arXiv subjects

Christian Blohmann

Publications and source records attributed to Christian Blohmann.

At least 19 recordsLinked to original sources

arXiv2026

Cartan calculus of cubical forms in tangent categories

We construct the Cartan calculus of differential cubical forms on an object $X$ of a cartesian tangent category with a scalar multiplication by a commutative ring object $R$. Borrowing the terminology of cubical singular homology, we define an (infinitesimal) cubical $n$-form to be a morphism $\omega: T^n X \to R$ on the iterated tangent bundle that is antisymmetric and $R$-linear in every factor of $T^n$. We call $\omega$ differential if, in addition, it is its own derivative in the fiber directions. (This notion of forms is a special case of the singular forms of Cruttwell and Lucyshyn-Wright.) We prove that the differential cubical forms are naturally equipped with the structure of a commutative differential graded algebra (CDGA). Then we show that, if the ring object has no $2$-torsion, this CDGA together with the Lie algebra of vector fields and the inner derivatives constitutes a Cartan calculus. It lies between their initial Cartan calculus of algebraic K\"ahler forms and the terminal one of Lie-Rinehart forms. We give a number of examples. In particular, on affine schemes over a field of characteristic other than two and on elastic diffeological spaces we retrieve the usual de Rham complex, which is generally different from the complex of Lie-Rinehart forms.

math.CT↗

arXiv2026

Cartan calculus in tangent categories

We determine the structure needed in a tangent category in the sense of Rosick\'y and Cockett-Cruttwell to construct the Cartan calculus on all objects. The missing ingredient is a scalar multiplication by a commutative ring object $R$, playing the role of the smooth real line, which equips the tangent bundle of every object with the structure of an $R$-module compatible with the tangent structure. We show that under these axioms the Lie algebra of vector fields acts by derivations on the ring of $R$-valued functions and satisfies the Leibniz rule. In other words, the tangent bundle is an abstract Lie algebroid, so that the Lie algebra of vector fields is a Lie-Rinehart algebra over the ring of functions. Consequently, every object carries a Cartan calculus of Lie-Rinehart forms, given by the Chevalley-Eilenberg complex together with its differential, inner derivative, and Lie derivative. Examples include the tangent categories of smooth manifolds, $G$-manifolds, Lie groupoids, log manifolds, pro-manifolds, elastic diffeological spaces, affine and general schemes, graded manifolds, and affine $C^\infty$-schemes.

math.CT↗

arXiv2025

Functoriality of bornological groupoid convolution

We show that the complete bornological convolution algebras of Lie groupoids and convolution bimodules of groupoid bibundles define a monoidal functor from the 2-category of differentiable stacks to the Morita 2-category of complete bornological algebras. The convolution algebras are generally non-unital, but are shown to possess one-sided approximate units such that the multiplication operators Mackey converge in the functional bornology of endomorphisms. This implies that the convolution algebras are self-induced and the convolution modules are smooth in the sense of R. Meyer. We also show that Lie groupoid actions that are submersive, proper, and transitive have projective convolution modules. This implies that all convolution algebras are quasi-unital. We provide a long list of examples and applications, such as to bornological noncommutative tori, which are Hopf monoids in the Morita category.

math.DG↗

arXiv2024

Differentiable groupoid objects and their abstract Lie algebroids

The infinitesimal counterpart of a Lie groupoid is its Lie algebroid. As a vector bundle, it is given by the source vertical tangent bundle restricted to the identity bisection. Its sections can be identified with the invariant vector fields on the groupoid, which are closed under the Lie bracket. We generalize this differentiation procedure to groupoid objects in any category with an abstract tangent structure in the sense of Rosick\'{y} and a scalar multiplication by a ring object that plays the role of the real numbers. We identify the categorical conditions that the groupoid object must satisfy to admit a natural notion of invariant vector fields. Then we show that invariant vector fields are closed under the Lie bracket defined by Rosick\'{y} and satisfy the Leibniz rule with respect to ring-valued morphisms on the base of the groupoid. The result is what we define axiomatically as an abstract Lie algebroid, by generalizing the underlying vector bundle to a module object in the slice category over its base. Examples include diffeomorphism groups, bisection groups of Lie groupoids, the diffeological symmetry groupoids of general relativity (Blohmann/Fernandes/Weinstein), symmetry groupoids in Lagrangian Field Theory, holonomy groupoids of singular foliations, elastic diffeological groupoids, groupoid objects in differentiable stacks, and affine groupoid schemes.

math.CT↗

arXiv2023

Hamiltonian Lie algebroids over Poisson manifolds

We extend to Poisson manifolds the theory of hamiltonian Lie algebroids originally developed by two of the authors for presymplectic manifolds. As in the presymplectic case, our definition, involving a vector bundle connection on the Lie algebroid, reduces to the definition of hamiltonian action for an action Lie algebroid with the trivial connection. The clean zero locus of the momentum section of a hamiltonian Lie algebroid is an invariant coisotropic submanifold, the distribution being given by the image of the anchor. We study some basic examples: bundles of Lie algebras with zero anchor and cotangent and tangent Lie algebroids. Finally, we discuss a suggestion by Alejandro Cabrera that the conditions for a Lie algebroid $A$ to be hamiltonian may be expressed in terms of two bivector fields on $A^*$, the natural Poisson structure on the dual of a Lie algebroid and the horizontal lift by the connection of the given Poisson structure on the base.

math.SG↗

arXiv2023

A Lie-Rinehart algebra in general relativity

We construct a Lie-Rinehart algebra over an infinitesimal extension of the space of initial value fields for Einstein's equations. The bracket relations in this algebra are precisely those of the constraints for the initial value problem. The Lie-Rinehart algebra comes from a slight generalization of a Lie algebroid in which the algebra consists of sections of a sheaf rather than a vector bundle. (An actual Lie algebroid had been previously constructed by Blohmann, Fernandes, and Weinstein over a much larger extension.) The construction uses the BV-BFV (Batalin-Fradkin-Vilkovisky) approach to boundary value problems, starting with the Einstein equations themselves, to construct an $L_\infty$-algebroid over a graded manifold which extends the initial data. The Lie-Rinehart algebra is then constructed by a change of variables. One of the consequences of the BV-BFV approach is a proof that the coisotropic property of the constraint set follows from the invariance of the Einstein equations under space-time diffeomorphisms.

math-ph↗

arXiv2023

Elastic diffeological spaces

We introduce a class of diffeological spaces, called elastic, on which the left Kan extension of the tangent functor of smooth manifolds defines an abstract tangent functor in the sense of Rosicky. On elastic spaces there is a natural Cartan calculus, consisting of vector fields and differential forms, together with the Lie bracket, de Rham differential, inner derivative, and Lie derivative, satisfying the usual graded commutation relations. Elastic spaces are closed under arbitrary coproducts, finite products, and retracts. Examples include manifolds with corners and cusps, diffeological groups and diffeological vector spaces with a mild extra condition, mapping spaces between smooth manifolds, and spaces of sections of smooth fiber bundles. This paper is a condensed preview of a longer work, explaining its motivation, main concepts, and results, but omitting most of the proofs.

math.DG↗

arXiv2021

Hamiltonian Lie algebroids

In previous work with M.C. Fernandes, we found a Lie algebroid symmetry for the Einstein evolution equations of general relativity. The present work was motivated by the effort to explain the coisotropic structure of the constraint subset for the initial value problem by extending the notion of hamiltonian structure from Lie algebra actions to general Lie algebroids over presymplectic manifolds. After comparing possible compatibility conditions between the anchor $A\to TM$ and the presymplectic structure on the base $M$, we choose the most natural of them, given by a suitably chosen connection on $A$. We define a notion of momentum section of $A^*$ and a condition for compatibility with the Lie bracket. A Lie algebroid over a presymplectic manifold with compatible anchor and momentum section is then called hamiltonian. For an action Lie algebroid, we retrieve the conditions of a hamiltonian action. The clean zero locus of the momentum section of a hamiltonian Lie algebroid is a coisotropic submanifold. We show that a bracket-compatible momentum map is equivalent to a closed basic extension of the presymplectic form, within the generalization of the BRST model of equivariant cohomology to Lie algebroids. We construct groupoids by reduction of an action Lie groupoid $G\times M$ by a subgroup $H$ of $G$ which is not necessarily normal, and we find conditions which imply that a hamiltonian structure descends to their Lie algebroids. We consider many examples and, in particular, find that the tangent Lie algebroid over a symplectic manifold is hamiltonian with respect to some connection if and only if the symplectic structure has a nowhere vanishing primitive. Recent results of Stratmann and Tang show that this is the case whenever the symplectic structure is exact.

math.SG↗

arXiv2021

The homotopy momentum map of general relativity

We show that the action of spacetime vector fields on the variational bicomplex of general relativity has a homotopy momentum map that extends the map from vector fields to conserved currents given by Noether's first theorem to a morphism of $L_\infty$-algebras.

math.SG↗

arXiv2016

Removable presymplectic singularities and the local splitting of Dirac structures

We call a singularity of a presymplectic form $ω$ removable in its graph if its graph extends to a smooth Dirac structure over the singularity. An example for this is the symplectic form of a magnetic monopole. A criterion for the removability of singularities is given in terms of regularizing functions for pure spinors. All removable singularities are poles in the sense that the norm of $ω$ is not locally bounded. The points at which removable singularities occur are the non-regular points of the Dirac structure for which we prove a general splitting theorem: Locally, every Dirac structure is the gauge transform of the product of a tangent bundle and the graph of a Poisson structure. This implies that in a neighborhood of a removable singularity $ω$ can be split into a non-singular presymplectic form and a singular presymplectic form which is the partial inverse of a Poisson bivector that vanishes at the singularity. An interesting class of examples is given by log-Dirac structures which generalize log-symplectic structures. The analogous notion of removable singularities of Poisson structures is also studied.

math.SG↗

arXiv2011

Groupoid symmetry and constraints in general relativity

When the vacuum Einstein equations are cast in the form of hamiltonian evolution equations, the initial data lie in the cotangent bundle of the manifold MΣ of riemannian metrics on a Cauchy hypersurface Σ. As in every lagrangian field theory with symmetries, the initial data must satisfy constraints. But, unlike those of gauge theories, the constraints of general relativity do not arise as momenta of any hamiltonian group action. In this paper, we show that the bracket relations among the constraints of general relativity are identical to the bracket relations in the Lie algebroid of a groupoid consisting of diffeomorphisms between space-like hypersurfaces in spacetimes. A direct connection is still missing between the constraints themselves, whose definition is closely related to the Einstein equations, and our groupoid, in which the Einstein equations play no role at all. We discuss some of the difficulties involved in making such a connection.

math.DG↗

arXiv2008

Stacky Lie groups

Presentations of smooth symmetry groups of differentiable stacks are studied within the framework of the weak 2-category of Lie groupoids, smooth principal bibundles, and smooth biequivariant maps. It is shown that principality of bibundles is a categorical property which is sufficient and necessary for the existence of products. Stacky Lie groups are defined as group objects in this weak 2-category. Introducing a graphic notation, it is shown that for every stacky Lie monoid there is a natural morphism, called the preinverse, which is a Morita equivalence if and only if the monoid is a stacky Lie group. As example we describe explicitly the stacky Lie group structure of the irrational Kronecker foliation of the torus.

math.DG↗

arXiv2007

Hopfish structure and modules over irrational rotation algebras

Inspired by the group structure on $S^1/ \bbZ$, we introduce a weak hopfish structure on an irrational rotation algebra $A$ of finite Fourier series. We consider a class of simple $A$-modules defined by invertible elements, and we compute the tensor product between these modules defined by the hopfish structure. This class of simple modules turns out to generate an interesting commutative unital ring.

math.QA↗

arXiv2007

Group-like objects in Poisson geometry and algebra

A group, defined as set with associative multiplication and inverse, is a natural structure describing the symmetry of a space. The concept of group generalizes to group objects internal to other categories than sets. But there are yet more general objects that can still be thought of as groups in many ways, such as quantum groups. We explain some of the generalizations of groups which arise in Poisson geometry and quantization: the germ of a topological group, Poisson Lie groups, rigid monoidal structures on symplectic realizations, groupoids, 2-groups, stacky Lie groups, and hopfish algebras.

math.SG↗

arXiv2005

A Gravity Theory on Noncommutative Spaces

A deformation of the algebra of diffeomorphisms is constructed for canonically deformed spaces with constant deformation parameter theta. The algebraic relations remain the same, whereas the comultiplication rule (Leibniz rule) is different from the undeformed one. Based on this deformed algebra a covariant tensor calculus is constructed and all the concepts like metric, covariant derivatives, curvature and torsion can be defined on the deformed space as well. The construction of these geometric quantities is presented in detail. This leads to an action invariant under the deformed diffeomorphism algebra and can be interpreted as a theta-deformed Einstein-Hilbert action. The metric or the vierbein field will be the dynamical variable as they are in the undeformed theory. The action and all relevant quantities are expanded up to second order in theta.

hep-th↗

arXiv2005

Separation of noncommutative differential calculus on quantum Minkowski space

Noncommutative differential calculus on quantum Minkowski space is not separated with respect to the standard generators, in the sense that partial derivatives of functions of a single generator can depend on all other generators. It is shown that this problem can be overcome by a separation of variables. We study the action of the universal L-matrix, appearing in the coproduct of partial derivatives, on generators. Powers of he resulting quantum Minkowski algebra valued matrices are calculated. This leads to a nonlinear coordinate transformation which essentially separates the calculus. A compact formula for general derivatives is obtained in form of a chain rule with partial Jackson derivatives. It is applied to the massive quantum Klein-Gordon equation by reducing it to an ordinary q-difference equation. The rest state solution can be expressed in terms of a product of q-exponential functions in the separated variables.

math.QA↗

arXiv2004

Reconstruction of universal Drinfeld twists from representations

Universal Drinfeld twists are inner automorphisms which relate the coproduct of a quantum enveloping algebra to the coproduct of the undeformed enveloping algebra. Even though they govern the deformation theory of classical symmetries and have appeared in numerous applications, no twist for a semi-simple quantum enveloping algebra has ever been computed. It is argued that universal twists can be reconstructed from their well known representations. A method to reconstruct an arbitrary element of the enveloping algebra from its irreducible representations is developed. For the twist this yields an algebra valued generating function to all orders in the deformation parameter, expressed by a combination of basic and ordinary hypergeometric functions. An explicit expression for the universal twist of su(2) is given up to third order.

math.QA↗

arXiv2004

Realization of q-deformed spacetime as star product by a Drinfeld twist

Covariance ties the noncommutative deformation of a space into a quantum space closely to the deformation of the symmetry into a quantum symmetry. Quantum deformations of enveloping algebras are governed by Drinfeld twists, inner automorphisms which relate the deformed to the undeformed coproduct. While Drinfeld twists naturally define a covariant star product on the space algebra, this product is in general not associative and does not yield a quantum space. It is reported that, nevertheless, there are certain Drinfeld twists which realize the quantum plane, quantum Euclidean 4-space, and quantum Minkowski space.

math.QA↗