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General Lagrangian theory of even and odd fields on an arbitrary smooth manifold is considered. Its non-trivial reducible gauge symmetries and their algebra are defined in this very general setting by means of the inverse second Noether…
We prove a Galois-type correspondence between compositions of purely inseparable field extensions (including infinite ones) and subalgebras of differential operators. This correspondence can be utilized to establish a connection between…
Within the framework of a Kaluza-Klein theory, we provide the geometrization of a generic (Abelian and non-Abelian) gauge coupling, which comes out by choosing a suitable matter fields dependence on the extra-coordinates. We start by the…
Genetic and evolution algebras arise naturally from applied probability and stochastic processes. Gibbs measures describe interacting systems commonly studied in thermodynamics and statistical mechanics with applications in several fields.…
These lectures present an elementary introduction to quantum gauge fields. The first aim is to show how, in the tree approximation, gauge invariance follows from covariance and unitarity. This leads to the standard construction of the…
In this note we relate the valuations of the algebras appearing in the non-commutative geometry of quantized algebras to properties of sub-lattices in some vector spaces. We consider the case of algebras with $PBW$-bases and prove that…
This is a survey of results on positivity of vector bundles, inspired by the Brunn-Minkowski and Pr\'ekopa theorems. Applications to complex analysis, K\"ahler geometry and algebraic geometry are also discussed.
Gauge theories are important descriptions for many physical phenomena and systems in quantum computation. Automorphism of gauge group naturally gives global symmetries of gauge theories. In this work we study such symmetries in gauge…
We extend the theory of Interacting Hopf algebras with an order primitive, and give a sound and complete axiomatisation of the prop of polyhedral cones. Next, we axiomatise an affine extension and prove soundness and completeness for the…
We describe derivations of the Clifford algebra of a nondegenerate quadratic form on a countable dimensional vector space over an algebraically closed field of characteristic not equal to $2$. We also construct an algebraic automorphism of…
We present a method for classifying conformal field theories based on Coulomb gases (bosonic free-field construction). Given a particular geometric configuration of the screening charges, we give necessary conditions for the existence of…
We deform the anti-de Sitter algebra by adding additional generators $\mathcal{Z}_{ab}$, forming in this way the negative cosmological constant counterpart of the Maxwell algebra. We gauge this algebra and construct a dynamical model with…
We formulate gauge theories based on Leibniz(-Loday) algebras and uncover their underlying mathematical structure. Various special cases have been developed in the context of gauged supergravity and exceptional field theory. These are based…
A generalization of classical gauge theory is presented, in the framework of a noncommutative-geometric formalism of quantum principal bundles over smooth manifolds. Quantum counterparts of classical gauge bundles, and classical gauge…
The usual prescription for constructing gauge-invariant Lagrangian is generalized to the case where a Lagrangian contains second derivatives of fields as well as first derivatives. Symmetric tensor fields in addition to the usual vector…
We study the transformation properties of a scalar-tensor theory, coupled to fermions, under the Weyl rescaling associated with a transition from the Jordan to the Einstein frame. We give a simple derivation of the corresponding…
We consider the functions in two variables on an arbitrary poset, for which the convolution operation is defined. We obtain the generalization of incidence algebra and describe its properties: invertibility, the Jackobson radical,…
In this article we summarize and describe the recently found transforms for theories of connections modulo gauge transformations associated with compact gauge groups. Specifically, we put into a coherent picture the so-called loop…
We consider the classical \w42 algebra from the integrable system viewpoint. The integrable evolution equations associated with the \w42 algebra are constructed and the Miura maps , consequently modifications, are presented. Modifying the…
We introduce the natural lift of spacetime diffeomorphisms for conformal gravity and discuss the physical equivalence between the natural and gauge natural structure of the theory. Accordingly, we argue that conformal transformations must…