Related papers: The PBW Filtration, Demazure Modules and Toroidal …

In \cite{BigAlg-3gen}, an explicit description of bi-quadratic algebras on three generators with PBW basis was obtained. There are four classes: I-IV. The aim of the paper is to study algebras that belong to one of the classes: class II.1.…

This thesis studies the representation theory and linear structures of $\mathcal{Q}$-manifolds and higher Lie algebroids. We introduce differential graded modules (or for short DG-modules) of $\mathcal{Q}$-manifolds and the equivalent…

Let $k$ be an algebraically closed field. Let $B$ be the Borel subgroup of $\mGL_n(k)$ consisting of nonsingular upper triangular matrices. Let $\frb = \mLie B$ be the Lie algebra of upper triangular $n \times n$ matrices and $\fru$ the Lie…

We present a proof of an upper bound for the lengths of finite dimensional representations of algebras obeying a modified PBW property, including Lie algebras and quantum groups. The sharpness of the bound is proved and discussed.

We consider an "orientifold" generalization of Khovanov-Lauda-Rouquier algebras, depending on a quiver with an involution and a framing. Their representation theory is related, via a Schur-Weyl duality type functor, to Kac-Moody quantum…

This paper exhibits fundamental structure underlying Lie algebra homology with coefficients in tensor products of the adjoint representation, mostly focusing upon the case of free Lie algebras. The main result yields a DG category that is…

The Weyl algebra (W_{2m}[h]; *) is the algebra generated by u=(u_1,...,u_m,v_1,.....,v_m) over C with the fundamental commutation relation [u_i,v_j]=-ih\delta_{ij}, where h is a positive constant. The Heisenberg algebra (\Cal H_{2m}[nu];*)…

In Fialowski's classification for algebras of maximal class, there are three Lie algebras of maximal class with 1-dimensional homogeneous components: $\mathfrak{m}_0$, $L_1$ and $\mathfrak{m}_2$. In this paper, we studied their…

The quon algebra is an approach to particle statistics in order to provide a theory in which the Pauli exclusion principle and Bose statistics are violated by a small amount. The quons are particles whose annihilation and creation operators…

We consider the problem of representing in Hilbert space commutation relations of the form $$ a_ia_j^*=\delta_{ij}{\bold1} + \sum_{k\ell}T_{ij}^{k\ell} a_\ell^*a_k \quad,$$ where the $T_{ij}^{k\ell}$ are essentially arbitrary scalar…

Let $U_q'(\mathfrak{g})$ be a quantum affine algebra of arbitrary type and let $\mathcal{C}_{\mathfrak{g}}$ be Hernandez-Leclerc's category. We can associate the quantum affine Schur-Weyl duality functor $F_D$ to a duality datum $D$ in…

We consider the finite $W$-algebra $U(\g,e)$ associated to a nilpotent element $e \in \g$ in a simple complex Lie algebra $\g$ of exceptional type. Using presentations obtained through an algorithm based on the PBW-theorem, we verify a…

We introduce a new elliptic quantum toroidal algebra $U_{q,t,p}(gl_{1,tor})$. Various representations in the quantum toroidal algebra $U_{q,t}(gl_{1,tor})$ are extended to the elliptic case including the level (0,0) representation realized…

Let $A(1):=k[X]/(X^p)$ be the natural representation of the Witt algebra $W(1)$ over an algebraically closed field of prime characteristic $p>3$. In this note, we decompose the $W(1)$-module $A(1)\otimes A(1)$ into two invariant subspaces,…

Let $\Lambda$ be a finite dimensional algebra over an algebraically closed field. Criteria are given which characterize existence of a fine or coarse moduli space classifying, up to isomorphism, the representations of $\Lambda$ with fixed…

Denote $\fm_2$ the infinite dimensional $\N$-graded Lie algebra defined by the basis $e_i$ for $i\geq 1$ and by relations $[e_1,e_i]=e_{i+1}$ for all $i\geq 2$, $[e_2,e_j]=e_{j+2}$ for all $j\geq 3$. We compute in this article the bracket…

Let $p$ be a prime. We solve two problems in the mod $p$ representation theory of $\mathrm{GL}_2(\mathbb{F}_{q})$ where $q=p^f$. We first prove a Clebsch-Gordan decomposition theorem for the tensor product of two mod $p$ representations of…

We find the free field construction of the basic $W$-current and screening currents for the deformed $W$-superalgebra ${\cal W}_{q,t}\bigl(A(M,N)\bigr)$ associated with Lie superalgebra of type $A(M,N)$. Using this free field construction,…

Let $G$ be a semisimple algebraic group over an algebraically closed field $k$ of positive characteristic $p$. Under some restrictions on the size of $p$, the present paper establishes new results on the $G$-module structure of…

Let $G$ be a simply connected semi-simple complex algebraic group. Fix a maximal torus $T$ and a Borel subgroup $B$ such that $T\subset B\subset G$. Let $W$ the Weyl group of $G$ relative to $T$. For any $w$ in $W$, let $X_w=\bar {BwB/B}$…