arXiv research

Paper defines and computes a new weight system for gl_N Lie algebra.

In this paper we define an explicit basis for the $\mathfrak{gl}_n$-web algebra $H_n(\vec{k})$ (the $\mathfrak{gl}_n$ generalization of Khovanov's arc algebra) using categorified $q$-skew Howe duality. Our construction is a $\mathfrak{gl}_n$-web version of Hu--Mathas' graded cellular basis and has two major application…

Researchers compute $\mathfrak{gl}_2$-skein modules for lens spaces.

We show that the algebra of functions on the Grassmann supergroup Gr$_q(1| 1)$ has a (graded) Hopf algebra structure related to GL$_q(1| 1)$.

Study of generalized Legendrian racks and their GL-structures.

Foams have Lie algebra symmetries that simplify web state spaces.

The paper connects GL-racks to knot coloring invariants.

Irreducible skew-Berger algebras $\g\subset\gl(n,\Co)$, i.e. algebras spanned by the images of the linear maps $R:\odot^2\Co^n\to\g$ satisfying the Bianchi identity, are classified. These Lie algebras can be interpreted as irreducible complex Berger superalgebras contained in $\gl(0|n,\Co)$.

Extends gl(m|k) construction using Hilbert scheme of points.

New structure for quantum algebra representations.

We define parameter dependent $\mathfrak{gl}_2$-foams and their associated web and arc algebras, and verify that they specialize to several known $\mathfrak{sl}_2$ or $\mathfrak{gl}_2$ constructions related to higher link and tangle invariants. Moreover, we show that all these specializations are equivalent, and we ded…

The two-parametric quantum deformation of the algebra of coordinate functions on the supergroup GL$(1| 1)$ via a contraction of GL$_{p,q}(1| 1)$ is presented. Related differential calculus on the quantum superplane is introduced.

The paper constructs compatible Poisson brackets on gl(N).

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

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

We develop several applications of the fact that the Yokonuma--Hecke algebra of the general linear group GL is isomorphic to a direct sum of matrix algebras associated to Iwahori--Hecke algebras of type A. This includes a description of the semisimple and modular representation theory of the Yokonuma--Hecke algebras of…

A group of matrices $G$ with entries in a number field $K$ is defined to be numerical if $G$ has a finite index subgroup of matrices whose entries are algebraic integers. It is shown that an irreducible or completely reducible subgroup of $GL(n,K)\subset GL(n,\mathbb{C})$ is numerical if and only if the traces of its e…

For every oriented surface of finite type, we construct a functorial Khovanov homology for links in a thickening of the surface, which takes values in a categorification of the corresponding gl(2) skein module. The latter is a mild refinement of the Kauffman bracket skein algebra, and its categorification is constructe…

We introduce three non-trivial 2-cocycles $c_k$, k=0,1,2, on the Lie algebra $S^3H=Map(S^3,H)$ with the aid of the corresponding basis vector fields on $S^3$, and extend them to 2-cocycles on the Lie algebra $S^3gl(n,H)=S^3H \otimes gl(n,C)$. Then we have the corresponding central extension $S^3gl(n,H)\oplus \oplus_k (…

Let $S$ be a closed connected oriented surface of genus $g>0$. We study a Poisson subalgebra $W_1(g)$ of $C^{\infty}(\mathrm{Hom}(π_1(S), \mathrm{GL}(1, \mathbb{R}))/\mathrm{GL}(1, \mathbb{R}))$, the smooth functions on the moduli space of flat $\mathrm{GL}(1, \mathbb{R})$-bundles over $S$. There is a surjective Lie al…

New algebras and maps defined in knot Floer homology for trivalent vertices.

Researchers found sub-Lorentzian geodesics on a specific Lie subgroup.

The paper studies algebraic structures related to quantum groups.

Oleg Viro studied in arXiv:math/0204290 two interpretations of the (multivariable) Alexander polynomial as a quantum link invariant: either by considering the quasi triangular Hopf algebra associated to $U_q sl(2)$ at fourth roots of unity, or by considering the super Hopf algebra $U_q gl(1|1)$. In this paper, we show …

New algebraic description connects Fukaya category to bordered Floer homology.

We identify the Grothendieck group of the tangle Floer dg algebra with a tensor product of certain $U_q(gl(1|1))$ representations. Under this identification, up to a scalar factor, the map on the Grothendieck group induced by the tangle Floer dg bimodule associated to a tangle agrees with the Reshetikhin-Turaev homomor…

New basis for quantum gl_N invariants derived from Macdonald polynomials.

Develops quantum character theory for complex reductive groups.

We compute the algebraic hull of the Kontsevich-Zorich cocycle over any GL^+_2(R) invariant subvariety of the Hodge bundle, and derive from this finiteness results on such subvarieties.

Motivated by Demailly's strategy towards the Kobayashi hyperbolicity conjecture, we study the action on the k-jets of germs of holomorphic discs in a complex manifold X of the reparametrization group of k-jets of germs of biholomorphisms of the source. This reparametrization group is a subgroup of the general linear gr…

Possible irreducible holonomy algebras $\g\subset\sp(2m,\Real)$ of odd Riemannian supermanifolds and irreducible subalgebras $\g\subset\gl(n,\Real)$ with non-trivial first skew-symmetric prolongations are classified. An approach to the classification of some classes of the holonomy algebras of Riemannian supermanifolds…

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 relate decategorifications of Ozsváth-Szabó's new bordered theory for knot Floer homology to representations of $\mathcal{U}_q(\mathfrak{gl}(1|1))$. Specifically, we consider two subalgebras $\mathcal{C}_r(n,\mathcal{S})$ and $\mathcal{C}_l(n,\mathcal{S})$ of Ozsváth- Szabó's algebra $\mathcal{B}(n,\mathcal{S})$, an…

An almost Clifford and an almost Cliffordian manifold is a $G$--structure based on the definition of Clifford algebras. An almost Clifford manifold based on $\mathcal O:= \cc l (s,t)$ is given by a reduction of the structure group $GL(km, \mathbb R)$ to $GL(m, {\mathcal O})$, where $k=2^{s+t}$ and $m \in \mathbb N$. An…

Innovative rack theory applied to Legendrian links.

The totality Lie(V) of all Lie algebra structures on a vector space V over a field F is an algebraic variety over F on which the group GL(V) acts naturally. We give an explicit description of Lie(V) for dim V=3 which is based on the notion of compatibility of Lie algebra structures.

Rewriting theory applied to diagrammatic algebras for categorification.

We use super $q$-Howe duality to provide diagrammatic presentations of an idempotented form of the Hecke algebra and of categories of $\mathfrak{gl}_N$-modules (and, more generally, $\mathfrak{gl}_{N|M}$-modules) whose objects are tensor generated by exterior and symmetric powers of the vector representations. As an ap…

Researchers calculate dimensions of skein modules for 2-torus mapping tori.

Using the representation of the isometries as 2x2 invertible matrices over the division algebra $\H$ of quaternions, we give an algebraic characterization of the dynamical types of the orientation-preserving isometries of the hyperbolic 5-space. We also determine the conjugacy classes and the conjugacy classes of centr…

The rank $n$ swapping algebra is the Poisson algebra defined on the ordered pairs of points on a circle using the linking numbers, where a subspace of $(\mathbb{K}^n \times \mathbb{K}^{n*})^r/\operatorname{GL}(n,\mathbb{K})$ is its geometric mode. In this paper, we find an injective Poisson homomorphism from the Poisso…

Apart from global topological problems an affine homogeneous space is locally described by its curvature, its torsion and a slightly less tangible object called its connection in a given base point. Using this description of the local geometry of an affine homogeneous space we construct an algebraic variety $\mathfrak{…

We give a criterion of (micro-)kroneckerity of the linear Poisson pencil on $\frak{g}^*$ related to an algebraic Nijenhuis operator $N:\frak{g}\to \frak{g}$ on a finite-dimensional Lie algebra $\frak{g}$. As an application we get a series of examples of completely integrable systems on semisimple Lie algebras related t…

An algebra $A$ with identity $(a\circ b)\circ c-a\circ(b\circ c)=(a\circ c)\circ b-a\circ(c\circ b),$ is called right-symmetric. Cohomology and deformation theory for right-symmetric algebras are developed. Cohomologies of $gl_n$ and half-Witt algebras $W_n^{rsym}, p=0,$ $W_n^{rsym}({\bf m}), p>0,$ are calculated. In p…

We prove that the dimension of the space of primitive Vassiliev invariants of degree n grows - as n tends to infinity - faster than Exp(c Sqrt(n)) for any c < Pi Sqrt (2/3). The proof relies on the use of the weight systems coming from the Lie algebra gl(N). In fact, we show that our bound is - up to multiplication wit…

Witt algebra acts on categorified quantum groups in type A.

We use an isomorphism between the space of valence two Killing tensors on an n-dimensional constant sectional curvature manifold and the irreducible GL(n+1)-representation space of algebraic curvature tensors in order to translate the Nijenhuis integrability conditions for a Killing tensor into purely algebraic integra…

In this note we prove the following three algebraic facts which have applications in the theory of holonomy groups and homogeneous spaces: Any irreducibly acting connected subgroup $G \subset Gl(n,\rr)$ is closed. Moreover, if $G$ admits an invariant bilinear form of Lorentzian signature, $G$ is maximal, i.e. it is con…