arXiv research

The study describes the free Lie-Yamaguti algebra.

In this paper we first state the classification of the prolongations of complex free fundamental graded Lie algebras. Next we introduce the notion of free pseudo-product fundamental graded Lie algebras and study the prolongations of complex free pseudo-product fundamental graded Lie algebras. Furthermore we investigate…

We determine the abelianizations of the following three kinds of graded Lie algebras in certain stable ranges: derivations of the free associative algebra, derivations of the free Lie algebra and symplectic derivations of the free associative algebra. In each case, we consider both the whole derivation Lie algebra and …

New groups found that don't virtually algebraically fiber, related to mapping class group orbits.

New Lie-group methods preserve geometric divergence-free features on manifolds.

In this work we study the problem of existence of symplectic structures on free nilpotent Lie algebras. Necessary and sufficient conditions are given for even dimensional ones. The one dimensional central extension for odd dimensional free nilpotent Lie algebras is also considered.

Researchers solve a 25-year-old conjecture about vector fields.

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.

Proves incoherence of free-by-free and surface-by-free groups, solving two problems.

Constructive approach to Lie algebra gradings, computing maximal and enumerating all gradings.

Poly-free groups are constructed as iterated semidirect products of free groups. The class of poly-free groups includes the classical pure braid groups, fundamental groups of fiber-type hyperplane arrangements, and certain subgroups of the automorphism groups of free groups. The purpose of this article is to compute ce…

We present structural properties of Lie algebras admitting symmetric, invariant and nondegenerate bilinear forms. We show that these properties are not satisfied by nilradicals of parabolic subalgebras of real split forms of complex simple Lie algebras, neither by 2-step nilpotent Lie algebras associated with graphs, w…

Develops a new method to study algebraic tangent cones of sheaves using valuations.

A new algebra for Frobenius manifolds solves PDEs and constraints.

Found a basis and presentation for a specific algebra.

In this review paper, we present several results on central extensions of the Lie algebra of symplectic (Hamiltonian) vector fields, and compare them to similar results for the Lie algebra of (exact) divergence free vector fields. In particular, we comment on universal central extensions and integrability to the group …

We construct an abelian quotient of the symplectic derivation Lie algebra $\mathfrak{h}_{g,1}$ of the free Lie algebra generated by the fundamental representation of $\mathrm{Sp}(2g,\mathbb{Q})$. More specifically, we show that the weight $12$ part of the abelianization of $\mathfrak{h}_{g,1}$ is $1$-dimensional for $g…

We show that the algebraic intersection number of Scott and Swarup for splittings of free groups coincides with the geometric intersection number for the sphere complex of the connected sum of copies of $S^2\times S^1$.

We classify solvable Lie groups with a free nilradical admitting an Einstein left-invariant metric. Any such group is essentially determined by the nilradical of its Lie algebra, which is then called an Einstein nilradical. We show that among the free Lie algebras, there are very few Einstein nilradicals. Except for th…

Develops a spectral sequence for Lie group actions on manifolds.

Let X be the moduli space of SL(3,C) representations of a free group of rank r. In this paper we describe maximal algebraically independent subsets of certain minimal sets of coordinate functions on X. These subsets locally parametrize the moduli space.

The paper studies algebraic integer relations and sequences converging to 4.

The aim of our paper is to construct pseudo $H$-type algebras from the covering free nilpotent two-step Lie algebra as the quotient algebra by an ideal. We propose an explicit algorithm of construction of such an ideal by making use of a non-degenerate scalar product. Moreover, as a bypass result, we recover the existe…

Efficient algorithms decide algebraic constraints of causal graphs.

We prove that any countable discrete and torsion free subgroup of a general linear group over an arbitrary field or a similar subgroup of an almost connected Lie group satisfies the integral algebraic K-theoretic (split) Novikov conjecture over \cpt and §, where \cpt denotes the C^*-algebra of compact operators and §de…

The study classifies complex parallelisable nilmanifolds with unobstructed deformations.

Characterizes almost Abelian Lie algebras with special $H$-structures.

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

Motivated by the Moore-Segal axioms for an open-closed topological field theory, we consider planar open string topological field theories. We rigorously define a category 2Thick whose objects and morphisms can be thought of as open strings and diffeomorphism classes of planar open string worldsheets. Just as the categ…

We construct an action of the free group $F_n$ on the homotopy category of projective modules over a finite dimensional zigzag algebra. The main theorem in the paper is that this action is faithful. We describe the relationship between homotopy classes of paths in the punctured disc and complexes of projective zigzag m…

Researchers describe Casimir functions for 3- and 4-step nilpotent Lie groups.

Normal forms and moduli stacks for flat connections on complex manifolds.

Study homotopy types of free racks and quandles, proving analogs of Milnor's theorem.

In this paper we present a new characterization of free group actions (in classical differential geometry), involving dynamical systems and representations of the corresponding transformation groups. In fact, given a dynamical system, we provide conditions including the existence of "sufficiently many" representations …

We classify the holonomy algebras of manifolds admitting an indecomposable torsion free $G_2^*$-structure, i.e. for which the holonomy representation does not leave invariant any proper non-degenerate subspace. We realize some of these Lie algebras as holonomy algebras of left-invariant metrics on Lie groups.

That announcement gives the structure of totally reducible linear Lie algebras which are the Lie algebra of the holonomy group of (at least) one torsion-free connection. The result uses the (already known) classi cation of the irreducible ones and some previous (unpublished) works by the author giving the classi cation…

We study the 8 natural GL equivariant geometric realization questions for the space of generalized algebraic curvature tensors. All but one of them is solvable; a non-zero projectively flat Ricci antisymmetric generalized algebraic curvature is not geometrically realizable by a projectively flat Ricci antisymmetric tor…

We examine free orientation-reversing group actions on orientable handlebodies, and free actions on nonorientable handlebodies. A classification theorem is obtained, giving the equivalence classes and weak equivalence classes of free actions in terms of algebraic invariants that involve Nielsen equivalence. This is app…

We study a natural Lie algebra structure on the free vector space generated by all rooted planar trees as the associated Lie algebra of the nonsymmetric operad (non-$Σ$ operad, preoperad) of rooted planar trees. We determine whether the Lie algebra and some related Lie algebras are finitely generated or not, and prove …

In this note we present a description of wave front evolving from an algebraic hypersurface by means of a pull-back of the discriminantal loci of a tame polynomial via a polynomial mapping. As an application we give examples of wave fronts which define free/almost free divisors near the focal point.

We study the natural map eta between a group of binary planar trees whose leaves are labeled by elements of a free abelian group H and a certain group D(H) derived from the free Lie algebra over H. Both of these groups arise in several different topological contexts. The map eta is known to be an isomorphism over Q, bu…

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 examine questions of geometric realizability for algebraic structures which arise naturally in affine and Riemannian geometry. Suppose given an algebraic curvature operator R at a point P of a manifold M and suppose given a real analytic (resp. C-k for finite k at least 2) pseudo-Riemannian metric on M defined near …

Let X be the moduli space of SL(n,C), SU(n), GL(n,C), or U(n)-valued representations of a rank r free group. We classify the algebraic singular stratification of X. This comes down to showing that the singular locus corresponds exactly to reducible representations if there exist singularities at all. Then by relating a…

A deformation of the Orlik-Solomon algebra of a matroid M is defined as a quotient of the free associative algebra over a commutative ring R with 1. It is shown that the given generators form a Groebner basis and that after suitable homogenization the deformation and the Orlik-Solomon have the same Hilbert series as R-…

The goal of this work is to study the ideals of the Goldman Lie algebra $S$. To do so, we construct an algebra homomorphism from $S$ to a simpler algebraic structure, and focus on finding ideals of this new structure instead. The structure $S$ can be regarded as either a $\mathbb{Q}$-module or a $\mathbb{Q}$-module gen…

The concordance group of algebraically slice knots is the subgroup of the classical knot concordance group formed by algebraically slice knots. Results of Casson and Gordon and of Jiang showed that this group contains in infinitely generated free (abelian) subgroup. Here it is shown that the concordance group of algebr…

New findings on algebraic structure of hyperbolic graph braid groups.