Related papers: Quantized coordinate rings, PBW-type bases and $q$…
In this paper, we present explicit actions of braid group on the universal enveloping superalgebra ${\boldsymbol U}(\mathfrak{{q}}_n)$ and the quantum queer superalgebra ${\boldsymbol U}_{\!{v}}(\mathfrak{{q}}_{n})$. Then we provide a new…
In his work on crystal bases \cite{Kas}, Kashiwara introduced a certain degeneration of the quantized universal enveloping algebra of a semi-simple Lie algebra $\mathfrak g$, which he called a quantum boson algebra. In this paper, we…
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…
We first construct an action of the extended double affine braid group $\mathcal{\ddot{B}}$ on the quantum toroidal algebra $U_{q}(\mathfrak{g}_{\mathrm{tor}})$ in untwisted and twisted types. As a crucial step in the proof, we obtain a…
A $q$-analogue of the Higgs algebra, which describes the symmetry properties of the harmonic oscillator on the $2$-sphere, is obtained as the commutant of the $\mathfrak{o}_{q^{1/2}}(2) \oplus \mathfrak{o}_{q^{1/2}}(2)$ subalgebra of…
We define two subalgebras which can be seen as the quantization of the coordinate rings of the unipotent radical of the standard positive (respectively negative) Borel subgroup of $SL_{n+1}$. We give a presentation for these algebras and…
For an acyclic quiver, we establish a connection between the cohomology of quiver Grassmannians and the dual canonical bases of the algebra $U_q^-(\mathfrak{g})$, where $U_q^-(\mathfrak{g})$ is the negative half of the quantized enveloping…
We introduce the notion of integrable modules over $\imath$quantum groups (a.k.a. quantum symmetric pair coideal subalgebras). After determining a presentation of such modules, we prove that each integrable module over a quantum group is…
We prove that the deformed oscillator superalgebra $W_q(n)$ (which in the Fock representation is generated essentially by $n$ pairs of $q$-bosons) is a factor algebra of the quantized universal enveloping algebra $U_q[osp(1/2n)]$. We write…
The braided approach to q-deformation (due to the author and collaborators) gives natural algebras $R_{21}u_1Ru_2=u_2R_{21}u_1R$ and $R_{21}x_1x_2=x_2x_1R$ for q-Minkowski and q-Euclidean spaces respectively. These algebras are covariant…
Let ${\mathbf U}^-_q$ be the negative part of the quantized enveloping algebra associated to a Kac-Moody algebra ${\mathfrak g}$ of symmetric type, and $\underline{\mathbf U}^-_q$ the algebra corresponding to the orbit algebra ${\mathfrak…
Let $\mathfrak{g}$ be a semi-simple Lie algebra with fixed root system, and $U_q(\mathfrak{g})$ the quantization of its universal enveloping algebra. Let $\mathcal{S}$ be a subset of the simple roots of $\mathfrak{g}$. We show that the…
The $q$-Onsager algebra $O_q$ is presented by two generators $W_0$, $W_1$ and two relations, called the $q$-Dolan/Grady relations. Recently Baseilhac and Koizumi introduced a current algebra $\mathcal A_q$ for $O_q$. Soon afterwards,…
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…
The quantum duality principal (QDP) by Drinfeld predicts a connection between the quantized universial enveloping algebras and the quantized coordinate algebras, where the underlying classical objects are related by the duality in Poisson…
As a natural generalization of ordinary Lie algebras we introduce the concept of quantum Lie algebras ${\cal L}_q(g)$. We define these in terms of certain adjoint submodules of quantized enveloping algebras $U_q(g)$ endowed with a quantum…
We discuss the second quantization of scalar field theory on the q-deformed fuzzy sphere S^2_{q,N} for q \in \R, using a path-integral approach. We find quantum field theories which are manifestly covariant under U_q(su(2)), have a smooth…
Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra and let $U_q(\mathfrak{g})$ denote the corresponding quantized enveloping algebra. In the present paper we show that quantum symmetric pair coideal subalgebras $B_{c,s}$ of…
The quantum complex Grassmannian U_q/K_q of rank l is the quotient of the quantum unitary group U_q=U_q(n) by the quantum subgroup K_q=U_q(n-l)xU_q(l). We show that (U_q,K_q) is a quantum Gelfand pair and we express the zonal spherical…
We introduce the notion of pure Q-solvable algebra. The quantum matrices, quantum Weyl algebra, U_q(n) are the examples. It is proved that the skew field of fractions of pure Q-solvable algebra is isomorphic to the skew field of twisted…