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Papers by Melchior Grützmann

Journal de Mathématiques Pures et Appliquées, 2021

Soit S un fibré spinoriel d'un fibré vectoriel pseudo-Euclidien (E, g) de rang pair. Nous introdu... more Soit S un fibré spinoriel d'un fibré vectoriel pseudo-Euclidien (E, g) de rang pair. Nous introduisons une nouvelle filtration de l'algèbre D(M, S) des opérateurs différentiels sur S. Pour cette filtration, l'algèbre graduée associée gr D(M, S) s'avèreêtre isomorpheà l'algèbre O(M) des fonctions lisses sur M, la variété graduée symplectique de degré 2 canoniquement associéeà (E, g). En conséquence, nous construisons la quantification de Weyl de M comme une application WQ : O(M) → D(M, S), et montrons que WQ satisfait toutes les propriétés voulues d'une quantification. En application, nous obtenons une bijection entre les structures d'algébroïdes de Courant (E, g, ρ, •, •), caractérisées de manièreéquivalentes par des fonctions hamiltoniennes génératrices sur la variété graduée symplectique M, et les opérateurs de Dirac générateurs anti-symétriques D ∈ D(M, S). L'opérateur D 2 est un nouvel invariant de (E, g, ρ, •, •), qui généralise la norme au carré de la 3-forme de Cartan d'une algèbre de Lie quadratique. Nousétudions cet invariant en détail dans le cas particulier où E est le double d'un bi-algébroïde de Lie (A, A *).

International Journal of Geometric Methods in Modern Physics, 2015

Starting with minimal requirements from the physical experience with higher gauge theories, i.e. ... more Starting with minimal requirements from the physical experience with higher gauge theories, i.e. gauge theories for a tower of differential forms of different form degrees, we discover that all the structural identities governing such theories can be concisely recombined into what is called a Q-structure or, equivalently, an L∞-algebroid. This has many technical and conceptual advantages: complicated higher bundles become just bundles in the category of Q-manifolds in this approach (the many structural identities being encoded in the one operator Q squaring to zero), gauge transformations are generated by internal vertical automorphisms in these bundles and even for a relatively intricate field content the gauge algebra can be determined in some lines and is given by what is called the derived bracket construction. This paper aims equally at mathematicians and theoretical physicists; each more physical section is followed by a purely mathematical one. While the considerations are va...

We generalize Hansen--Strobl's definition of HHH-twisted Courant algebroid such that the twis... more We generalize Hansen--Strobl's definition of HHH-twisted Courant algebroid such that the twist HHH of the Jacobi identity is a 4-form in the kernel of the anchor map and is closed under a naturally occurring exterior covariant derivative. We give examples and define a cohomology.

We give numerous examples of almost Lie algebroids arising as Dirac structures in pre-Courant alg... more We give numerous examples of almost Lie algebroids arising as Dirac structures in pre-Courant algebroids, e.g. from twisted Poisson structures, as well as from twisted actions of a Lie algebra. We moreover define a cohomology for them, motivated by a Q-structure, that is trivial for (bundles of) Lie algebras but characterizes the underlying integrable distribution the image of the anchor map.

Journal of Symplectic Geometry, 2009

In this paper we study the cohomology H • st (E) of a Courant algebroid E. We prove that if E is ... more In this paper we study the cohomology H • st (E) of a Courant algebroid E. We prove that if E is transitive, H • st (E) coincides with the naive cohomology H • naive (E) of E as conjectured by Stiénon and Xu. For general Courant algebroids E we define a spectral sequence converging to H • st (E). If E is with split base, we prove that there exists a natural transgression homomorphism T 3 (with image in H 3 naive (E)) which, together with H • naive (E), gives all H • st (E). For generalized exact Courant algebroids, we give an explicit formula for T 3 depending only on theŠevera characteristic clas of E.

Journal of Geometry and Physics, 2011

We define a new kind of algebroid which fulfills a Leibniz rule, a Jacobi identity twisted by a 3... more We define a new kind of algebroid which fulfills a Leibniz rule, a Jacobi identity twisted by a 3-form H with values in the kernel of the anchor map, and the twist is closed under a naturally occurring exterior covariant derivative. We give examples and define three kinds of cohomologies, two via realizations as Q-structures on graded manifolds. The paper

We introduce Courant algebroids, providing definitions, some historical notes, and some elementar... more We introduce Courant algebroids, providing definitions, some historical notes, and some elementary properties. Next, we summarize basic properties of graded manifolds. Then, drawing on the work by Roytenberg and others, we introduce the graded or supergraded language demostrating a cochain complex/ cohomology for (general) Courant algebroids. We review spectral sequences and show how this tool is used to compute cohomology for regular

Indagationes Mathematicae, 2014

We introduce the notion of matched pairs of Courant algebroids and give several examples arising ... more We introduce the notion of matched pairs of Courant algebroids and give several examples arising naturally from complex manifolds, holomorphic Courant algebroids, and certain regular Courant algebroids. We consider the matched sum of two Dirac subbundles, one in each of two Courant algebroids forming a matched pair.

Journal de Mathématiques Pures et Appliquées, 2021

Soit S un fibré spinoriel d'un fibré vectoriel pseudo-Euclidien (E, g) de rang pair. Nous introdu... more Soit S un fibré spinoriel d'un fibré vectoriel pseudo-Euclidien (E, g) de rang pair. Nous introduisons une nouvelle filtration de l'algèbre D(M, S) des opérateurs différentiels sur S. Pour cette filtration, l'algèbre graduée associée gr D(M, S) s'avèreêtre isomorpheà l'algèbre O(M) des fonctions lisses sur M, la variété graduée symplectique de degré 2 canoniquement associéeà (E, g). En conséquence, nous construisons la quantification de Weyl de M comme une application WQ : O(M) → D(M, S), et montrons que WQ satisfait toutes les propriétés voulues d'une quantification. En application, nous obtenons une bijection entre les structures d'algébroïdes de Courant (E, g, ρ, •, •), caractérisées de manièreéquivalentes par des fonctions hamiltoniennes génératrices sur la variété graduée symplectique M, et les opérateurs de Dirac générateurs anti-symétriques D ∈ D(M, S). L'opérateur D 2 est un nouvel invariant de (E, g, ρ, •, •), qui généralise la norme au carré de la 3-forme de Cartan d'une algèbre de Lie quadratique. Nousétudions cet invariant en détail dans le cas particulier où E est le double d'un bi-algébroïde de Lie (A, A *).

International Journal of Geometric Methods in Modern Physics, 2015

Starting with minimal requirements from the physical experience with higher gauge theories, i.e. ... more Starting with minimal requirements from the physical experience with higher gauge theories, i.e. gauge theories for a tower of differential forms of different form degrees, we discover that all the structural identities governing such theories can be concisely recombined into what is called a Q-structure or, equivalently, an L∞-algebroid. This has many technical and conceptual advantages: complicated higher bundles become just bundles in the category of Q-manifolds in this approach (the many structural identities being encoded in the one operator Q squaring to zero), gauge transformations are generated by internal vertical automorphisms in these bundles and even for a relatively intricate field content the gauge algebra can be determined in some lines and is given by what is called the derived bracket construction. This paper aims equally at mathematicians and theoretical physicists; each more physical section is followed by a purely mathematical one. While the considerations are va...

We generalize Hansen--Strobl's definition of HHH-twisted Courant algebroid such that the twis... more We generalize Hansen--Strobl's definition of HHH-twisted Courant algebroid such that the twist HHH of the Jacobi identity is a 4-form in the kernel of the anchor map and is closed under a naturally occurring exterior covariant derivative. We give examples and define a cohomology.

We give numerous examples of almost Lie algebroids arising as Dirac structures in pre-Courant alg... more We give numerous examples of almost Lie algebroids arising as Dirac structures in pre-Courant algebroids, e.g. from twisted Poisson structures, as well as from twisted actions of a Lie algebra. We moreover define a cohomology for them, motivated by a Q-structure, that is trivial for (bundles of) Lie algebras but characterizes the underlying integrable distribution the image of the anchor map.

Journal of Symplectic Geometry, 2009

In this paper we study the cohomology H • st (E) of a Courant algebroid E. We prove that if E is ... more In this paper we study the cohomology H • st (E) of a Courant algebroid E. We prove that if E is transitive, H • st (E) coincides with the naive cohomology H • naive (E) of E as conjectured by Stiénon and Xu. For general Courant algebroids E we define a spectral sequence converging to H • st (E). If E is with split base, we prove that there exists a natural transgression homomorphism T 3 (with image in H 3 naive (E)) which, together with H • naive (E), gives all H • st (E). For generalized exact Courant algebroids, we give an explicit formula for T 3 depending only on theŠevera characteristic clas of E.

Journal of Geometry and Physics, 2011

We define a new kind of algebroid which fulfills a Leibniz rule, a Jacobi identity twisted by a 3... more We define a new kind of algebroid which fulfills a Leibniz rule, a Jacobi identity twisted by a 3-form H with values in the kernel of the anchor map, and the twist is closed under a naturally occurring exterior covariant derivative. We give examples and define three kinds of cohomologies, two via realizations as Q-structures on graded manifolds. The paper

We introduce Courant algebroids, providing definitions, some historical notes, and some elementar... more We introduce Courant algebroids, providing definitions, some historical notes, and some elementary properties. Next, we summarize basic properties of graded manifolds. Then, drawing on the work by Roytenberg and others, we introduce the graded or supergraded language demostrating a cochain complex/ cohomology for (general) Courant algebroids. We review spectral sequences and show how this tool is used to compute cohomology for regular

Indagationes Mathematicae, 2014

We introduce the notion of matched pairs of Courant algebroids and give several examples arising ... more We introduce the notion of matched pairs of Courant algebroids and give several examples arising naturally from complex manifolds, holomorphic Courant algebroids, and certain regular Courant algebroids. We consider the matched sum of two Dirac subbundles, one in each of two Courant algebroids forming a matched pair.