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We study bounded domains $\Omega\subset\mathbb{C}^n$ whose Bergman metric is locally symmetric, i.e. its Riemannian curvature tensor is parallel with respect to the Levi-Civita connection. Following the strategy developed in…
The Weyl-Wigner prescription for quantization on Euclidean phase spaces makes essential use of Fourier duality. The extension of this property to more general phase spaces requires the use of Kac algebras, which provide the necessary…
The Hilbert manifold $\Sigma$ consisting of positive invertible (unitized) Hilbert-Schmidt operators has a rich structure and geometry. The geometry of unitary orbits $\Omega\subset \Sigma$ is studied from the topological and metric…
Let $\mathcal K$ be a complete quasivariety of completely regular universal topological algebras of continuous signature $\mathcal E$ (which means that $\mathcal K$ is closed under taking subalgebras, Cartesian products, and includes all…
We investigate $\mathcal F$-Borel topological spaces. We focus on finding out how a~complexity of a~space depends on where the~space is embedded. Of a~particular interest is the~problem of determining whether a~complexity of given space $X$…
We present a unified and completely general formulation of extended geometry, characterised by a Kac-Moody algebra and a highest weight coordinate module. Generalised diffeomorphisms are constructed, as well as solutions to the section…
Feigin and Fuchs have given a well-known construction of intertwining operators between "Fock-type" modules over the Virasoro algebra. The intertwiners are obtained via contour integration of certain "screening operators" over top homology…
Let $A$ be a finite dimensional associative $\mathbb{K}$-algebra over an algebraically closed field $\mathbb{K}$ of characteristic zero. To $A$, we can associate its basic form that is given by a quiver $Q = (Q_0, Q_1)$ with an admissible…
For any finite dimensional Lie superalgebra $\dot{\mathfrak{g}}$ (maybe a Lie algebra) with an even derivation $d$ and a finite order automorphism $\sigma$ that commutes with $d$, we introduce the $(d,\sigma)$-twisted Affine-Virasoro…
In this paper we generalize classical results on Lie algebras and universal enveloping algebras of Lie algebras to Lie-Rinehart algebras. We define for any Lie-Rinehart algebra $L$ and any cocycle $f$ in $Z^2(L,B)$, a universal enveloping…
We construct super Hamiltonian integrable systems within the theory of Supersymmetric Poisson vertex algebras (SUSY PVAs). We provide a powerful tool for the understanding of SUSY PVAs called the super master formula. We attach some Lie…
We propose an extension of the definition of vertex algebras in arbitrary space-time dimensions together with their basic structure theory. An one-to-one correspondence between these vertex algebras and axiomatic quantum field theory (QFT)…
Algebraic quantum field theory and prefactorization algebra are two mathematical approaches to quantum field theory. In this monograph, using a new coend definition of the Boardman-Vogt construction of a colored operad, we define homotopy…
We present a universal normal algebra suitable for constructing and classifying Calabi-Yau spaces in arbitrary dimensions. This algebraic approach includes natural extensions of reflexive weight vectors to higher dimensions, related to…
We study conformal symmetry breaking differential operators which map differential forms on $\mathbb{R}^n$ to differential forms on a codimension one subspace $\mathbb{R}^{n-1}$. These operators are equivariant with respect to the conformal…
In the paper there are investigated various approximate representations of the infinite dimensional $\Bbb Z$--graded Lie algebras: the Witt algebra of all Laurent polynomial vector fields on a circle and its one-dimensional nontrivial…
This paper investigates the relationship between algebraic quantum field theories and factorization algebras on globally hyperbolic Lorentzian manifolds. Functorial constructions that map between these two types of theories in both…
Since the time when the first optical instruments have been invented, an idea that the visible image of an object under observation depends on tools of observation became commonly assumed in physics. A way to formalize it in mathematics is…
We describe a global approach to the study of duality transformations between antisymmetric fields with transitions and argue that the natural geometrical setting for the approach is that of gerbes, these objects are mathematical…
Starting from a purely algebraic procedure based on the commutant of a subalgebra in the universal enveloping algebra of a given Lie algebra, the notion of algebraic Hamiltonians and the constants of the motion generating a polynomial…