arXiv research

Study of coloured invariants of torus knots using $\mathcal{W}$ algebras.

The trace of the affine Hecke category is compared with the elliptic Hall algebra.

Develops quantum character theory for complex reductive groups.

Let $\{{\cdot},{\cdot}\}_{\boldsymbol{\mathcal{P}}}$ be a variational Poisson bracket in a field model on an affine bundle $π$ over an affine base manifold $M^m$. Denote by $\times$ the commutative associative multiplication in the Poisson algebra $\boldsymbol{\mathcal{A}}$ of local functionals $Γ(π)\to\Bbbk$ that take…

The paper studies a special Grassmannian space and shows it's an orbit of a unitary group.

For compact complex manifolds with vanishing first Chern class that are compact torus principal bundles over Kähler manifolds, we prove that all holomorphic geometric structures on them, of affine type, are locally homogeneous. For a compact simply connected complex manifold in Fujiki class $\mathcal C$, whose dimensio…

Constructs a new graded variety from algebraic data.

Develops homotopies for Lagrangian field theory using advanced algebraic structures.

In this paper we construct a homomorphism of the affine braid group $Br_n^{aff}$ in the convolution algebra of the equivariant matrix factorizations on the space $\overline{\mathcal{X}}_2=\mathfrak{b}_n\times GL_n\times\mathfrak{n}_n$ considered in the earlier paper of the authors. We explain that the pull-back on the …

Let $M$ be a Riemannian manifold. For $p\in M$, the tensor algebra of the negative part of the (complex) affinization of the tangent space of $M$ at $p$ has a natural structure of a meromorphic open-string vertex algebra. These meromorphic open-string vertex algebras form a vector bundle over $M$ with a connection. We …

For six dimensional nilmanifolds we build a module $\mathcal{H}$ of an affine Kac Moody vertex algebras. Then, we associate some logarithmic fields for the module $\mathcal{H}$ and we study their singularities. We also presented a physics motivation behind this construction. We study a particular case, we show that whe…

If $\mathcal{M}=(M,\nabla)$ is an affine surface, let $\mathcal{Q}(\mathcal{M}):=\ker(\mathcal{H}+\frac1{m-1}ρ_s)$ be the space of solutions to the quasi-Einstein equation for the crucial eigenvalue. Let $\tilde{\mathcal{M}}=(M,\tilde\nabla)$ be another affine structure on $M$ which is strongly projectively flat. We sh…

Paper studies symmetries in singular foliations using Lie $\infty$-morphisms.

Let $\widetilde{J}$ be the canonical para-complex structure on $\mathbb{R}^4$. In this paper we study $3$-dimensional centro-affine hypersurfaces with a $\widetilde{J}$-tangent centro-affine vector field (sometimes called $\widetilde{J}$-tangent centro-affine hypersurfaces) as well as $3$-dimensional $\widetilde{J}$-ta…

The paper characterizes and studies compact subsets of complex projective space with specific line intersection properties.

Let $\mathcal{M}$ be a Type $\mathcal{A}$ affine surface. We show that $\mathcal{M}$ is linearly strongly projectively flat. We use the quasi-Einstein equation together with the condition that $\mathcal{M}$ is strongly projectively flat to examine to examine the geodesic completeness of $\mathcal{M}$.

The thesis explores kinematical symmetries beyond Lorentzian spacetime.

Consider the $3$-dimensional $\mathcal N=4$ supersymmetric gauge theory associated with a compact Lie group $G_c$ and its quaternionic representation $\mathbf M$. Physicists study its Coulomb branch, which is a noncompact hyper-Kähler manifold with an $\mathrm{SU}(2)$-action, possibly with singularities. We give a math…

Let $A$ be an expanding $d\times d$ matrix with integer entries and ${\mathcal D}\subset {\mathbb Z}^d$ be a finite digit set. Then the pair $(A, {\mathcal D})$ defines a unique integral self-affine set $K=A^{-1}(K+{\mathcal D})$. In this paper, by replacing the Euclidean norm with a pseudo-norm $w$ in terms of $A$, we…

In this paper we study some affine structures on nilpotent Lie algebras endowed with a contact form. These affine structures are constructed from an affine structure on a symplectic Lie algebra by a central extension.

We obtain a characterization of the real Lie algebras admitting abelian complex structures in terms of certain affine Lie algebras $\frak a \frak f \frak f (A)$, where $A$ is a commutative algebra. These affine Lie algebras are natural generalizations of $\frak a \frak f \frak f (\Bbb C)$ and the corresponding Lie grou…

For product manifolds, cohomologically calibrated affine connections are geometrically irreducible.

New complex surfaces found with interesting geometric properties.

We realise Stroppel's extended arc algebra in the Fukaya-Seidel category of a natural Lefschetz fibration on the generic fiber of the adjoint quotient map on a type $A$ nilpotent slice with two Jordan blocks, and hence obtain a symplectic interpretation of certain parabolic two-block versions of Bernstein-Gelfan'd-Gelf…

In this paper, we consider the connectedness of planar self-affine set $T(A,\mathcal{D})$ arising from an integral expanding matrix $A$ with characteristic polynomial $f(x)=x^2+bx+c$ and a digit set $\mathcal{D}=\{0,1,\dots, m\}v$. The necessary and sufficient conditions only depending on $b,c,m$ are given for the $T(A…

Algebraic geometry replaces manifolds in differential geometry.

Analytic structure found on manifold of idempotent operators.

Weil algebra morphism induce natural transformations between Weil bundles. In some well known cases, a natural transformation is endowed with a canonical structure of affine bundle. We show that this structure arises only when the Weil algebra morphism is surjective and its kernel has null square. Moreover, in some cas…

Abstract M5 branes on ADE singularities yields BPS spectrum and partition functions.

Real analytic ($\mathcal{C}^ω$) surfaces $S^2$ in $\mathbb{R}^3 \ni (x,y,u)$ graphed as $\big\{ u = F(x,y) \big\}$ with $F_{xx} \neq 0$ whose Gaussian curvature vanishes identically: \[ 0 \,\equiv\, F_{xx}\,F_{yy} - F_{xy}^2, \] possess, under the action of the affine transformation group ${\sf Aff}_3(\mathbb{R}) = {\s…

Affine $λ$-equidistants of convex polygons with parallel opposite sides have applications to isoperimetric inequalities.

Affine deformations of cotangent groupoids

Self-affine tiles homeomorphic to a ball proven for a specific digit set.

We present a construction of invariants for links using an isomorphism theorem for affine Yokonuma--Hecke algebras. The isomorphism relates affine Yokonuma--Hecke algebras with usual affine Hecke algebras. We use it to construct a large class of Markov traces on affine Yokonuma--Hecke algebras, and in turn, to produce …

Riemannian symmetric spaces are fundamental objects in finite dimensional differential geometry. An important problem is the construction of symmetric spaces for generalizations of simple Lie groups, especially their closest infinite dimensional analogues known as Kac-Moody groups. We solve this problem and construct a…

Study on affine surfaces with specific algebraic properties.

This paper studies Hopf algebras from dg manifolds.

This paper deals with affine connections on real manifolds. We give a new characterization of flat affine connections on real manifolds by means of certain affine representations of the Lie group of automorphisms preserving the connection. Then we specialize the characterization to the case of a left invariant connecti…

Let us denote by $\mathcal K_n$ the hyperspace of all convex bodies of $\mathbb R^n$ equipped with the Hausdorff distance topology. An affine invariant point $p$ is a continuous and Aff(n)-equivariant map $p:\mathcal K_n\to \mathbb R^n$, where Aff(n) denotes the group of all nonsingular affine maps of $\mathbb R^n$. Fo…

Develops a theory for mth order p-affine capacity for convex bodies containing the origin.

We analyze the moduli space of non-flat homogeneous affine connections on surfaces. For Type $\mathcal{A}$ surfaces, we write down complete sets of invariants that determine the local isomorphism type depending on the rank of the Ricci tensor and examine the structure of the associated moduli space. For Type $\mathcal{…

Study numerically flat bundles on Fujiki manifolds using algebraic groups.

New algebraic structure characterizes vector bundles without module requirement.

In this note we prove that every non characteristically filiform Lie algebra is endowed with an affine structure.

Study special affine connections on symmetric spaces and their products.

The ordinary (or classical) Birman-Wenzl-Murakami algebras were initially conceived as an algebraic framework for the Kauffman link invariant. They also appear as centralizer algebras for representations of quantum universal enveloping algebras of orthogonal or symplectic types. It was shown by Morton and Wassermann th…

New algebra for twice-punctured torus curves.

Builds geometric structures for algebraic groups over real closed fields.