arXiv research

New Lie algebras from knot homology.

Extends Loday-Quillen-Tsygan theorem to bornological Lie algebra homology.

Research connects Lie algebras to configuration space (co)homology.

Study uses superalgebra homology to classify 4D Engel-like Lie algebras.

We determine the second homology group of the homological Goldman Lie algebra for an oriented surface.

The study explores Lie superalgebras constructed from Lie algebras using Schouten-like brackets.

We determine the minimal number of generators of the homological Goldman Lie algebra of a surface consisting of elements of the first homology group of the surface.

Let S be a compact connected oriented surface, whose boundary is connected or empty. A homology cylinder over the surface S is a cobordism between S and itself, homologically equivalent to the cylinder over S. The Y-filtration on the monoid of homology cylinders over S is defined by clasper surgery. Using a functorial …

We determine all the ideals of the homological Goldman Lie algebra, which reflects the structure of an oriented surface.

The paper explains why a specific type of link homology is useful.

Researchers determined the second homology group of a specific symplectic derivation Lie algebra.

We show that a certain symmetry exists in the stable irreducible decomposition of the Lie algebra consisting of symplectic derivations of the free Lie algebra generated by the first homology group of compact oriented surfaces.

New method computes automorphisms of surface groups using skein algebras.

Defines Lie and Courant algebroids over Lie groupoids using homological vector fields.

Quantum groups created from disk configuration space homologies.

In this paper we will prove a super-analogue of a well-known result by Kontsevich which states that the homology of a certain complex which is generated by isomorphism classes of oriented graphs can be calculated as the Lie algebra homology of an infinite-dimensional Lie algebra of symplectic vector fields.

Researchers establish a connection between knot homology and Lie algebra actions.

Homological blocks match Witten-Reshetikhin-Turaev invariants for Seifert fibered 3-spheres.

New knot invariants discovered using mirror symmetry.

An $L_\infty$-algebra is built on symplectic manifold homology.

According to seminal work of Kontsevich, the unstable homology of the mapping class group of a surface can be computed via the homology of a certain lie algebra. In a recent paper, S. Morita analyzed the abelianization of this lie algebra, thereby constructing a series of candidates for unstable classes in the homology…

In 1999 Chas and Sullivan showed that the homology of the free loop space of an oriented manifold admits the structure of a Batalin-Vilkovisky algebra. In this paper we give a direct description of this Batalin-Vilkovisky algebra in the case that the manifold is a compact Lie group G. Our answer is phrased in terms of …

We study the hairy graph homology of a cyclic operad; in particular we show how to assemble corresponding hairy graph cohomology classes to form cocycles for ordinary graph homology, as defined by Kontsevich. We identify the part of hairy graph homology coming from graphs with cyclic fundamental group as the dihedral h…

Odd Khovanov homology gets a new algebraic action from super foams.

Lie's third theorem proven for Lie ∞-algebras.

Let K be a compact Lie group. We compute the abelianization of the Lie algebra of equivariant vector fields on a smooth K-manifold X. We also compute the abelianization of the Lie algebra of strata preserving smooth vector fields on the quotient X/K.

We consider the Lie algebra consisting of all derivations on the free associative algebra, generated by the first homology group of a closed oriented surface, which kill the symplectic class. We find the first non-trivial abelianization of this Lie algebra and discuss its relation to unstable cohomology classes of the …

We develop a relative version of Kostant's harmonic theory and use this to prove a relative version of Kostant's theorem on Lie algebra (co)homology. These are associated to two nested parabolic subalgebras in a semisimple Lie algebra. We show how relative homology groups can be used to realize representations with low…

We show that the Gerstenhaber algebra of the 1-jet Lie algebroid of a Jacobi manifold has a canonical exact generator, and discuss duality between its homology and the Lie algebroid cohomology. We also discuss a new example of a Lie bialgebroid on Poisson manifolds.

Let M be a closed, oriented, n -manifold, and LM its free loop space. Chas and Sullivan defined a commutative algebra structure in the homology of LM, and a Lie algebra structure in its equivariant homology. These structures are known as the string topology loop product and string bracket, respectively. In this paper w…

Lifts an $\mathfrak{sl}_2$ action to annular Khovanov homology's stable refinement.

We associate Hamiltonian homological evolutionary vector fields --which are the non-Abelian variational Lie algebroids' differentials-- with Lie algebra-valued zero-curvature representations for partial differential equations.

Smooth groupoid algebras are H-unital, with implications for algebraic and homological properties.

We expose a K-theoretic approach to study group C*-algebras and C*-algebraic compact quantum groups: 1. The conception of multidimensional geometric quantization and the index of group C*-algebras; 2. the entire homology of noncommutative de Rham currents and the noncommutative Chern characters, and their computation f…

In two seminal papers Kontsevich used a construction called_graph homology_ as a bridge between certain infinite dimensional Lie algebras and various topological objects, including moduli spaces of curves, the group of outer automorphisms of a free group, and invariants of odd dimensional manifolds. In this paper, we s…

Twilled L(ie)-R(inehart) algebas generalize, in the Lie-Rinehart context, complex structures on smooth manifolds. An almost complex manifold determines an almost twilled pre-LR algebra, which is a true twilled LR-algebra iff the almost complex structure is integrable. We characterize twilled LR-structures in terms of c…

Twilled L(ie-)R(inehart)-algebras generalize, in the Lie-Rinehart context, complex structures on smooth manifolds. An almost complex manifold determines an "almost twilled pre-LR algebra", which is a true twilled LR-algebra iff the almost complex structure is integrable. We characterize twilled LR structures in terms o…

A general construction of an sh Lie algebra from a homological resolution of a Lie algebra is given. It is applied to the space of local functionals equipped with a Poisson bracket, induced by a bracket for local functions along the lines suggested by Gel'fand, Dickey and Dorfman. In this way, higher order maps are con…

This paper upgrades Khovanov homology to an L-infinity module structure.

We consider homotopy actions of a Lie algebroid on a graded manifold, defined as suitable $L_{\infty}$-algebra morphisms. On the "semi-direct product" we construct a homological vector field that projects to the Lie algebroid. Our main theorem states that this construction is a bijection. Since several classical geomet…

We define the notion of action of an L-infinity algebra $g$ on a graded manifold $M$, and show that such an action corresponds to a homological vector field on $g[1] \times M$ of a specific form. This generalizes the correspondence between Lie algebra actions on manifolds and transformation Lie algebroids. In particula…

Lie groups of automorphisms of cotangent bundles of Lie groups are completely characterized and interesting results are obtained. We give prominence to the fact that the Lie groups of automorphisms of cotangent bundles of Lie groups are super symmetric Lie groups. In the cases of orthogonal Lie lgebras, semi-simple Lie…

Let R be a commutative ring, and let A be a Poisson algebra over R. We construct an (R,A)-Lie algebra structure, in the sense of Rinehart, on the A-module of Kähler differentials of A depending naturally on A and the Poisson bracket. This gives rise to suitable algebraic notions of Poisson homology and cohomology for a…

Lichnerowicz-Jacobi cohomology and homology of Jacobi manifolds are reviewed. We present both in a unified approach using the representation of the Lie algebra of functions on itself by means of the hamiltonian vector fields. The use of the associated Lie algebroid allows to prove that the Lichnerowicz-Jacobi cohomolog…

We construct a solution of the master equation by means of standard tools from homological perturbation theory under just the hypothesis that the ground field be of characteristic zero, thereby avoiding the formality assumption of the relevant Lie algebra. To this end we endow the homology H(g) of any differential grad…

We introduce a notion of duality for a Lie-Rinehart algebra giving certain bilinear pairings in its cohomology generalizing the usual notions of Poincaré duality in Lie algebra cohomology and de Rham cohomology. We show that the duality isomorphisms can be given by a cap product with a suitable fundamental class and he…

Two references added and the introduction slightly expanded. We show that the tree-level part of a recent theory of invariants of 3-manifolds (due, independently, to Goussarov and Habiro) is essentially given by classical algebraic topology in terms of the Johnson homomorphism and Massey products, for arbitrary 3-manif…

We present an alternative definition for the Goussarov--Habiro filtration of the Z-module freely generated by oriented integral homology 3-spheres, by means of Lagrangian-preserving homology handlebody replacements (LP-surgeries). Garoufalidis, Goussarov and Polyak proved that the graded space (G_n)_n associated to thi…