraic varieties.

Proves effective Chen ranks conjecture for Koszul modules.

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…

Researchers compute link surgery modules for 2-component L-space links.

In this note, we study the Koszul-Brylinski homology of holomorphic Poisson manifolds. We show that it is isomorphic to the cohomology of a certain smooth complex Lie algebroid with values in the Evens-Lu-Weinstein duality module. As a consequence, we prove that the Evens-Lu-Weinstein pairing on Koszul-Brylinski homolo…

This paper constructs Koszul duals for Heegaard Floer Dehn surgery formulas.

In this paper we aim to understand the category of stable-Yetter-Drinfeld modules over enveloping algebra of Lie algebras. To do so, we need to define such modules over Lie algebras. These two categories are shown to be isomorphic. A mixed complex is defined for a given Lie algebra and a stable-Yetter-Drinfeld module o…

The semi-classical data attached to stacks of algebroids in the sense of Kashiwara and Kontsevich are Maurer-Cartan elements on complex manifolds, which we call extended Poisson structures as they generalize holomorphic Poisson structures. A canonical Lie algebroid is associated to each Maurer-Cartan element. We study …

Defines Jacobi-Koszul-Vinberg structures on Jacobi-left-symmetric algebroids.

Holomorphic Koszul-Brylinski homology studied via Dolbeault cohomology.

Formula derived for holomorphic Poisson blow-ups.

The paper studies affine connections on singular warped products and their curvature.

The paper introduces new structures for left-symmetric algebroids.

For each integer $k\geq 4$ we describe diagrammatically a positively graded Koszul algebra $\mathbb{D}_k$ such that the category of finite dimensional $\mathbb{D}_k$-modules is equivalent to the category of perverse sheaves on the isotropic Grassmannian of type ${\rm D}_k$ or ${\rm B}_{k-1}$, constructible with respect…

The Hesse-Koszul flow converges to the Hesse-Einstein metric on compact Hessian manifolds.

Characterizes real left symmetric algebras with positive definite Koszul form and related Kähler-Einstein structures.

A Lie-admissible algebra gives by anticommutativity a Lie algebra. In this work we study remarkable classes of Lie-admissible algebras such as Vinberg, PreLie algebras. We compute the corresponding binary quadratic operads and study their Koszul duality. Considering Lie algebras as Lie-admissible algebras we can define…

Paper proves Koszul duality for weighted A-infinity algebras.

Our main objective is to demonstrate how homological perturbation theory (HPT) results over the last 40 years immediately or with little extra work give some of the Koszul duality results that have appeared in the last decade. Higher homotopies typically arise when a huge object, e. g. a chain complex defining various …

We study the deformation theory of pre-symplectic structures, i.e. closed two-forms of fixed rank. The main result is a parametrization of nearby deformations of a given pre-symplectic structure in terms of an $L_\infty$-algebra, which we call Koszul $L_\infty$-algebra. This $L_\infty$-algebra is a cousin of the Koszul…

Quantizes the relationship between Koszul and Schouten brackets in Poisson geometry.

It is a classical fact in Poisson geometry that the cotangent bundle of a Poisson manifold has the structure of a Lie algebroid. Manifestations of this structure are the Lichnerowicz differential on multivector fields (calculating Poisson cohomology) and the Koszul bracket of differential forms. "Raising indices" by th…

Study submanifolds in Koszul-Vinberg geometry, a blend of Poisson and pseudo-Riemannian structures.

Constructs Koszul dual algebras for star-shaped diagrams in 3-manifolds.

Introduces a new operator generating higher Koszul brackets on differential forms.

The Koszul-Tate resolution is described in the context of the geometry of jet spaces and differential equations. The application due to Barnich, Brandt, and Henneaux of this resolution to computing the horizontal cohomology is analyzed. Relations with the Vinogradov spectral sequence are discussed.

Let G be a general (not necessarily finite dimensional compact) Lie group, let g be its Lie algebra, let Cg be the cone on g in the category of differential graded Lie algebras, and consider the functor which assigns to a chain complex V the V-valued total de Rham complex of G. We describe the G-equivariant de Rham coh…

We deal with smooth real manifolds as well as complex analytic manifolds as well. It is well known that the concept of star product is powerful enough to produce all Poisson structures on real manifolds. According to [BdM] it is not known whether holomorphic star products exist on complex analytic manifolds. The main p…

Research connects geometric structures to knot theory and algebraic combinatorics.

We show that Verdier duality for certain sheaves on the moduli spaces of graphs associated to Koszul operads corresponds to Koszul duality of operads. This in particular gives a conceptual explanation of the appearance of graph cohomology of both the commutative and Lie types in computations of the cohomology of the ou…

Proves properties of Torelli Lie algebra for surfaces.

The paper computes KV cochain differentials and their geometric implications.

Finite presentations for skein algebras linked to gauge field theory.

The abstract discusses cohomologies and deformations of Rota-Baxter operators on Lie algebroids and Koszul-Vinberg structures.

Study classifies 3D Hessian manifolds, proving their topology.

Constructs a new graded variety from algebraic data.

We show that if a generator of a differential Gerstenhaber algebra satisfies certain Cartan-type identities, then the corresponding Lie bracket is formal. Geometric examples include the shifted de Rham complex of a Poisson manifold and the subcomplex of differential forms on a symplectic manifold vanishing on a Lagrang…

Inspired by recent works of Zang Liu, Alan Weinstein and Ping Xu, we introduce the notions of CC algebroids and non asymmetric Courant algebroids and study these structures. It is shown that CC algebroids of rank greater than 3 are the same as Courant algebroids up to a constant factor, though the definition of CC alge…

Let X and Y be finite-type CW-complexes (X connected, Y simply connected), such that the rational cohomology ring of Y is a k-rescaling of the rational cohomology ring of X. Assume H^*(X,Q) is a Koszul algebra. Then, the homotopy Lie algebra pi_*(Omega Y) tensor Q equals, up to k-rescaling, the graded rational Lie alge…

Normal forms for Q-structures on graded manifolds explained.

The McCool group, denoted $PΣ_n$, is the group of pure symmetric automorphisms of a free group of rank $n$. The cohomology algebra $H^*(PΣ_n, \mathbb{Q})$ was determined by Jensen, McCammond and Meier. We prove that $H^*(PΣ_n, \mathbb{Q})$ is a non-Koszul algebra for $n \geq 4$, which answers a question of Cohen and Pr…

We give a combinatorial proof of the quasi-invertibility of $\widehat{CFDD}(\mathbb{I}_\mathcal{Z})$ in bordered Heegaard Floer homology, which implies a Koszul self-duality on the dg-algebra $\mathcal{A}(\mathcal{Z})$, for each pointed matched circle $\mathcal{Z}$. This is done by giving an explicit description of a r…

Study shows looping a 6-manifold over a 4-manifold results in a product of loops on spheres.

Study on deformation of affine structures on Lie groups using cohomology.

The cohomology of affine structures on Lie groups is compared with that of Koszul-Vinberg algebras.

The paper constructs a Saito basis for a specific class of divisors and applies it to logarithmic Poisson geometry.

Study of Batalin-Vilkovisky algebra on Poisson manifolds with diagonalizable modular symmetry.