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Let $\mathcal{H}$ be a complex, separable Hilbert space (of finite or infinite dimension), and let $\mathcal{U}(\mathcal{H})$ denote the group of unitary operators on $\mathcal{H}$. A symmetry is, by definition, a unitary operator $J$ with…

Let $H$ be the space of all Hermitian matrices of infinite order and $U(\infty)$ be the inductive limit of the chain $U(1)\subset U(2)\subset...$ of compact unitary groups. The group $U(\infty)$ operates on the space $H$ by conjugations,…

We introduce the set of framed convex polyhedra with N faces as the symplectic quotient C^2N//SU(2). A framed polyhedron is then parametrized by N spinors living in C^2 satisfying suitable closure constraints and defines a usual convex…

Let $\mathbf{G}=\mathrm{U}(2,n)$ be the unitary group associated to a Hermitian space over a quadratic imaginary number field $E$. We assume that 2 is unramified in $E$, and the Hermitian space splits at all finite places and has signature…

There is a natural action of SL(2,R) on the moduli space of translation surfaces, and this yields an action of the unipotent subgroup $U = {\begin{pmatrix} 1 & * 0 & 1 \end{pmatrix}}$. We classify the U-invariant ergodic measures on certain…

Consider the example of the relationship of the group O(3) of rotations in 3-space to the special unitary group SU(2). Given other unitary groups, what transformations can we find? In this paper we describe a method of constructing…

$\text{SO}(1, d+1)$ is the isometry group of $(d+1)$-dimensional de Sitter spacetime $\text{dS}_{d+1}$ and the conformal group of $\mathbb{R}^{d}$. This note gives a pedagogical introduction to the representation theory of $\text{SO}(1,…

For an $n$-dimensional Leibniz/Lie algebra $\mathfrak{h}$ over a field $k$ we introduce a new invariant ${\mathcal A}(\mathfrak{h})$, called the \emph{universal algebra} of $\mathfrak{h}$, as a quotient of the polynomial algebra $k[X_{ij}…

We study the SO(4)xSU(2) invariant and N=(1,0) supersymmetry-preserving nilpotent (non-anticommutative) Moyal deformation of hypermultiplets interacting with an abelian gauge multiplet, starting from their off-shell formulation in Euclidean…

Rigorous justification of the Hubbard-Stratonovich transformation for the Pruisken-Sch\"afer type of parameterisations of real hyperbolic O(m,n)-invariant domains remains a challenging problem. We show that a naive choice of the volume…

It is shown that extending Osp(4/1) supergravity to quaternion co-ordinates to get HU(4/1) gives the standard model with 3 generations of fermions, U(1)xSU(2)xSU(3) bosons plus 2 gravitinos, 1 graviton, and nothing else. It is shown using…

We study the SO(4)x SU(2) invariant Q-deformation of Euclidean N=(1,1) gauge theories in the harmonic superspace formulation. This deformation preserves chirality and Grassmann harmonic analyticity but breaks N=(1,1) to N=(1,0)…

We show that $U(k)$-invariant hypercomplex structures on (open subsets) of regular semisimple adjoint orbits in $\mathfrak{gl}(k,{\mathbb C})$ correspond to algebraic curves $C$ of genus $(k-1)^2$, equipped with a flat projection…

The paper contains essentially two new results. Physically, a deformation of the parastatistics in a sense of quantum groups is carried out. Mathematically, an alternative to the Chevalley description of the quantum orthosymplectic…

Each square complex matrix is unitarily similar to an upper triangular matrix with diagonal entries in any prescribed order. Let A and B be upper triangular n-by-n matrices that (i) are not similar to direct sums of matrices of smaller…

We consider the target space theory of bosonic and heterotic string theory to first order in $\alpha'$ compactified to three dimensions, using a formulation that is manifestly T-duality invariant under ${\rm O}(d,d,\mathbb{R})$ with $d=23$…

We study invariant and bi-invariant metrics on groups focusing on finite groups $G$. We show that non-equivalent (bi) invariant metrics on $G$ are in 1-1 correspondence with unitary symmetric (conjugate) partitions on $G$. To every metric…

We give a complete characterization of invariant subspaces for $(M_{z_1}, \ldots, M_{z_n})$ on the Hardy space $H^2(\mathbb{D}^n)$ over the unit polydisc $\mathbb{D}^n$ in $\mathbb{C}^n$, $n >1$. In particular, this yields a complete set of…

For a compact Riemann surface $X$ of genus $g > 1$, $\Hom(\pi_1(X), U(p,1))/U(p,1)$ is the moduli space of flat $\U(p,1)$-connections on $X$. There is an integer invariant, $\tau$, the Toledo invariant associated with each element in…

The properties of Hopf star operations and twisted Hopf stars operations on quantum groups are discussed in relation with the theory of representations (star representations). Invariant Hermitian sesquilinear forms (scalar products) on…