arXiv research

In this paper, we present a study on the prolongations of representations of Lie algebras. We show that a tangent bundle of a given Lie algebra attains a Lie algebra structure. Then, we prove that this tangent bundle is algebraically isomorphic to the Lie algebra of a tangent bundle of a Lie group. Using these, we defi…

Survey on stated skein algebras and their representations.

The concept of F-algebra and its representation can be extended to an arbitrary bundle. We define operations of fibered F-algebra in fiber. The paper presents the representation theory of of fibered F-algebra as well as a comparison of representation of F-algebra and of representation of fibered F-algebra.

Using crossed homomorphisms, we show that the category of weak representations (resp. admissible representations) of Lie-Rinehart algebras (resp. Leibniz pairs) is a left module category over the monoidal category of representations of Lie algebras. In particular, the corresponding bifunctor of monoidal categories is e…

We prove that a polar orthogonal representation of a real reductive algebraic group has the same closed orbits as the isotropy representation of a pseudo-Riemannian symmetric space. We also develop a partial structural theory of polar orthogonal representations of real reductive algebraic groups which slightly generali…

Heegaard Floer homology connects to polynomial representations of Hecke algebras.

We obtain minimal dimension matrix representations for each indecomposable five-dimensional Lie algebra over $\R$ and justify in each case that they are minimal. In each case a matrix Lie group is given whose matrix Lie algebra provides the required representation.

These lectures given in Montreal in Summer 1997 are mainly based on, and form a condensed survey of, the book by N. Chriss and V. Ginzburg: `Representation Theory and Complex Geometry', Birkhauser 1997. Various algebras arising naturally in Representation Theory such as the group algebra of a Weyl group, the universal …

Researchers present and compare different representations of dissipative Hamiltonian DAE systems.

Maximal Laplacian algebras applied to invariant theory solved inverse problems.

Study quandle modules over geometric quandles and their relation to Lie-Yamaguti representations.

Researchers create explicit representations for skein algebras of small surfaces, revealing their Azumaya loci.

Study mapping class groups and their representations linking to algebraic structures.

Representations of two bridge knot groups in the isometry group of some complete Riemannian 3-manifolds as $E^{3}$ (Euclidean 3-space), $H^{3}$ (hyperbolic 3-space) and $E^{2,1}$ (Minkowski 3-space), using quaternion algebra theory, are studied. We study the different representations of a 2-generator group in which th…

Constructs a representation of the string 2-group on a von Neumann algebra.

Researchers use Mellin-Barnes integrals to study trinomial equations and their braids.

New 2-representations link spectral enhancements in link homology.

Introduces a new algebra from loop braid groups with properties similar to Hecke algebras.

We study the semidirect product of a Lie algebra with a representation up to homotopy and provide various examples coming from Courant algebroids, string Lie 2-algebras, and omni-Lie algebroids. In the end, we study the semidirect product of a Lie group with a representation up to homotopy and use it to give an integra…

Researchers define new quantum representations for a Lorentz algebra and study their Clebsch-Gordan decomposition.

Constructs positive energy representations from Toda equations Stokes data.

In order to understand the structure of the cohomologies involved in the study of projectively equivariant quantizations, we introduce a notion of affine representation of a Lie algebra.We show how it is related to linear representations and 1-cohomology classes of the algebra. We classify the affine representations of…

The paper connects two skein algebras and characterizes their representations.

Classifies prime algebraic tangles up to 14 crossings.

The general method of Reshetikhin and Turaev is followed to develop topological invariants of closed, connected, orientable 3-manifolds from a new class of algebras called pseudo-modular Hopf algebras. Pseudo-modular Hopf algebras are a class of Z_2-graded ribbon Hopf algebras that generalise the concept of a modular H…

Given a Lie group G whose Lie algebra is endowed with a nondegenerate invariant symmetric bilinear form, we construct a Poisson algebra of continuous functions on a certain open subspace R of the space of representations in G of the fundamental group of a compact connected orientable topological surface with finitely m…

The paper studies properties of stated SL(n)-skein algebras and their centers.

Classifies Zariski closures of positive representations in Lie groups.

This thesis explores DAHA representations using stated skein theory.

New knot invariants derived using quantum cluster algebras.

The paper formalizes how concepts are encoded in text-guided generative models and provides a method to manipulate them.

We present in this paper the algebra of fused permutations and its deformation the fused Hecke algebra. The first one is defined on a set of combinatorial objects that we call fused permutations, and its deformation is defined on a set of topological objects that we call fused braids. We use these algebras to prove a S…

Floer homology connects to quiver Hecke algebras in Coulomb branches.

New dg-algebras link graph colorings to sheaves.

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 …

This is the third article in the series begun with [BonWon3, BonWon4], devoted to finite-dimensional representations of the Kauffman bracket skein algebra of an oriented surface $S$. In [BonWon3] we associated a classical shadow to an irreducible representation $ρ$ of the skein algebra, which is a character $r_ρ\in \ma…

Representations of $C^*$-algebras are realized on section spaces of holomorphic homogeneous vector bundles. The corresponding section spaces are investigated by means of a new notion of reproducing kernel, suitable for dealing with involutive diffeomorphisms defined on the base spaces of the bundles. Applications of th…

Paper constructs representations for virtual braids and flat braids.

New structure for quantum algebra representations.

Importance of theorem dedicated to isomorphisms consist in statement that they allow to identify different mathematical objects which have something common from the point of view of certain model. This paper considers morphisms of \Ts representation of $\mathfrak{F}$\Hyph algebra and morphisms of \Ts representation of …

We provide an explicit algorithm to calculate invariant tensors for the adjoint representation of the simple Lie algebra $sl(n)$, as well as arbitrary representation in terms of roots. We also obtain explicit formulae for the adjoint representations of the orthogonal and symplectic Lie algebras $so(n)$ and $sp(n)$.

The study examines representations of compactly supported diffeomorphisms with a positive energy condition.

The objective of this paper is to determine the finite dimensional, indecomposable representations of the algebra that is generated by two complex structures over the real numbers. Since the generators satisfy relations that are similar to those of the infinite dihedral group, we give the algebra the name iD-infinity.

Measuring comodules are defined and shown to provide a useful generalization of the set of maps between modules with a broad range of applications. Three applications are described. Connections on bundles are described in terms of measuring comodules, enabling curvature to be defined under general algebraic circumstanc…

We propose a new non-commutative generalization of the representation variety and the character variety of a knot group. Our strategy is to reformulate the construction of the algebra of functions on the space of representations in terms of Hopf algebra objects in a braided category (braided Hopf algebra). The construc…

We give criteria for real, complex and quaternionic representations to define s-representations, focusing on exceptional Lie algebras defined by spin representations. As applications, we obtain the classification of complex representations whose second exterior power is irreducible or has an irreducible summand of co-d…

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 $\g$,…

The thesis explores centralisers and Hecke algebras in representation theory with applications to knots and physics.