arXiv research

New representation for braid groups and surface braid groups, extending Lawrence-Krammer-Bigelow.

Every finite dimensional real representation of a compact real semisimple Lie algebra determines a metric 2-step nilpotent Lie algebra and a corresponding simply connected metric 2-step nilpotent Lie group N. We study the differential geometry of N using representation theory of the complexified complex semisimple Lie …

The study calculates Poincaré series for character varieties of nilpotent groups.

The main goal of this paper is to compute $μ(\g)$ and $μ_{nil}(\g)$ for each nilpotent Lie algebra $\g$ of dimension 6 over a field of characteristic zero $\k$. Here $μ(\g)$ and $μ_{nil}(\g)$ is the minimal dimension of a faithful representation of $\g$ and the minimal dimension of a faithful nilrepresentation of $\g$,…

Researchers compute determinants and torsions of Rumin complex in specific Lie group representations.

Extends Floquet-Bloch theory to nilpotent groups for geometric applications.

The Gaiotto locus for Sp(2n) is shown to lie in the nilpotent cone.

New bi-Hamiltonian systems found on specific Lie groups.

Researchers identify the quiver structure at the bottom of the twisted nilpotent cone on $\mathbb{P}^1$.

We give some examples of non-complete invariant affine connections on nilpotent and filiform Lie groups. This permits to describe non-nilpotent faithful representations on the model of filiform n-dimensional Lie algebras and, in particular, on the 3 dimensional Heisenberg algebra.

Study sigma-models on flag manifolds with curvature zero and relate to chiral models.

The paper classifies vector fields on 5D nilpotent Lie groups.

Quantum group invariants from Lie superalgebra representations at roots of unity.

Study prolongations of nilpotent Lie algebras with specific structural subalgebras.

The starting point of our analysis is an old idea of writing an eigenfunction expansion for a heat kernel considered in the case of a hypoelliptic heat kernel on a nilpotent Lie group $G$. One of the ingredients of this approach is the generalized Fourier transform. The formula one gets using this approach is explicit …

The paper establishes analogs of Stallings' theorem for group homomorphisms and their nilpotent quotients.

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

We study the conditions for a nilpotent Lie group to be foliated into subgroups that have square integrable (relative discrete series) unitary representations, that fit together to form a filtration by normal subgroups. Then we use that filtration to construct a class of "stepwise square integrable" representations on …

Let G be a complex reductive linear algebraic group and let K be a maximal compact subgroup of G. Given a nilpotent group Γgenerated by r elements, we consider the representation spaces Hom(Γ,G) and Hom(Γ,K) with the natural topology induced from an embedding into G^r and K^r respectively. The goal of this paper is to …

The paper calculates heat kernel and closed geodesic asymptotics for nilpotent coverings.

A 2-step nilpotent Lie algebra n is called nonsingular if ad(X): n --> [n,n] is onto for any X not in [n,n]. We explore nonsingular algebras in several directions, including the classification problem (isomorphism invariants), the existence of canonical inner products (nilsolitons) and their automorphism groups (maxima…

For a stratified symplectic space, a suitable concept of stratified Kaehler polarization, defined in terms of an appropriate Lie-Rinehart algebra, encapsulates Kaehler polarizations on the strata and the behaviour of the polarizations across the strata and leads to the notion of stratified Kaehler space. This notion es…

Study describes signatures of Ricci curvature on nilmanifolds.

In the classification theorems of Vinberg and Yakimova for commutative nilmanifolds, the relevant nilpotent groups have a very surprising analytic property. The manifolds are of the form $G/K = N \rtimes K/K$ where, in all but three cases, the nilpotent group $N$ has irreducible unitary representations whose coefficien…

Characterizes holonomies of convex projective cusps.

Study on special Lie groups with Lorentzian metrics.

We continue the study of the distribution of closed geodesics on nilmanifolds constructed from a simply connected 2-step nilpotent Lie group with a left invariant metric and a lattice. We consider a Lie group with an associated 2-step nilpotent Lie algebra constructed from an irreducible representation of a compact sem…

The paper connects Hodge theory and modular forms to prove inequalities.

Classifies real trivectors in 9D using Galois cohomology.

Nilpotent quandles have simple characterizations and are closely related to nilpotency.

For a lattice $Γ$ of a simply connected solvable Lie group $G$, we describe the analytic germ in the variety of representations of $Γ$ at the trivial representation as an analytic germ which is linearly embedded in the analytic germ associated with the nilpotent Lie algebra determined by $G$. By this description, under…

This thesis was inspired by work of M. Cowling, F. De Mari, A. Koranyi and M. Reimann, who studied multicontact structures for the homogeneous manifolds G/P, where G is a semisimple Lie group and P is the minimal parabolic subgroup of G. The multicontact structure here arises naturally by the nilpotent component N of t…

Study harmonic maps into Cayley plane using twistor theory.

Constructs Lie groups with negative Ricci curvature.

Nilpotent Lie algebras obtained from ordered sets and quivers are algebraic Ricci solitons.

The paper explores conditions for non-abelian tensor products to be linear.

New method constructs nilpotent Lie algebras from quivers.

Study knot invariants using automorphism groups of free nilpotent groups.

This paper deals with naturally reductive pseudo-Riemannian 2-step nilpotent Lie groups $(N, \la \,,\,\ra_N)$, such that $\la \,,\,\ra_N$ is invariant under a left action. The case of nondegenerate center is completely characterized. In fact, whenever $\la \,,\, \ra_N$ restricts to a metric in the center it is proved h…

Nilpotent Sasakian manifolds are Heisenberg nilmanifolds.

We exhibit pseudo Riemannian manifolds which are Szabó nilpotent of arbitrary order, or which are Osserman nilpotent of arbitrary order, or which are Ivanov-Petrova nilpotent of order 3.

Classifies complex structures on specific nilpotent Lie algebras.

Researchers extend pseudodifferential calculus on filtered manifolds using fixed point algebras.

We compute the characteristic varieties and the Alexander polynomial of a finitely generated nilpotent group. We show that the first characteristic variety may be used to detect nilpotence. We use the Alexander polynomial to deduce that the only torsion-free, finitely generated nilpotent groups with positive deficiency…

The paper studies how geometric transformations affect semi-classical operators on specific Lie groups.

Study on invariant structures on 7D nilpotent Lie groups, focusing on Sasaki and K-contact.

We study direct limits $(G,K) = \varinjlim (G_n,K_n)$ of Gelfand pairs of the form $G_n = N_n\rtimes K_n$ with $N_n$ nilpotent, in other words pairs $(G_n,K_n)$ for which $G_n/K_n$ is a commutative nilmanifold. First, we extend the criterion of \cite{W3} for a direct limit representation to be multiplicity free. Then w…

This paper classifies Ricci soliton subgroups in a specific type of nilpotent group.