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Martin Bordemann

Publications and source records attributed to Martin Bordemann.

At least 19 recordsLinked to original sources
arXiv2026

Free Reductive Lie Algebra Pairs of Lie-Yamaguti algebras

The goal of this article is to show the categorical links between on the one hand the category of reductive Lie algebra pairs $\mathcal{RLP}$ and on the other hand the category of Lie-Yamaguti algebras $\mathcal{LY}$. The fact that the well-known construction of an enveloping algebra associating to a Lie-Yamaguti algebra a reductive Lie algebra pair is not functorial leads us to the main construction of the article, namely a left adjoint to the natural restriction functor $G:\mathcal{RLP}\to\mathcal{LY}$. As a final result we observe that the construction of the enveloping algebra becomes functorial when one restricts the morphisms of the categories $\mathcal{RLP}$ and $\mathcal{LY}$ to the surjective ones. Then it becomes a right adjoint to the restriction functor.

math.RA↗
arXiv2023

A gentle introduction to Drinfel'd associators

In this note we give an introduction to Drinfel'd's associator coming from the Knizhnik-Zamolodchikov connections and a self-contained proof of the hexagon and pentagon equations by means of minimal amounts of analysis or differential geometry: we rather use limits of concrete parallel transports.

math.QA↗
arXiv2021

Noncommutative localization in smooth deformation quantization

In this paper we shall show the equivalence of algebraic and analytic localisation for algebras of smooth deformation quantization for several situations. The proofs are based on old work by Whitney, Malgrange and Tougeron on the commutative algebra of smooth function rings from the 60's and 70's.

math.QA↗
arXiv2020

The Minimality of Determinantal Varieties

The determinantal variety $Σ_{pq}$ is defined to be the set of all $p\times q$ real matrices with $p\geq q$ whose ranks are strictly smaller than $q$. It is proved that $Σ_{pq}$ is a minimal cone in $\mathbb R^{pq}$ and all its strata are regular minimal submanifolds.

math.DG↗
arXiv2018

Structure theory of Rack-Bialgebras

In this paper we focus on a certain self-distributive multiplication on coalgebras, which leads to so-called rack bialgebra. Inspired by semi-group theory (adapting the Suschkewitsch theorem), we do some structure theory for rack bialgebras and cocommutative Hopf dialgebras. We also construct canonical rack bialgebras (some kind of enveloping algebras) for any Leibniz algebra and compare to the existing constructions. We are motivated by a differential geometric procedure which we call the Serre functor: To a pointed differentible manifold with multiplication is associated its distribution space supported in the chosen point. For Lie groups, it is well-known that this leads to the universal enveloping algebra of the Lie algebra. For Lie racks, we get rack-bialgebras, for Lie digroups, we obtain cocommutative Hopf dialgebras.

math.QA↗
arXiv2018

Algebraic deformation quantization of Leibniz algebras

In this paper we focus on a certain self-distributive multiplication on coalgebras, which leads to so-called rack bialgebra. We construct canon-ical rack bialgebras (some kind of enveloping algebras) for any Leibniz algebra. Our motivation is deformation quantization of Leibniz algebras in the sense of [6]. Namely, the canonical rack bialgebras we have constructed for any Leibniz algebra lead to a simple explicit formula of the rack-star-product on the dual of a Leibniz algebra recently constructed by Dherin and Wagemann in [6]. We clarify this framework setting up a general deformation theory for rack bialgebras and show that the rack-star-product turns out to be a deformation of the trivial rack bialgebra product.

math.AT↗
arXiv2018

L-infinity Formality check for the Hochschild Complex of Certain Universal Enveloping Algebras

We study the L-infinity-formality problem for the Hochschild complex of the universal enveloping algebra of some examples of Lie algebras such as Cartan-3-regular quadratic Lie algebras (for example semisimple Lie algebras and in more detail so(3)), and free Lie algebras generated by a vector space of dimension at least 2. We show that for these examples formality in Kontsevich's sense does NOT hold, although some of them allow unconditioned deformability. We compute the L-infinity-structure on the cohomology given by homotopy transfer in certain cases.

math.QA↗
arXiv2016

A dirty integration of Leibniz algebras

In this article, we present an integration of any real finite-dimensional Leibniz algebra as a Lie rack which reduces in the particular case of a Lie algebra to the ordinary connected simply connected Lie group. The construction is not functorial.

math.DG↗
arXiv2012

Twisting Poisson algebras, coPoisson algebras and Quantization

The purpose of this paper is to study twistings of Poisson algebras or bialgebras, coPoisson algebras or bialgebras and star-products. We con- sider Hom-algebraic structures generalizing classical algebraic structures by twisting the identities by a linear self map. We summarize the results on Hom-Poisson algebras and introduce Hom-coPoisson algebras and bialge- bras. We show that there exists a duality between Hom-Poisson bialgebras and Hom-coPoisson bialgebras. A relationship between enveloping Hom- algebras endowed with Hom-coPoisson structures and corresponding Hom- Lie bialgebra structures is studied. Moreover we set quantization problems and generalize the notion of star-product. In particular, we characterize the twists for the Moyal-Weyl product for polynomials of several variables.

math.RA↗
arXiv2009

A quasi-Lie bialgebra formulation of the Pohlmeyer-Rehren Poisson algebra

We present a quasi-Lie bialgebra (QLBA) quantization problem which comes from an algebraic reformulation of the Nambu-Goto string theory and invariant charges by Pohlmeyer and Rehren. This QLBA structure depends on a symmetric bivector (coming from a Minkowski metric) and is built on the free Lie algebra on a finite dimensional vector space. We solve this problem when the bivector has rank 1 or 2.

math.QA↗
arXiv2007

Deformation Quantization of Surjective Submersions and Principal Fibre Bundles

In this paper we establish a notion of deformation quantization of a surjective submersion which is specialized further to the case of a principal fibre bundle: the functions on the total space are deformed into a right module for the star product algebra of the functions on the base manifold. In case of a principal fibre bundle we require in addition invariance under the principal action. We prove existence and uniqueness of such deformations. The commutant within all differential operators on the total space is computed and gives a deformation of the algebra of vertical differential operators. Applications to noncommutative gauge field theories and phase space reduction of star products are discussed.

math.QA↗
arXiv2007

Noncommutative Riemann Surfaces

We introduce C-Algebras of compact Riemann surfaces $Σ$ as non-commutative analogues of the Poisson algebra of smooth functions on $Σ$. Representations of these algebras give rise to sequences of matrix-algebras for which matrix-commutators converge to Poisson-brackets as $N\to\infty$. For a particular class of surfaces, nicely interpolating between spheres and tori, we completely characterize (even for the intermediate singular surface) all finite dimensional representations of the corresponding C-algebras.

math-ph↗
arXiv2006

A homological approach to singular reduction in deformation quantization

We use the method of homological quantum reduction to construct a deformation quantization on singular symplectic quotients in the situation, where the coefficients of the moment map define a complete intersection. Several examples are discussed, among others one where the singularity type is worse than an orbifold singularity.

math-ph↗
arXiv2005

Formalite $G_\infty$ adaptee et star-representations sur des sous-varietes coisotropes

Let X be a Poisson manifold and C a coisotropic submanifold and let I be the vanishing ideal of C. In this work we want to construct a star product * on X such that I[[lambda]] is a left ideal for *. Thus we obtain a representation of the star product algebra A[[lambda]] = C^\infty(X)[[lambda]] on B[[lambda]] = A[[lambda]] / I[[lambda]] deforming the usual representation of A on the functions on C. The result follows from a generalization of Tamarkin's formality adapted to the submanifold C. We show that in the case X = R^n and C = R^{n-l} with l > 1 there are no obstructions to this formality.

math.QA↗
arXiv2004

(Bi)modules, morphismes et réduction des star-produits: le cas symplectique, feuilletages et obstructions

(Bi)modules, morphisms and reduction of star-products are studied in a framework of multidifferential operators along maps: morphisms deform Poisson maps and representations on functions spaces deform coisotropic maps. If a star-product is representable on a coisotropic submanifold, is is equivalent to a star-product for which the vanishing ideal is a left ideal. If the reduced phase space exists, a star-product with suitable Deligne class is representable and the reduced algebra is the commutant of this module (hence a bimodule). Obstructions to representability to third order are related to the Atiyah-Molino class of the foliation of the coisotropic submanifold, and the same kind of obstructions occurs for the quantisation of a Poisson map between symplectic manifolds. For vanishing Atiyah-Molino class, the representation and morphism problem is solvable.

math.QA↗
arXiv2002

Sur l'existence d'une prescription d'ordre naturelle projectivement invariante

P.Lecomte has proposed to take into account the covariant derivatives used to build ordering prescriptions for the naturality of transformation properties and has conjectured that there exists an natural ordering prescription for differential operators of any orders between density bundles which in addition is invariant under projective changes of the covariant derivatives. We prove this conjecture by constructing a projectively invariant lift of a torsion-free connexion to a torsion-free connexion on (the positive part of) the total space of the bundle of all $a$-densities for nonzero $a$, by lifting the symbols in a projectively invariant way (they turn out to be in bijection to the space of all $\real^+$-equivariant divergence-free symmetric tensor fields on the total space), and by using the standard ordering procedure (`all the covariant derivatives to the right') on the total space. For Ricci-flat manifolds we show that this ordering prescription coincides --with the appropiate replacements-- with an explicit formula in $\real^m$ obtained by Duval, Lecomte and Ovsienko.

math.DG↗
arXiv2002

Traces for star products on the dual of a Lie algebra

In this paper, we describe all traces for the BCH star-product on the dual of a Lie algebra. First we show by an elementary argument that the BCH as well as the Kontsevich star-product are strongly closed if and only if the Lie algebra is unimodular. In a next step we show that the traces of the BCH star-product are given by the $\ad$-invariant functionals. Particular examples are the integration over coadjoint orbits. We show that for a compact Lie group and a regular orbit one can even achieve that this integration becomes a positive trace functional. In this case we explicitly describe the corresponding GNS representation. Finally we discuss how invariant deformations on a group can be used to induce deformations of spaces where the group acts on.

math.QA↗