Related papers: The classical limit for stated $SL_n$-skein module…

This is the first of a series of papers devoted to certain pairs of commuting nilpotent elements in a semisimple Lie algebra that enjoy quite remarkable properties and which are expected to play a major role in Representation theory. The…

In this paper we present recent results on the computation of skein modules of 3-manifolds using braids and appropriate knot algebras. Skein modules generalize knot polynomials in $S^3$ to knot polynomials in arbitrary 3-manifolds and they…

The skein module for a d-dimensional manifold is a vector space spanned by embedded framed graphs decorated by a category A with suitable extra structure depending on the dimension d, modulo local relations which hold inside d-balls. For a…

We give an $SL_3$ analogue of the triangular decomposition of the Kauffman bracket stated skein algebras described by Le. To any punctured bordered surface, we associate an $SL_3$ stated skein algebra which contains the $SL_3$ skein algebra…

This paper gives an explicit structure theorem for the symmetric group acting on the symmetric algebra of its natural module. Let $G$ be the symmetric group on $x_1,..., x_n$ and let $d_i$ be the $i^{\text{th}}$ elementary symmetric…

We introduce a quantum cluster algebra structure $\mathscr A_\omega(\mathfrak{S})$ inside the skew-field fractions ${\rm Frac}\bigl(\widetilde{\mathscr{S}}_\omega(\mathfrak{S})\bigr)$ of the projected stated ${\rm SL}_n$-skein algebra…

Let $R_n$ denote the KLR algebra of type $A^{(1)}_{e-1}$. Using the presentation of Specht modules given by Kleschev-Mathas-Ram, Loubert completely determined $\hom_{R_n}(S^\mu,S^\lambda)$ where $\mu$ is an arbitrary partition, $\lambda$ is…

A nilmanifold is a (left) quotient of a nilpotent Lie group by a cocompact lattice. A hypercomplex structure on a manifold is a triple of complex structure operators satisfying the quaternionic relations. A hypercomplex nilmanifold is a…

We explicitly construct, in terms of Gelfand--Tsetlin tableaux, a new family of simple positive energy representations for the simple affine vertex algebra V_k(sl_{n+1}) in the minimal nilpotent orbit of sl_{n+1}. These representations are…

The purpose of this paper is to prove that we can construct all finite dimensional irreducible nilpotent modules of type 1 inductively by using Schnizer homomorphisms for quantum algebra at roots of unity of type A, B, C, D or G.

We interpret Manolescu-Neithalath's cabled Khovanov homology formula for computing Morrison-Walker-Wedrich's $\mathrm{KhR}_2$ skein lasagna module as a homotopy colimit (mapping telescope) in a completion of the category of complexes over…

Let $n$ be a positive integer, and let $k$ be a field (of arbitrary characteristic) accessible to symbolic computation. We describe an algorithmic test for determining whether or not a finitely presented $k$-algebra $R$ has infinitely many…

By applying Fourier transformations to the natural orthogonal oscillator representations of special linear Lie algebras, Luo and the second author (2013) obtained a large family of infinite-dimensional irreducible representations of the…

Skein lasagna modules are a recent tool developed for the study of 4-manifolds. We provide general formula for 1-, 2-, and 3-handle attachments for skein modules defined with any functorial link theory in $S^3 \times I$ generalizing…

We study the affine variety $L_{n}(\mathfrak{g})$ of Lie algebra representations, the collection of all homomorphisms from an arbitrary $n$-dimensional Lie algebra into a fixed real semi-simple Lie algebra $\mathfrak{g}$. Using techniques…

In this paper we show that the projective cover of the trivial irreducible module of a finite-dimensional solvable restricted Lie algebra is induced from the one-dimensional trivial module of a maximal torus. As a consequence, we obtain…

A real Lie algebra is said to be characteristically solvable if its derivation algebra is solvable. We explicitly determine the moduli space of left-invariant metrics, up to isometric automorphism, for $6$-dimensional nilmanifolds whose…

For every involution $\mathbf{w}$ of the symmetric group $S_n$ we establish, in terms ofa special canonical quotient of the dominant Verma module associated with $\mathbf{w}$, an effective criterion, which allows us to verify whether the…

Let g be a semisimple Lie algebra over an algebraically closed field K of characteristic 0 and O be a nilpotent orbit in g. Then Orb is a symplectic algebraic variety and one can ask whether it is possible to quantize $\Orb$ (in an…

We prove that for a noetherian semilocal ring $R$ with exactly $k$ isomorphism classes of simple right modules the monoid $V^*(R)$ of isomorphism classes of countably generated projective right (left) modules, viewed as a submonoid of…