arXiv research

The paper classifies orbits of semisimple elements in real semisimple Lie algebras.

In this note we propose a method to classify homogeneous nilpotent elements in a real $Z_m$-graded semisimple Lie algebra $g$. Using this we describe the set of orbits of homogeneous elements in a real $Z_2$-graded semisimple Lie algebra. A classification of 4-vectors (resp. 4-forms) on $R^8$ can be given using this me…

We define the notion of a Kirby element of a ribbon category C (not necessarily semisimple). Kirby elements lead to 3-manifolds invariants. We characterize (in terms of the structure maps of some categorical Hopf algebra) a set of Kirby elements of C which is sufficiently large to recover the known quantum invariants c…

We formulate and prove that there are "abundant" in nilpotent orbits in real semisimple Lie algebras, in the following sense. If S denotes the collection of hyperbolic elements corresponding the weighted Dynkin diagrams coming from nilpotent orbits, then S span the maximally expected space, namely, the (-1)-eigenspace …

Study limits of adjoint orbits for Lie groups, describing nilpotent orbits.

We characterise the canonical elements, in the sense of Burstall--Rawnsley \cite{BurRaw90}, of a compact semisimple Lie algebra and discuss the case of $\mathfrak{so}(n)$ in detail. In so doing, we correct two errors in Burstall et al. \cite{BurEscFerTri04}.

Semisimple (co)adjoint orbits through real hyperbolic elements are well-known to be symplectomorphic to cotangent bundles. We provide a new proof of this fact based on elementary results on both Lie theory and symplectic geometry. Our proof establishes a new connection between the Iwasawa horospherical projection and t…

This paper concerns the topology of isospectral real manifolds of certain Jacobi elements associated with real split semisimple Lie algebras. The manifolds are related to the compactified level sets of the generalized (nonperiodic) Toda lattice equations defined on the semisimple Lie algebras. We then give a cellular d…

We construct families of functions in involution for transverse Poisson structures at nilpotent elements of Lie-Poisson structures on simple Lie algebras by using the argument shift method. Examples show that these families contain completely integrable systems that consist of polynomial functions. We provide a uniform…

The study shows that certain spacetimes are isospectrally rigid.

We investigate the properties of principal elements of Frobenius Lie algebras, following the work of M. Gerstenhaber and A. Giaquinto. We prove that any Lie algebra with a left symmetric algebra structure can be embedded, in a natural way, as a subalgebra of some sl(m,K), for K= R or C. Hence, the work of Belavin and D…

Let G_R be a Lie group acting on an oriented manifold M, and let $ω$ be an equivariantly closed form on M. If both G_R and M are compact, then the integral $\int_M ω$ is given by the fixed point integral localization formula (Theorem 7.11 in [BGV]). Unfortunately, this formula fails when the acting Lie group G_R is not…

New relation between quantum groups and BPS series.

The purpose of this paper is to extend the explicit geometric evaluation of semisimple orbital integrals for smooth kernels for the Casimir operator obtained by the first author to the case of kernels for arbitrary elements in the center of the enveloping algebra.

Establishes geometric properties of elements in the positive semigroup of a general real semisimple Lie group.

We consider Lie groups ${\rm SU}(n,1)$ and ${\rm Sp}(n,1)$ that act as the isometries of the complex and quaternionic hyperbolic spaces respectively. We classify pairs of semisimple elements in ${\rm Sp}(n,1)$ and ${\rm SU}(n,1)$ up to conjugacy. This gives local parametrization of the representations $ρ$ in $Hom(F_2, …

We show that a basis of a semisimple Lie algebra of compact type, for which any diagonal left-invariant metric has a diagonal Ricci tensor, is characterized by the Lie algebraic condition of being "nice". Namely, the bracket of any two basis elements is a multiple of another basis element. This extends the work of Laur…

We develop an inductive approach to the representation theory of the Yokonuma-Hecke algebra ${\rm Y}_{d,n}(q)$, based on the study of the spectrum of its Jucys-Murphy elements which are defined here. We give explicit formulas for the irreducible representations of ${\rm Y}_{d,n}(q)$ in terms of standard $d$-tableaux; w…

Consider a proper, isometric action by a unimodular locally compact group $G$ on a Riemannian manifold $M$ with boundary, such that $M/G$ is compact. For an equivariant, elliptic operator $D$ on $M$, and an element $g \in G$, we define a numerical index $\operatorname{index}_g(D)$, in terms of a parametrix for $D$ and …

Constructs bihamiltonian structures from Lie algebras for specific types of nilpotent elements.

Let $G$ be a semisimple Lie group with discrete series. We use maps $K_0(C^*_rG)\to \mathbb{C}$ defined by orbital integrals to recover group theoretic information about $G$, including information contained in $K$-theory classes not associated to the discrete series. An important tool is a fixed point formula for equiv…

The paper counts conjugacy classes of loxodromic elements in Anosov subgroups with a power saving error term.

Method finds MAEs on contactified para-Kähler manifolds.

A formula connects two algebraic structures derived from a category.

We consider Drinfeld-Sokolov bihamiltonian structure associated to a distinguished nilpotent elements of semisimple type and the space of common equilibrium points defined by its leading term. On this space, we construct a local bihamiltonian structure which form an exact Poisson pencil, defines an algebraic classical …

Defines rho numbers for metrics with positive scalar curvature.

A method of constructing a class of bihamiltonian structures is presented. Elements of this class are generalizations of the so-called bihamiltonian structures of general position on odd-dimensional manifolds. The method consists in a simultaneous reduction of both the real and imaginary parts of a complex symplectic f…

A knot in the 3-sphere in genus-1 1-bridge position (called a (1,1)-position) can be described by an element of the braid group of two points in the torus. Our main results tell how to translate between a braid group element and the sequence of slope invariants of the upper and lower tunnels of the (1,1)-position. Afte…

The paper calculates the full asymptotics of analytic torsions for compact orbifolds.

A para-Kähler manifold can be defined as a pseudo-Riemannian manifold $(M,g)$ with a parallel skew-symmetric para-complex structures $K$, i.e. a parallel field of skew-symmetric endomorphisms with $K^2 = \mathrm{Id}$ or, equivalently, as a symplectic manifold $(M,ω)$ with a bi-Lagrangian structure $L^\pm$, i.e. two c…

We prove sharp limit theorems on random walks on graphs with values in finite groups. We then apply these results (together with some elementary algebraic geometry, number theory, and representation theory) to finite quotients of lattices in semisimple Lie groups (specifically SL(n,Z) and Sp(2n, Z) to show that a ``ran…

Uniform systole bounds for arithmetic orbifolds and number fields.

The HKR (Hennings-Kauffman-Radford) framework is used to construct invariants of 4-thickenings of 2-dimensional CW complexes under 2-deformations (1- and 2- handle slides and creations and cancellations of 1-2 handle pairs). The input of the invariant is a finite dimensional unimodular ribbon Hopf algebra A and an elem…

New skein categories for non-semisimple settings, extending existing theory.

Given a semisimple stable autonomous tensor category over a field $K$, to any group presentation with finite number of generators we associate an element $Q(P)\in K$ invariant under the Andrews-Curtis moves. We show that in fact, this is the same invariant as the one produced by the algorithm of Frank Quinn. The new de…

ETQFTs created from non-semisimple modular categories.

We study post-Lie algebra structures on pairs of Lie algebras (g,n), and prove existence results for the case that one of the Lie algebras is semisimple. For semisimple g and solvable n we show that there exist no post-Lie algebra structures on (g,n). For semisimple n and certain solvable g we construct canonical post-…

For finitely generated groups $G$ and $H$ equipped with word metrics, a translation-like action of $H$ on $G$ is a free action where each element of $H$ moves elements of $G$ a bounded distance. Translation-like actions provide a geometric generalization of subgroup containment. Extending work of Cohen, we show that co…

Semisimple Lie groups act transitively on pseudo-Riemannian manifolds, making them flat.

A recent preprint of Csikós, Pyber and Szabó (arXiv:1411.7524) proves that the diffeomorphism group of $T^2\times S^2$ is not Jordan. The purpose of this paper is to generalize the arguments of Csikós, Pyber and Szabó in order to obtain many other examples of compact manifolds whose diffeomorphism group fails to be Jor…

Study non-semisimple TQFT for Burau representation density and unitarity.

Formulae for special almost-complex structures on Vogan diagrams.

We introduce the notion of a conformal pseudo-subriemannian fundamental graded Lie algebra of semisimple type. Moreover we give a classification of conformal pseudo-subriemannian fundamental graded Lie algebras of semisimple type and their prolongations.

Constructs maps on skein modules using non-semisimple quantum invariants.

New pseudo-Hermitian models from non-semisimple TQFTs.

Proves Witten-Reshetikhin-Turaev 3-TQFT as a boundary condition of Crane-Yetter 4-TQFT.

Study para-Sasakian φ-symmetric spaces using Boothby-Wang fibration.

New non-semisimple Ising anyons enable robust universal quantum computation.