Related papers: Local coordinates for complex and quaternionic hyp…

We propose to parametrize the configuration space of one-dimensional quantum systems of N identical particles by the elementary symmetric polynomials of bosonic and fermionic coordinates. It is shown that in this parametrization the…

In this paper, we obtain analogues of Jorgensen's inequality for non-elementary groups of isometries of quaternionic hyperbolic $n$-space generated by two elements, one of which is loxodromic. Our result gives some improvement over earlier…

Let $G = GL_N$ over an algebraically closed field of odd characteristic, and $\theta$ an involutive automorphism on $G$ such that $H = (G^{\theta})^0$ is isomorphic to $SO_N$. Then $G^{\iota\theta} = \{ g \in G \mid \theta(g) = g^{-1} \}$…

Let $V$ be a vector space of dimension $n+1$. We demonstrate that $n$-component third-order Hamiltonian operators of differential-geometric type are parametrised by the algebraic variety of elements of rank $n$ in $S^2(\Lambda^2V)$ that lie…

Let $\Sigma_{g,n}$ be a compact oriented surface of genus $g$ with $n$ open disks removed. The graph algebra $\mathcal{L}_{g,n}(H)$ was introduced by Alekseev--Grosse--Schomerus and Buffenoir--Roche and is a combinatorial quantization of…

Let g be a semisimple Lie algebra with h a Cartan subalgebra. The orbit method attempts to assign representations of g to orbits in g*. Orbital varieties are particular Lagrangian subvarieties of such orbits which should lead to highest…

Let $F$ be a non-archimedean local field of characteristic different from $2$ and $G$ be either an odd special orthogonal group ${\rm SO}_{2r+1}(F)$ or a symplectic group ${\rm Sp}_{2r}(F)$. In this paper, we establish the local converse…

Let X denote either CP^m or C^m. We study certain analytic properties of the space E^n of ordered geometrically generic n-point configurations in X. This space consists of all q=(q_1,...,q_n) in X^n such that no m+1 of the points…

Let $G$ be a group. An element $g$ in $G$ is called reversible if it is conjugate to $g^{-1}$ within $G$, and called strongly reversible if it is conjugate to its inverse by an order two element of $G$. Let $\textbf{H}_{\mathbb H}^n$ be the…

In this paper we enumerate and classify the ``simplest'' pairs (M,G) where M is a closed orientable 3-manifold and G is a trivalent graph embedded in M. To enumerate the pairs we use a variation of Matveev's definition of complexity for…

We study the geometry and spectral theory of Weil-Petersson random surfaces with genus-$g$ and $n$ cusps in the large-$n$ limit. We show that for a random hyperbolic surface in $\mathcal{M}_{g,n}$ with $n$ large, the number of small…

Orbits of coadjoint representations of classical compact Lie groups have a lot of applications. They appear in representation theory, geometrical quantization, theory of magnetism, quantum optics etc. As geometric objects the orbits were…

We give a survey of several models of irreducible complementary series representations and their limits, special representations, for the groups SU(n,1) and SO(n,1), including new ones. These groups, whose geometrical meaning is well known,…

Let G be a Lie group, $g = Lie(G)$ - its Lie algebra, $g*$ - the dual vector space and $\widehat G$ - the set of equivalence classes of unitary irreducible representations of $G$. The orbit method [1] establishes a correspondence between…

We use our new type of bounded locally homeomorphic quasiregular mappings in the unit 3-ball to address long standing problems for such mappings. The construction of such mappings comes from our construction of non-trivial compact…

The set $\Cal C(G)$ of closed subgroups of a locally compact group $G$ has a natural topology which makes it a compact space. This topology has been defined in various contexts by Vietoris, Chabauty, Fell, Thurston, Gromov, Grigorchuk, and…

Superspace is considered as space of parameters of the supercoherent states defining the basis for oscillator-like unitary irreducible representations of the generalized superconformal group SU(2m,2n/2N) in the field of quaternions H. The…

Let $S$ be a semigroup. The elements $a,b\in S$ are called primarily conjugate if $a=xy$ and $b=yx$ for certain $x,y\in S$. The relation of conjugacy is defined as the transitive closure of the relation of primary conjugacy. In the case…

A new approach towards the composite structure of quarks and leptons in the context of the higher dimensional unified theories is proposed. Owing to the certain strong dynamics, much like an ordinary QCD, every possible vectorlike…

It is of interest to characterize algebraically the dynamical types of isometries of the complex and quaternionic hyperbolic planes. In the complex case, such a characterization is known from the work of Giraud-Goldman. In this paper, we…