arXiv research

Researchers extend geometric quantization to complex Abelian Lie supergroups.

It is well known that the category of real Lie supergroups is equivalent to the category of the so-called (real) Harish-Chandra pairs. That means that a Lie supergroup depends only on the underlying Lie group and its Lie superalgebra with certain compatibility conditions. More precisely, the structure sheaf of a Lie su…

The paper studies invariant Einstein metrics on Lie supergroups.

From the four normed division algebras--the real numbers, complex numbers, quaternions and octonions, of dimension k=1, 2, 4 and 8, respectively--a systematic procedure gives a 3-cocycle on the Poincare superalgebra in dimensions k+2=3, 4, 6 and 10, and a 4-cocycle on the Poincare superalgebra in dimensions k+3=4, 5, 7…

We study actions of Lie supergroups, in particular, the hitherto elusive notion of orbits through odd (or more general) points. Following categorical principles, we derive a conceptual framework for their treatment and therein prove general existence theorems for the isotropy (or stabiliser) supergroups and orbits thro…

We review recent works concerning deformation quantization of abelian supergroups. Indeed, we expose the construction of an induced representation of the Heisenberg supergroup and an associated pseudodifferential calculus by using Kirillov's orbits method. Then, a star-product is built on the abelian supergroup R^{m|n}…

Generalizes Lie supergroups to Lie superheaps.

Recent work applying higher gauge theory to the superstring has indicated the presence of `higher symmetry'. Infinitesimally, this is realized by a `Lie 2-superalgebra' extending the Poincare superalgebra in precisely the dimensions where the classical supersymmetric string makes sense: 3, 4, 6 and 10. In the previous …

With a view towards applications in the theory of infinite-dimensional representations of finite-dimensional Lie supergroups, we introduce a new category of supermanifolds. In this category, supermanifolds of `maps' and `fields' (fibre bundle sections) exist. In particular, loop supergroups can be realised globally in …

It is known that there exists a natural functor $Φ$ from Lie supergroups to super Harish-Chandra pairs. A functor going backwards, that associates a Lie supergroup with each super Harish-Chandra pair, yielding an equivalence of categories, was found by Koszul [18]; this result was later extended by other authors, to di…

Recent work applying higher gauge theory to the superstring has indicated the presence of 'higher symmetry', and the same methods work for the super-2-brane. In the previous paper in this series, we used a geometric technique to construct a 'Lie 2-supergroup' extending the Poincare supergroup in precisely those spaceti…

Local positive definite Z2^n-superfunctions can be extended.

We give a new and self-contained proof of the existence and unicity of the flow for an arbitrary (not necessarily homogeneous) smooth vector field on a real supermanifold, and extend these results to the case of holomorphic vector fields on complex supermanifolds. Furthermore we discuss local actions associated to supe…

We show that every graph product of finitely generated abelian groups acts properly and cocompactly on a CAT(0) cubical complex. The complex generalizes (up to subdivision) the Salvetti complex of a right-angled Artin group and the Coxeter complex of a right-angled Coxeter group. In the right-angled Artin group case it…

The abstract generalizes a construction for splitting supermanifolds and studies Lie supergroup cases.

Harmonic almost complex structures on specific Lie groups and solvmanifolds identified.

The complex Lie superalgebras $\mathfrak{g}$ of type $D(2,1;a)$ - also denoted by $\mathfrak{osp}(4,2;a)$ - are usually considered for "non-singular" values of the parameter $a$, for which they are simple. In this paper we introduce five suitable integral forms of $\mathfrak{g}$, that are well-defined at singular valu…

Classifies complex symplectic structures on Lie algebras with large abelian ideals.

The paper studies a flow on complex Lie groups, showing convergence to solitons.

Unique complex structures on specific Lie algebras.

Study of complex and Hermitian structures on specific Lie groups.

Extending previous work that involved D3-branes ending on a fivebrane with $θ_{\mathrm{YM}}\not=0$, we consider a similar two-sided problem. This construction, in case the fivebrane is of NS type, is associated to the three-dimensional Chern-Simons theory of a supergroup U$(m|n)$ or OSp$(m|2n)$ rather than an ordinary …

We obtain a characterization of the real Lie algebras admitting abelian complex structures in terms of certain affine Lie algebras $\frak a \frak f \frak f (A)$, where $A$ is a commutative algebra. These affine Lie algebras are natural generalizations of $\frak a \frak f \frak f (\Bbb C)$ and the corresponding Lie grou…

Study balanced Hermitian structures on almost abelian Lie algebras, classifying six-dimensional cases.

We study the boundary conditions in the topologically twisted Chern-Simons matter theories with the Lie 3-algebraic structure. We find that the supersymmetric boundary conditions and the gauge invariant boundary conditions can be unified as the complexified gauge invariant boundary conditions which lead to the supergro…

New invariants for 3-manifolds derived from supergroup representations.

Characterizes almost abelian Lie algebras with integrable complex structure

Study pseudo-Kähler structures on almost abelian solvmanifolds.

We study the structure of Lie groups admitting left invariant abelian complex structures in terms of commutative associative algebras. If, in addition, the Lie group is equipped with a left invariant Hermitian structure, it turns out that such a Hermitian structure is Kähler if and only if the Lie group is the direct p…

Study of abelian structures on odd-dimensional Lie algebras and their geometric properties.

It is a classical result that any complex analytic Lie supergroup $\mathcal{G}$ is split \cite{kosz}, that is its structure sheaf is isomorphic to the structure sheaf of a certain vector bundle. However, there do exist non-split complex analytic homogeneous supermanifolds. We study the question how to find out whether …

For a Lie group $G$ and a closed Lie subgroup $H\subset G$, it is well known that the coset space $G/H$ can be equipped with the structure of a manifold homogeneous under $G$ and that any $G$-homogeneous manifold is isomorphic to one of this kind. An interesting problem is to find an analogue of this result in the case…

The paper classifies invariant structures on complex almost Abelian groups.

We classify the 6-dimensional Lie algebras that can be endowed with an abelian complex structure and parameterize, on each of these algebras, the space of such structures up to holomorphic isomorphism.

Algebraic dimension is zero for generic complex structures on hypercomplex nilmanifolds.

In this paper we generalize some results of Richard Palais to the case of Lie supergroups and Lie superalgebras. More precisely, let $G$ be a Lie supergroup, $\mathfrak g$ its Lie superalgebra and let $ρ$ be an infinitesimal action (a representation) of $\mathfrak g$ on a supermanifold $M$. We will show that there alwa…

In the present paper, we describe two geometric notions, holomorphic Norden structures and Kähler-Norden structures on Hom-Lie groups, and prove that on Hom-Lie groups in the left invariant setting, these structures are related to each other. We study Kähler-Norden structures with abelian complex structures and give th…

Develops Lie algebraic approach for compact complex homogeneous manifolds.

Motivated by a paper of Zirnbauer, we develop a theory of Riemannian supermanifolds up to a definition of Riemannian symmetric superspaces. Various fundamental concepts needed for the study of these spaces both from the Riemannian and the Lie theoretical viewpoint are introduced, e.g. geodesics, isometry groups and inv…

Characterizes complex structures on specific Lie groups.

Abstract classifies Lie algebras with complex or symplectic structures.

The Streets-Tian conjecture is confirmed for Lie algebras with specific abelian ideals.

We study Lie algebras endowed with an abelian complex structure which admit a symplectic form compatible with the complex structure. We prove that each of those Lie algebras is completely determined by a pair (U,H) where U is a complex commutative associative algebra and H is a sesquilinear hermitian form on U which ve…

Let G be a Lie supergroup and H a closed subsupergroup. We study the unimodularity of the homogeneous supermanifold G/H, i.e. the existence of G-invariant sections of its Berezinian line bundle. To that end, we express this line bundle as a G-equivariant associated bundle of the principal H-bundle G over G/H. We also s…

Cocalibrated G_2-structures and cocalibrated G_2^*-structures are the natural initial values for Hitchin's evolution equations whose solutions define (pseudo)-Riemannian manifolds with holonomy group contained in Spin(7) or Spin_0(3,4), respectively. In this article, we classify which seven-dimensional real Lie algebra…

Characterizes almost Abelian Lie algebras with special $H$-structures.

Study locally conformally balanced metrics on specific Lie algebras.

Left-invariant metrics force 2-step nilpotent groups, preserving Kähler-like conditions.