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Jordan algebras arise naturally in (quantum) information geometry, and we want to understand their role and their structure within that framework. Inspired by Kirillov's discussion of the symplectic structure on coadjoint orbits, we provide…
The abstract theory of self-adjoint extensions of symmetric operators is used to construct self-adjoint realizations of a second-order elliptic operator on $\mathbb{R}^{n}$ with linear boundary conditions on (a relatively open part of) a…
Mean field theory has an unexpected group theoretic mathematical foundation. Instead of representation theory which applies to most group theoretic quantum models, Hartree-Fock and Hartree-Fock-Bogoliubov have been formulated in terms of…
The restricted Siegel disc is a homogeneous space related to the connected component $T_0(1)$ of the Universal Teichm\"uller space via the period mapping. In this paper we show that it is a coadjoint orbit of the universal central extension…
Quiver varieties have recently appeared in various different areas of Mathematics such as representation theory of Kac-Moody algebras and quantum groups, instantons on 4-manifolds, and resolutions Kleinian singularities. In this paper, we…
We study the orthogonal projections of symplectic balls in $\mathbb{R}^{2n}$ on complex subspaces. In particular we show that these projections are themselves symplectic balls under a certain complexity assumption. Our main result is a…
We investigate the SL(2,R) invariant geodesic curves with the as- sociated invariant distance function in parabolic geometry. Parabolic geom- etry naturally occurs in the study of SL(2,R) and is placed in between the elliptic and the…
For a given manifold $M$ we consider the non-linear Grassmann manifold $Gr_n(M)$ of $n$-dimensional submanifolds in $M$. A closed $(n+2)$-form on $M$ gives rise to a closed 2-form on $Gr_n(M)$. If the original form was integral, the 2-form…
We develop a mathematical concept towards gauge field theories based upon a Hilbert space endowed with a representation of a skew-adjoint Lie algebra and an action of a generalized Dirac operator. This concept shares common features with…
Kinematic space has been defined as the space of codimension-$2$ spacelike extremal surfaces in anti de Sitter (AdS$_{d+1}$) spacetime which, by the Ryu-Takayanagi proposal, compute the entanglement entropy of spheres in the boundary…
The aim of this note is to present some new explicit examples of $O(d,d)$-generalised Leibniz parallelisable spaces arising as the normal bundles of adjoint orbits $\mathcal{O}$ of some semi-simple Lie group $G$. Using this construction, an…
We construct start-products on the co-adjoint orbit of Lie group $\Aff({\bf C})$ of affine transformations of the complex straight line and apply them to obtain the irreducible unitary representations of this group. These results show…
We study the transverse Poisson structure to adjoint orbits in a complex semi-simple Lie algebra. The problem is first reduced to the case of nilpotent orbits. We prove then that in suitably chosen quasi-homogeneous coordinates the…
We consider canonical symplectic structure on the moduli space of flat ${\g}$-connections on a Riemann surface of genus $g$ with $n$ marked points. For ${\g}$ being a semisimple Lie algebra we obtain an explicit efficient formula for this…
Using the orbit method we attempt to reveal geometric and algebraic meaning of separation of variables for the integrable systems on coadjoint orbits in an $\mathfrak{sl}(3)$ loop algebra. We consider two types of generic orbits embedded…
It is known that the Schr\"odinger flow on a complex Grassmann manifold is equivalent to the matrix non-linear Schr\"odinger equation and the Ferapontov flow on a principal Adjoint U(n)-orbit is equivalent to the $n$-wave equation. In this…
In the present work we describe 3-dimensional complex SL_2-varieties where the generic SL_2-orbit is a surface. We apply this result to classify the minimal 3-dimensional projective varieties with Picard-number 1 where a semisimple group…
The adjoint action of a finite group of Lie type on its Lie algebra is studied. A simple formula is conjectured for the number of split semisimple orbits of a given genus. This conjecture is proved for type A, and partial results are…
We study the structure of the coadjoint orbits of the 2+1 Poincar\'e group, using a matricial representation of the group. We also obtain the orbits connected to irreducible representations of the group. Finally we obtain coherent states…
Let $k$ be a field of characteristic not 2 or 3. Let $V$ be the $k$-space of binary cubic polynomials. The natural symplectic structure on $k^2$ promotes to a symplectic structure $\omega$ on $V$ and from the natural symplectic action of…