Related papers: Elliptic Quantum Group U_{q,p}(B_N^{(1)}) and Vert…

We construct log-modular quantum groups at even order roots of unity, both as finite-dimensional ribbon quasi-Hopf algebras and as finite ribbon tensor categories, via a de-equivariantization procedure. The existence of such quantum groups…

We introduce a framework to define coalgebra and bialgebra structures on two-dimensional (2D) square lattices, extending the algebraic theory of Hopf algebras and quantum groups beyond the one-dimensional (1D) setting. Our construction is…

We initiate the study of p-adic algebraic groups G from the stability-theoretic and definable topological-dynamical points of view, that is, we consider invariants of the action of G on its space of types over Q_p in the language of fields.…

The search for elliptic quantum groups leads to a modified quantum Yang-Baxter relation and to a special class of quasi-triangular quasi Hopf algebras. This paper calculates deformations of standard quantum groups (with or without spectral…

Multi-parameter versions U_p(g) and C_p[G] of the standard quantum groups U_q(g) and C_q[G] are considered where G is a semi-simple connected complex algebraic group and g is the Lie algebra of G. The primitive spectrum of C_p[G] is…

Vertex operators associated with level two $U_q(\widehat{sl}_2)$ modules are constructed explicitly using bosons and fermions. An integral formula is derived for the trace of products of vertex operators. These results are applied to give…

$(L_p, L_q)$ estimates are obtained for oscillatory potentials $(K^\alphaf)(x)=\int\limits_{R^n}\frac{\exp(i|y|)}{|y|^{n-\alpha}}f(x-y)dy$, $0<\alpha<n$, $n\geq 2$, whose symbol has a singularity on the unit sphere. These potentials are…

We find the stable homotopy classification of elliptic operators on stratified manifolds. Namely, we establish an isomorphism of the set of elliptic operators modulo stable homotopy and the $K$-homology group of the singular manifold. As a…

By studying ``points of the underlying quantum groups''of coquasitriangular Hopf (face) algebras, we construct ribbon categories for each lattice models without spectral parameter of both vertex and face type. Also, we give a classification…

We obtain $L^p(L^q)$ maximal regularity estimates for time dependent second order elliptic operators in divergence form with rough dependencies in the spatial variables.

We realize the Hopf algebra $U_q({\mathfrak {sp}}_{2n})$ as an algebra of quantum differential operators on the quantum symplectic space $\mathcal{X}(f_s;\mathrm{R})$ and prove that $\mathcal{X}(f_s;\mathrm{R})$ is a…

We introduce certain correlation functions (graded $q$--traces) associated to vertex operator algebras and superalgebras which we refer to as $n$--point functions. These naturally arise in the studies of representations of Lie algebras of…

We prove uniform $L^p$ estimates for resolvents of higher order elliptic self-adjoint differential operators on compact manifolds without boundary, generalizing a corresponding resul of [3] in the case of Laplace-- Beltrami operators on…

We calculate commutation relations of vertex operators for the spin representation of $U_q(D_n^{(1)})$ by using recursive formulae of R-matrices. In quantum symmetry approach, we obtain the energy and momentum spectrum of the quantum spin…

The dually conjugate Hopf algebras $Fun_{p,q}(R)$ and $U_{p,q}(R)$ associated with the two-parametric $(p,q)$-Alexander-Conway solution $(R)$ of the Yang-Baxter equation are studied. Using the Hopf duality construction, the full Hopf…

We consider a class of degenerate Ornstein-Uhlenbeck operators in $\mathbb{R}^{N}$, of the kind [\mathcal{A}\equiv\sum_{i,j=1}^{p_{0}}a_{ij}(x) \partial_{x_{i}x_{j}}^{2}+\sum_{i,j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}%] where $(a_{ij})$ is…

We give one parameter deformation of level $k$ free field realization of the screening current of the elliptic algebra $U_{q,p}(sl_2^)$. By means of these free field realizations, we construct infinitly many commutative operators, which we…

We establish the $L^p$-$L^q$-boundedness of subelliptic pseudo-differential operators on a compact Lie group $G$. Effectively, we deal with the $L^p$-$L^q$-bounds for operators in the sub-Riemmanian setting because the subelliptic classes…

Shiraishi's two parameter generalization of $U_{q}(sl(2))$ to $U_{q,p}(sl(2))$ involving an elliptic function is considered. The generators are mapped non-linearly on those of $U_{q}(sl(2))$. This gives directly the irreducible…

We consider linear elliptic and parabolic equations with measurable coefficients and prove two types of $L_{p}$-estimates for their solutions, which were recently used in the theory of fully nonlinear elliptic and parabolic second order…