Related papers: On the multiplicativity of quantum cat maps
The action of ring automorphisms of the polynomial ring in two variables over the real numbers on real plane curves is considered. The orbits containing degree-three polynomials are computed, with one representative per orbit being…
We describe representation theory of the elliptic quantum group $E_{\tau,\eta}(sl_2)$. It turns out that the representation theory is parallel to the representation theory of the Yangian $Y(sl_2)$ and the quantum loop group $ U_q(\widetilde…
We construct a family of irreducible representations of the quantum plane and of the quantum Weyl algebra over an arbitrary field, assuming the deformation parameter is not a root of unity. We determine when two representations in this…
We compute the representation theory of two families of noncrossing partition quantum groups connected to amalgamated free products and free wreath products. This illustrates the efficiency of the methods developed in our previous joint…
We construct the positive principal series representations for $\mathcal{U}_q(\mathfrak{g}_\mathbb{R})$ where $\mathfrak{g}$ is of simply-laced type, parametrized by $\mathbb{R}_{\geq 0}^r$ where $r$ is the rank of $\mathfrak{g}$. We…
Building on the work [18], where some standard basis for the queer $q$-Schur superalgebra $\mathcal{Q}_q(n,r;R)$ is defined by a labelling set of matrices and their associated double coset representatives, we investigate the matrix…
We study beta-extensions in a p-adic classical group and we produce a relation between some beta-extensions by means of a Weil representation. We apply this to the study of reducibility points of some parabolically induced representations.
In this paper, we study the multiplicative behaviour of quantum channels, mathematically described by trace preserving, completely positive maps on matrix algebras. It turns out that the multiplicative domain of a unital quantum channel has…
We construct symmetric and exterior powers of the vector representation of the elliptic quantum groups $E_{\tau,\eta}(gl_N)$. The corresponding transfer matrices give rise to various integrable difference equations which could be solved in…
The notion of families of quantum invertible maps ($C^*$-algebra homomorphisms satisfying Podle\'s condition) is employed to strengthen and reinterpret several results concerning universal quantum groups acting on finite quantum spaces. In…
We show that the representations of the Cuntz C$^\ast$-algebras $O_n$ which arise in wavelet analysis and dilation theory can be classified through a simple analysis of completely positive maps on finite-dimensional space. Based on this…
In these notes we discuss the "self-reducibility property" of the Weil representation. We explain how to use this property to obtain sharp estimates of certain higher-dimensional exponential sums which originate from the theory of quantum…
We construct a special principal series representation for the modular double $U_{q\tilde{q}}(g_R)$ of type $A_r$ representing the generators by positive essentially self-adjoint operators satisfying the transcendental relations that also…
The theta characteristics on a Riemann surface are permuted by the induced action of the automorphism group, with the orbit structure being important for the geometry of the curve and associated manifolds. We describe two new methods for…
Representations of propagators by means of path integrals over velocities are discussed both in nonrelativistic and relativistic quantum mechanics. It is shown that all the propagators can only be expressed through bosonic path integrals…
We consider a category of continuous Hilbert space representations and a category of smooth Frechet representations, of a real Jacobi group $G$. By Mackey's theory, they are respectively equivalent to certain categories of representations…
We prove duality isomorphisms of certain representations of W-algebras which play an essential role in the quantum geometric Langlands Program and some related results.
Given two torus invariant Weil divisors $D$ and $D'$ on a two-dimensional cyclic quotient singularity $X$, the groups $\mathop{Ext}\nolimits^i_{X}(\mathcal{O}(D),\mathcal{O}(D'))$, $i>0$, are naturally $\mathbb{Z}^2$-graded. We interpret…
We give a formula for the modular operator and modular conjugation in terms of matrix coefficients of corepresentations of a quantum group in the sense of Kustermans and Vaes. As a consequence, the modular autmorphism group of a unimodular…
The aim of this paper is to survey some aspects of mapping class groups with focus on their finite dimensional representations arising in topological quantum field theory.