Related papers: The Equitable Basis for sl_2

The conformal bootstrap in physics has recently been adapted to prove remarkably sharp estimates on Laplace eigenvalues and triple correlations of automorphic forms on compact hyperbolic surfaces. These estimates derive from an infinite…

We consider the graded space $R$ of syzygies for the coordinate algebra $A$ of projective variety $X=G/P$ embedded into projective space as an orbit of the highest weight vector of an irreducible representation of semisimple complex Lie…

We study equivalence relations $\mathcal R(\Gamma\curvearrowright G)$ that arise from left translation actions of countable groups on their profinite completions. Under the assumption that the action $\Gamma\curvearrowright G$ is free and…

In this paper, we study gravitational symmetry algebras that live on 2-dimensional cuts $S$ of asymptotic infinity. We define a notion of wedge algebra $\mathcal{W}(S)$ which depends on the topology of $S$. For the cylinder $S=\mathbb{C}^*$…

A new highly symmetrical model of the compact Lie algebra $\mathfrak{g}^c_2$ is provided as a twisted ring group for the group $\mathbb{Z}_2^3$ and the ring $\mathbb{R}\oplus\mathbb{R}$. The model is self-contained and can be used without…

This is a sequel to \cite{osy} and \cite{sxy}. Associated with $G:=\GL_n$ and its rational representation $(\rho, M)$ over an algebraically closed filed $\bk$, we define an enhanced algebraic group $\uG:=G\ltimes_\rho M$ which is a product…

Let g = Lie(G) be the Lie algebra of a simple algebraic group G over an algebraically closed field of characteristic 0. Let e be a nilpotent element of g and let g_e = Lie(G_e) where G_e stands for the stabiliser of e in G. For g classical,…

Let $V$ be a simple vertex operator algebra which admits the continuous, faithful action of a compact Lie group $G$ of automorphisms. We establish a Schur-Weyl type duality between the unitary, irreducible modules for $G$ and the…

Inspired by Vershik and Okounkov's inductive and Lie-theoretic approach to the representation theory of the symmetric group, we extend their point of view to reducible $S_n$-modules. Using induced representations along Young's lattice, we…

Let $G \leq \mathrm{GL}(V)$ be a group with a unique subnormal quasisimple subgroup $E(G)$ that acts absolutely irreducibly on $V$. A base for $G$ acting on $V$ is a set of vectors with trivial pointwise stabiliser in $G$. In this paper we…

As a sequel of \cite{Ou}, in this shot note, we investigate block basis for coinvariants of finite modular pseudo-reflection groups. We are particularly interested in the case where $ G $ is a subgroup of the parabolic subgroups of $…

We study the GIT quotient of the symplectic grassmannian parametrizing lagrangian subspaces of \bigwedge^3{\mathbb C}^6 by the natural action of SL_6, call it M. This is a compactification of the moduli space of smooth double EPW-sextics.…

Landau-Ginzburg mirror symmetry studies isomorphisms between graded Frobenius algebras, known as A- and B-models. Fundamental to constructing these models is the computation of the finite, Abelian $\textit{maximal symmetry group}$…

For an infinite group $G$, the poset $\mathcal{L}_G$ of group topologies constitutes a complete lattice. Although $\mathcal{L}_G$ is modular when $G$ is abelian, this property fails to persist for nilpotent groups. Extending Arnautov's 2010…

We introduce novel polynomial deformations of the Lie algebra $sl(2)$. We construct their finite-dimensional irreducible representations and the corresponding differential operator realizations. We apply our results to a class of spin…

We define and study sl\_2-categorifications on abelian categories. We show in particular that there is a self-derived (even homotopy) equivalence categorifying the adjoint action of the simple reflection. We construct categorifications for…

We investigate the existence of left-invariant closed G$_2$-structures on seven-dimensional non-solvable Lie groups, providing the first examples of this type. When the Lie algebra has trivial Levi decomposition, we show that such a…

In this note, we construct new solutions to the heterotic $\mathrm{G}_2$-system with non-abelian gauge group, both compact and non-compact, on certain $2$-step nilmanifolds and $3$-Sasakian manifolds. Our approach is based on an ansatz that…

Let $G$ be a simple complex Lie group, $\alg{g}$ be its Lie algebra, $K$ be a maximal compact form of $G$ and $\alg{k}$ be a Lie algebra of $K$. We denote by $X\rightarrow \overline{X}$ the anti-involution of $\alg{g}$ which singles out the…

We consider the rational dynamical quantum group $E_y(gl_2)$ and introduce an $E_y(gl_2)$-module structure on $\oplus_{k=0}^n H^*_{GL_n\times\C^\times}(T^*Gr(k,n))'$, where $H^*_{GL_n\times\C^\times}(T^*Gr(k,n))'$ is the equivariant…