Related papers: Extension Quiver for Lie Superalgebra $\mathfrak{q…
Infinite dimensional representations of the real form U_q(u_{n,1}) of the Drinfeld--Jimbo algebra U_q(gl_{n+1}) are defined. The principal series of representations of U_q(u_{n,1}) is studied. Intertwining operators for pairs of the…
Let $p$ be a prime. Given a split semisimple group scheme $G$ over a normal integral domain $R$ which is a faithfully flat $\mathbb Z_{(p)}$-algebra, we classify all finite dimensional representations $V$ of the fiber $G_K$ of $G$ over…
We introduce a category of $q$-oscillator representations over the quantum affine superalgebras of type $D$ and construct a new family of its irreducible representations. Motivated by the theory of super duality, we show that these…
Block type Lie algebras have been studied by many authors in the latest twenty years. In this paper, we will study a class of more general Block type Lie algebra $\mathcal{B}(p,q)$, which is a class of infinite-dimensional Lie algebra by…
Reduction of the left regular representation of quantum algebra $sl_q(3)$ is studied and ~$q$-difference intertwining operators are constructed. The irreducible representations correspond to the spaces of local sections of certain line…
We present modules for the extended affine Lie algebra $\widetilde{\frak{gl}_{3}(\bc_q)}$ by using the idea of free fields. A necessary and sufficient condition for the modules being unitary is given.
In the derived category of mod-KQ for Dynkin quiver Q, we construct a full subcategory in a canonical way, so that its endomorphism algebra is a higher Auslander algebra of global dimension $3k+2$ for any $k\geq 1$. Furthermore, we extend…
We compute the Hilbert series of three-dimensional $\mathcal{N}=3$ quiver gauge theories by taking a specific limit of the superconformal index. Our approach introduces auxiliary fugacities associated with symmetries which, while not…
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…
In this paper we identify the cotangent to the derived stack of representations of a quiver $Q$ with the derived moduli stack of modules over the Ginzburg dg-algebra associated with $Q$. More generally, we extend this result to finite type…
In this paper we consider the following question: Is it possible to construct all real root representations of a given quiver Q by using universal extension functors, starting with a real Schur representation? We give a concrete example…
In this paper, we study the Whittaker modules for the quantum enveloping algebra $U_q(\sl_3)$ with respect to a fixed Whittaker function. We construct the universal Whittaker module, find all its Whittaker vectors and investigate the…
We study non-standard Verma type modules over the Kac-Moody queer Lie superalgebra $\mathfrak{q}(n)^{(2)}$. We give a sufficient condition under which such modules are irreducible. We also give a classification of all irreducible diagonal…
We calculate the first extension groups for finite-dimensional simple modules over an arbitrary generalized current Lie algebra, which includes the case of loop Lie algebras and their multivariable analogs.
The construction approach proposed in the previous paper Ref. 1 allows us there and in the present paper to construct at generic deformation parameter $q$ all finite--dimensional representations of the quantum Lie superalgebra…
We provide a classification of generalized tilting modules and full exceptional sequences for the dual extension algebra of the path algebra of a uniformly oriented linear quiver modulo the ideal generated by paths of length two with its…
Given a complex semisimple Lie algebra ${\mathfrak g}$ and a commutative ${\mathbb C}$-algebra $A$, let ${\mathfrak g}[A] = {\mathfrak g} \otimes A$ be the corresponding generalized current algebra. In this paper we explore questions…
We study the truncation functors and show the existence of projective cover of each irreducible module in parabolic BGG category $\mathcal O$ over infinite rank Lie algebra of types $\mathfrak{a,b,c,d}$. Moreover, $\mathcal O$ is a Koszul…
Quantum spaces with $\frak{su}(2)$ noncommutativity can be modelled by using a family of $SO(3)$-equivariant differential $^*$-representations. The quantization maps are determined from the combination of the Wigner theorem for $SU(2)$ with…
Let $\mathcal{B}_{\mathfrak{q}}$ be a finite-dimensional Nichols algebra of diagonal type corresponding to a matrix $\mathfrak{q} \in \mathbf{k}^{\theta \times \theta}$, where $\mathbf{k}$ is an algebraically closed field of characteristic…