Related papers: Quantum groups, Verma modules and $q$-oscillators:…
Quantum homogeneous supervector bundles arising from the quantum general linear supergoup are studied. The space of holomorphic sections is promoted to a left exact covariant functor from a category of modules over a quantum parabolic…
Let $\Uq$ be a quantum group. Regarding a (noncommutative) space with $\Uq$-symmetry as a $\Uq$-module algebra $A$, we may think of equivariant vector bundles on $A$ as projective $A$-modules with compatible $\Uq$-action. We construct an…
We formulate several basic properties of Verma supermodules over regular symmetrizable Kac--Moody Lie superalgebras, exhibiting $\mathfrak{gl}(1|1)$-nature as revealed through changing Borel subalgebras. We investigate variants of Verma…
The simplicity of the Kac modules for the quantum superalgebra U_q(gl(m,n)) is studied, and the relation between the representation of U_q(gl(m,n)) and that of U_q(g_{\0}) is investigated.
Quantum-mechanical concepts can be formulated in constructive finite terms without loss of their empirical content if we replace a general unitary group by a unitary representation of a finite group. Any linear representation of a finite…
In our previous paper, we constructed an explicit GL(n)-equivariant quantization of the Kirillov--Kostant-Souriau bracket on a semisimple coadjoint orbit. In the present paper, we realize that quantization as a subalgebra of endomorphisms…
We develop the representation theory of shifted quantum affine algebras $\mathcal{U}_q^\mu(\hat{\mathfrak{g}})$ and of their truncations which appeared in the study of quantized K-theoretic Coulomb branches of 3d $N = 4$ SUSY quiver gauge…
We consider a constructive modification of quantum-mechanical formalism. Replacement of a general unitary group by unitary representations of finite groups makes it possible to reproduce quantum formalism without loss of its empirical…
We construct, using the quantum dilogarithm, a series of *-representations of quantized cluster varieties. This includes a construction of infinite dimensional unitary projective representations of their discrete symmetry groups - the…
We construct the first examples of purely continuous, $q$-deformed Lie type locally compact quantum groups in higher rank. They arise from Drinfeld-Jimbo quantization, at unimodular deformation parameter, of the totally positive part of…
We start from the quantum Miura transformation [7] for the $W$-algebra associated with $GL(n)$ group and find an evident formula for quantum L-operator as well as for the action of $W_l$ currents (l=1,..,n) on elements of the completely…
Different generators of a deformed oscillator algebra give rise to one-parameter families of $q$-exponential functions and $q$-Hermite polynomials related by generating functions. Connections of the Stieltjes and Hamburger classical moment…
The operator algebras of a new family of relativistic geometric models of the relativistic oscillator are studied. It is shown that, generally, the operator of number of quanta and the pair of the shift operators of each model are the…
We realize geometrically a family of simple modules of (shifted) quantum loop groups including Kirillov-Reshetikhin and prefundamental representations. To do this, we introduce a new family of algebras attached to quivers with potentials,…
Shapovalov elements $\theta _{\beta,m}$ of the classical or quantized universal enveloping algebra of a simple Lie algebra $\mathfrak{g}$ are parameterized by a positive root $\beta$ and a positive integer $m$. They relate the highest…
The coherent state method has proved to be useful in quantum physics and mathematics. This method, more precisely, the vector coherent state method, has been used by some authors to construct representations of superalgebras but almost, to…
We discuss an approach to quantum gerbes over quantum groups in terms of q-deformation of transition functions for a loop group bundle. The case of the quantum group SUq(2) is treated in some detail.
The differential and variational calculus on the $SL_{q}(2,R)$ group is constructed. The spontaneous breaking symmetry in the WZNW model with $SL_{q}(2,R)$ quantum group symmetry and in the $\sigma$-models with ${SL_{q}(2,R)/U_{h}(1)}$…
Usually in quantum mechanics the Heisenberg algebra is generated by operators of position and momentum. The algebra is then represented on an Hilbert space of square integrable functions. Alternatively one generates the Heisenberg algebra…
A formulation of quantum mechanics with additive and multiplicative (q-)difference operators instead of differential operators is studied from first principles. Borel-quantisation on smooth configuration spaces is used as guiding…