Related papers: Equations for formal toric degenerations

Let $\pi_1,...,\pi_n$ be an irreducible finite-dimensional $\mathfrak{sl}_2$-modules. Using the theory of the representations of the current algebras, we introduce a several ways to construct a $q$-grading on $\pi_1\otimes...\otimes\pi_n$.…

Classes of algebraic structures that are defined by equational laws are called varieties or equational classes. A variety is finitely generated if it is defined by the laws that hold in some fixed finite algebra. We show that every…

We describe a class of toric varieties in the $N$-dimensional affine space which are minimally defined by no less than $N-2$ binomial equations.

For every natural number k we introduce the notion of k-th order convolution of functions on abelian groups. We study the group of convolution preserving automorphisms of function algebras in the limit. It turns out that such groups have…

Consider the algebraic dynamics on a torus T=G_m^n given by a matrix M in GL_n(Z). Assume that the characteristic polynomial of M is prime to all polynomials X^m-1. We show that any finite equivariant map from another algebraic dynamics…

Quantum superalgebras $su_{q}(m\mid n)$ are studied in the framework of $R$-matrix formalism. Explicit parametrization of $L^{(+)}$ and $L^{(-)}$ matrices in terms of $su_{q}(m\mid n)$ generators are presented. We also show that quantum…

A $(q,t)$-deformation of the 2d Toda integrable hierarchy is introduced by enhancing the underlying symmetry algebra $\mathfrak{gl}(\infty)\simeq \text{q-W}_{1+\infty}$ to the quantum toroidal $\mathfrak{gl}(1)$ algebra. The…

Given a unital $*$-algebra $\mathscr{A}$ together with a suitable positive filtration of its set of irreducible bounded representations, one can construct a C$^*$-algebra $A_0$ with a dense two-sided ideal $A_c$ such that $\mathscr{A}$ maps…

In this note, we study the arithmetic nature of values of modular functions, meromorphic modular forms and meromorphic quasi-modular forms with respect to arbitrary congruence subgroups, that have algebraic Fourier coefficients. This…

We study the structure of the generator of a symmetric, conservative quantum dynamical semigroup with norm-bounded generator on a von Neumann algebra equipped with a faithful semifinite trace. For von Neumann algebras with abelian commutant…

In the framework of C*-algebraic deformation quantization we propose a notion of deformation groupoid which could apply to known examples e.g. Connes' tangent groupoid of a manifold, its generalisation by Landsman and Ramazan, Rieffel's…

We investigate the notion of \textit{semi-nil clean} rings, defined as those rings in which each element can be expressed as a sum of a periodic and a nilpotent element. Among our results, we show that if $R$ is a semi-nil clean NI ring,…

We define a new grading, that we call the "level grading", on the algebra of polynomials generated by the derivatives $u_{k+i}=\partial^{k+i}u/\partial x^{k+i}$ over the ring $K^{(k)}$ of $C^{\infty}$ functions of $u,u_1,...,u_k$. This…

We prove the following result related to the inverse problem for universal deformation rings of group representations: Given a finite field k, denote by W(k) the ring of Witt vectors over k and by K the field of fractions of W(k). If a…

In this survey I summarize the constructions of toric degenerations obtained from valuations and Gr\"obner theory and describe in which sense they are equivalent. I show how adapted bases can be used to generalize the classical Newton…

We consider vector valued weak solutions $u:\Omega_T\to \mathbb{R}^N$ with $N\in \mathbb{N}$ of degenerate or singular parabolic systems of type \begin{equation*} \partial_t u - \mathrm{div} \, a(z,u,Du) = 0 \qquad\text{in}\qquad \Omega_T=…

Recently, a number of interesting relations have been discovered between generalised Pauli/Dirac groups and certain finite geometries. Here, we succeeded in finding a general unifying framework for all these relations. We introduce…

Let $K$ be a perfectoid field. We describe all quotient fields of the perfectoid Tate algebra\begin{equation*}T_{n,K}^{\text{perfd}}=K\langle X_{1}^{1/p^{\infty}},\dots, X_{n}^{1/p^{\infty}}\rangle\end{equation*}in any number $n\geq1$ of…

We classify positive energy representations with finite degeneracies of the Lie algebra $W_{1+\infty}\/$ and construct them in terms of representation theory of the Lie algebra $\hatgl ( \infty R_m )\/$ of infinite matrices with finite…

We introduce an algebraic method for describing the Hodge filtration of degenerating hypersurfaces in projective toric varieties. For this purpose, we show some fundamental properties of logarithmic differential forms on proper equivariant…