# Alexei KotovUniversity of Hradec Králové · Department of Mathematics

Alexei Kotov

PhD

## About

36

Publications

2,466

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480

Citations

Citations since 2016

Introduction

Super- and graded geometry, special Riemannian geometry, Lie algebroids and groupoids, geometry of PDEs, non-linear sigma models

## Publications

Publications (36)

In this paper we address several algebraic constructions in the context of groupoids, algebroids and $\mathbb Z$-graded manifolds. We generalize the results of integration of $\mathbb N$-graded Lie algebras to the honest $\mathbb Z$-graded case and provide some examples of application of the technique based on Harish-Chandra pairs. We extend the co...

In this paper we discuss the question of integrating differential graded Lie algebras (DGLA) to differential graded Lie groups (DGLG). We first recall the classical problem of integration in the context, and recollect the known results. Then, we define the category of differential graded Lie groups and study its properties. We show how to associate...

We discuss the notion of basic cohomology for Dirac structures and, more generally, Lie algebroids. We then use this notion to characterize the obstruction to a variational formulation of Dirac dynamics.

A bstract
Any local gauge theory can be represented as an AKSZ sigma model (upon parameterization if necessary). However, for non-topological models in dimension higher than 1 the target space is necessarily infinite-dimensional. The interesting alternative known for some time is to allow for degenerate presymplectic structure in the target space....

In this paper we discuss the categorical properties of $\mathbb Z$-graded manifolds. We start by describing the local model paying special attention to the differences in comparison to the $\mathbb{N}$-graded case. In particular we explain the origin of formality for the functional space and spell-out the structure of the power series. Then we make...

Any local gauge theory can be represented as an AKSZ sigma model (upon parameterization if necessary). However, for non-topological models in dimension higher than 1 the target space is necessarily infinite-dimensional. The interesting alternative known for some time is to allow for degenerate presymplectic structure in the target space. This leads...

We show that the data needed for the method of the embedding tensor employed in gauging supergravity theories are precisely those of a Leibniz algebra (with one of its induced quotient Lie algebras embedded into a rigid symmetry Lie algebra that provides an additional “representation constraint”). Every Leibniz algebra gives rise to a Lie n-algebra...

In this paper we discuss the question of integrating differential graded Lie algebras (DGLA) to differential graded Lie groups (DGLG). We first recall the classical problem of integration in the context, and present the construction for (non-graded) differential Lie algebras. Then, we define the category of differential graded Lie groups and study...

A gauge PDE is a natural notion which arises by abstracting what physicists call a local gauge field theory defined in terms of BV‐BRST differential (not necessarily Lagrangian). We study supergeometry of gauge PDEs paying particular attention to globally well‐defined definitions and equivalences of such objects. We demonstrate that a natural geome...

A gauge PDE is a natural notion which arises by abstracting what physicists call a local gauge field theory defined in terms of BV-BRST differential (not necessarily Lagrangian). We study supergeometry of gauge PDEs paying particular attention to globally well-defined definitions and equivalences of such objects. We demonstrate that a natural geome...

We show that the data needed for the method of the embedding tensor employed in gauging supergravity theories are precisely those of a Leibniz algebra (with one of its induced quotient Lie algebras embedded into a rigid symmetry Lie algebra that provides an additional “represtentation constraint”). Every Leibniz algebra gives rise to a Lie n-algebr...

The construction of gauge theories beyond the realm of Lie groups and algebras leads one to consider Lie groupoids and algebroids equipped with additional geometrical structures which, for gauge invariance of the construction, need to satisfy particular compatibility conditions. This paper is supposed to analyze these compatibilities from a mathema...

Consider an anchored bundle (E,ρ), i.e. a vector bundle E→M equipped with a bundle map ρ:E→TM covering the identity. M. Kapranov showed in the context of Lie–Rinehard algebras that there exists an extension of this anchored bundle to an infinite rank universal free Lie algebroid FR(E)⊃E. We adapt his construction to the case of an anchored bundle e...

Consider an anchored bundle (E, ρ), i.e. a vector bundle E → M equipped with a bundle map ρ : E → T M covering the identity. M. Kapranov showed in the context of Lie-Rinehard algebras that there exists an extension of this anchored bundle to an infinite rank universal free Lie algebroid F R(E) ⊃ E. We adapt his construction to the case of an anchor...

Cartan-Lie algebroids, i.e. Lie algebroids equipped with a compatible connection, permit the definition of an adjoint representation, on the fiber as well as on the tangent of the base. We call (positive) quadratic Lie algebroids, Cartan-Lie algebroids with ad-invariant (Riemannian) metrics on their fibers and base $\kappa$ and $g$, respectively. W...

The object of our study is a Lie algebroid $A$ or a Cartan-Lie algebroid $(A,\nabla)$ (a Lie algebroid with a compatible connection) over a base manifold $M$ equipped with appropriately compatible geometrical structures. The main focus is on a Riemannian base $(M,g)$, but we also consider symplectic and generalized Riemannian structures. For the Ri...

Established fundamental physics can be described by fields, which are maps. The source of such a map is space-time, which can be curved due to gravity. The map itself needs to be curved in its gauge field part so as to describe interaction forces like those mediated by photons and gluons. In the present article, we permit non-zero curvature also on...

We show that in the context of two-dimensional sigma models minimal coupling of an ordinary rigid symmetry Lie algebra g leads naturally to the appearance of the “generalized tangent bundle” \( \mathbb{T} \)
M≡TM⊕T
*
M by means of composite fields. Gauge transformations of the composite fields follow the Courant bracket, closing upon the choice of...

The gauge principle is at the heart of a good part of fundamental physics:
Starting with a group G of so-called rigid symmetries of a functional defined
over space-time Sigma, the original functional is extended appropriately by
additional Lie(G)-valued 1-form gauge fields so as to lift the symmetry to
Maps(Sigma,G). Physically relevant quantities...

We study the Lagrangian antifield BRST formalism, formulated in terms of exterior horizontal forms on the infinite order jet space of graded fields for topological field theories associated to \(Q\)-bundles. In the case of a trivial \(Q\)-bundle with a flat fiber and arbitrary base, we prove that the BRST cohomology are isomorphic to the cohomology...

Mapping spaces of supermanifolds are usually thought as exclusively in
functorial terms (i.e. trough the Grothendieck functor of points). In
this work we provide a geometric description of such mapping spaces in
terms of infinite-dimensional super-vector bundles.

We describe explicitly Lie superalgebra isomorphisms between the Lie superalgebras of first-order superdifferential operators on supermanifolds, showing in particular that any such isomorphism induces a diffeomorphism of the supermanifolds. We also prove that the group of automorphisms of such a Lie superalgebra is a semi-direct product of the subg...

In this contribution we review some of the interplay between sigma models in theoretical physics and novel geometrical structures such as Lie (n-)algebroids. The first part of the article contains the mathematical background, the definition of various algebroids as well as of Dirac structures, a joint generalization of Poisson, presymplectic, but a...

We prove a ‘superversion’ of Shanks and Pursell’s classical result stating that any isomorphism of the Lie algebras of compactly supported vector fields is implemented by a diffeomorphism of underlying manifolds. We thus provide a Lie algebraic characterization of supermanifolds and describe explicitly isomorphisms of the Lie superalgebras of super...

We investigate Atiyah algebroids, i.e. the infinitesimal objects of principal bundles, from the viewpoint of Lie algebraic approach to space. First we show that if the Lie algebras of smooth sections of two Atiyah algebroids are isomorphic, then the corresponding base manifolds are necessarily diffeomorphic. Further, we give two characterizations o...

During the last decades algebraization of space turned out to be a promising tool at the interface between Mathematics and Theoretical Physics. Starting with works by Gel'fand-Kolmogoroff and Gel'fand-Naimark, this branch developed as from the fortieth in two directions: algebraic characterization of usual geometric space on the one hand, and algeb...

A Q-manifold is a graded manifold endowed with a vector field of degree one squaring to zero. We consider the notion of a Q-bundle, that is, a fiber bundle in the category of Q-manifolds. To each homotopy class of ``gauge fields'' (sections in the category of graded manifolds) and each cohomology class of a certain subcomplex of forms on the fiber...

We introduce a new topological sigma model, whose fields are bundle maps from the tangent bundle of a 2-dimensional world-sheet to a Dirac subbundle of an exact Courant algebroid over a target manifold. It generalizes simultaneously the (twisted) Poisson sigma model as well as the G/G-WZW model. The equations of motion are satisfied, iff the corres...

Chern–Simons (CS) gauge theories in three dimensions and the Poisson sigma model (PSM) in two dimensions are examples of the same theory, if their field equations are interpreted as morphisms of Lie algebroids and their symmetries (on-shell) as homotopies of such morphisms. We point out that the (off-shell) gauge symmetries of the PSM in the litera...

In this paper we continue our study of the fourth order transgression on
hyper\"ahler manifolds introduced in the previous paper. We give a local
construction for the fourth-order transgression of the Chern character form of
an arbitrary vector bundle supplied with a self-dual connection on a four
dimensional hyperk\"ahler manifold. The constructio...

We propose a generalization of the Hodge dd
c
-lemma to the case of hyperkähler manifolds. As an application we derive a global construction of the fourth order transgression
of the Chern character forms of hyperholomorphic bundles over compact hyperkähler manifolds. In Section 3 we consider the
fourth order transgression for the infinite-dimensio...

In this paper we consider the Poisson algebraic structure associated with a
classical $r$-matrix, i.e. with a solution of the modified classical
Yang--Baxter equation. In Section 1 we recall the concept and basic facts of
the $r$-matrix type Poisson orbits. Then we describe the $r$-matrix Poisson
pencil (i.e the pair of compatible Poisson structure...

We check the Vaisman condition of geometric quantization for the R-matrix-type Poisson pencil on a coadjoint orbit of a compact
semisimple Lie group. It is shown that this condition is not satisfied for Hermitian symmetric spaces. We also construct some
examples where the Vaisman condition is satisfied.

We check The Vaisman condition of geometric quantization for R-matrix type Poisson pencil on a coadjoint orbit of a compact Lie group. It is shown that this condition isn't satisfied.

## Projects

Projects (2)

Higher gauge theories are gauge theories with in addition to the 1-form gauge fields also carry some of higher form degrees. Supergeometrical tools are very useful in this context, already before the BV formulation. In the near future we intend to construct several Yang-Mills type functionals of this sort and in parallel study the involved mathematics in its own right as well.

The standard model of elementary particles is described by particular Yang-Mills gauge theory (in particular, for the structure group U(1) times SU(2) times SU(3)), which in turn is based on principal bundles for Lie group actions. The Poisson sigma model can be viewed as a topological Chern-Simons type of gauge theory the Lie algebroid T*M of a Poisson manifold (M,Pi). In 2004 I made a first attempt to generalize this setting to non-topological theories (my article titled ''Algebroid Yang-Mills theories'', but cf also the preparational works performed together with Martin Bojowald, Melchior Grützman and in particular Alexei Kotov in the years around this one). In 2015, Alexei and I proposed a full-fleged extension of the standard model based on Lie groupoids or, more generally, Lie algebroids. This project is about developing this theory further, on the mathematical but eventually also on the physics level.