Related papers: Quantized enveloping superalgebra of type $P$

In this paper, we begin the study of highest weight representations of the quantized enveloping superalgebra ${\mathfrak U}_q {\mathfrak p}_n$ of type $P$. We introduce a Drinfeld-Jimbo representation and establish a…

We introduce a new family of superalgebras, the quantum walled Brauer-Clifford superalgebras ${\mathsf {BC}}_{r,s}(q)$. The superalgebra ${\mathsf {BC}}_{r,s}(q)$ is a quantum deformation of the walled Brauer-Clifford superalgebra ${\mathsf…

We define a new $q$-deformation of Brauer's centralizer algebra which contains Hecke algebras of type $A$ as unital subalgebras. We determine its generic structure as well as the structure of certain semisimple quotients. This is expected…

For a finite dimensional semisimple Lie algebra ${\frak{g}}$ and a root $q$ of unity in a field $k,$ we associate to these data a double quiver $\bar{\cal{Q}}.$ It is shown that a restricted version of the quantized enveloping algebras…

Let $U_\hbar\mathfrak{g}$ denote the Drinfeld-Jimbo quantum group associated to a complex semisimple Lie algebra $\mathfrak{g}$. We apply a modification of the $R$-matrix construction for quantum groups to the evaluation of the universal…

For quantized universal enveloping algebras we construct weight modules by inducing representations of the centralizer of the Cartan subalgebra in the quantized universal enveloping algebra. The induced modules arising from…

We introduce the quasi-partition algebra $QP_k(n)$ as a centralizer algebra of the symmetric group. This algebra is a subalgebra of the partition algebra and inherits many similar combinatorial properties. We construct a basis for…

We reconstruct the quantum enveloping superalgebra ${\bf U}(\mathfrak{gl}_{m|n})$ over $\mathbb Q(v)$ via (finite dimensional) quantum Schur superalgebras. In particular, we obtain a new basis containing the standard generators of ${\bf…

Let $\mathcal P:=\mathcal P_{m\times n}$ denote the quantized coordinate ring of the space of $m\times n$ matrices, equipped with natural actions of the quantized enveloping algebras $U_q(\mathfrak{gl}_m)$ and $U_q(\mathfrak{gl}_n)$. Let…

Lie antialgebras is a class of supercommutative algebras recently appeared in symplectic geometry. We define the notion of enveloping algebra of a Lie antialgebra and study its properties. We show that every Lie antialgebra is canonically…

Let $\frak{g}$ be a finite dimensional simple complex Lie algebra and $U=U_q(\frak{g})$ the quantized enveloping algebra (in the sense of Jantzen) with $q$ being generic. In this paper, we show that the center $Z(U_q(\frak{g}))$ of the…

Inspired by a result in [Ga], we locate two $ k[q,q^{-1}] $-integer forms of $ F_q[SL(n+1)] $, along with a presentation by generators and relations, and prove that for $ q=1 $ they specialize to $ U({\mathfrak{h}}) $, where $…

It is shown that the quantised enveloping algebra of sl(n) contains a quantum Lie algebra, defined by means of axioms similar to Woronowicz's., This gives rise to Lie algebra-like generators and relations for the locally finite part of the…

Take the matrix Lie superalgebra $gl_{N|N}$ with the standard generators $E_{ij}$ where $i,j=-N,...,-1,1,...,N$. Define an involutive automorphism of $gl_{N|N}$ by sending $E_{ij}$ to $E_{-i,-j}$. Then the corresponding twisted subalgebra…

We provide a novel construction of quantized universal enveloping $*$-algebras of real semisimple Lie algebras, based on Letzter's theory of quantum symmetric pairs. We show that these structures can be `integrated', leading to a…

We establish classical and categorical Howe dualities between the Lie superalgebras $\mathfrak{p}(m)$ and $\mathfrak{p}(n)$, for $m,n \geq 1$. We also describe a presentation via generators and relations as well as a Kostant…

The universal enveloping algebra $U(\mathfrak{g} )$ of a current (super)algebra or loop (super)algebra $\mathfrak{g} $ is considered over an algebraically closed field $\mathbb{K} $ with characteristic $p\ge 0$. This paper focuses on the…

The periplectic Lie superalgebra $\mathfrak{p}(n)$ is one of the most mysterious and least understood simple classical Lie superalgebras with reductive even part. We approach the study of its finite dimensional representation theory in…

It is known that a generalized $q$-Schur algebra may be constructed as a quotient of a quantized enveloping algebra $\UU$ or its modified form $\dot{\UU}$. On the other hand, we show here that both $\UU$ and $\dot{\UU}$ may be constructed…

We introduce the periplectic $q$-Brauer category over an integral domain of characteristic not $2$. This is a strict monoidal supercategory and can be considered as a $q$-analogue of the periplectic Brauer category. We prove that the…