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Since the gauge group underlying 2+1-dimensional general relativity is non-compact, certain difficulties arise in the passage from the connection to the loop representations. It is shown that these problems can be handled by appropriately…
An analogue of Burnside's Lemma for 2-transitive groups is shown to hold for a class of topological groups. If the group is compact the representation is finite and splits into an irreducible and the constant functions. If both the group…
Extending our earlier study of nonlinear Bogolyubov-Valatin transformations (canonical transformations for fermions) for one fermionic mode, in the present paper we perform a thorough study of general (nonlinear) canonical transformations…
This is mainly a lecture note taken by myself following Weinberg's book, but also contains some corrections to the abuse of mathematical treatment. This article discusses projective unitary representations of Poincare group on the single…
The orbits space of an irreducible representation of a finite group is a variety whose coordinate ring is finitely generated by homogeneous invariant polynomials. Boris Dubrovin showed that the orbits spaces of the reflection groups acquire…
The formal algebraic structures that govern higher-spin theories within the unfolded approach turn out to be related to an extension of the Kontsevich Formality, namely, the Shoikhet-Tsygan Formality. Effectively, this allows one to…
We give a proof of a conjecture of Lehrer and Shoji regarding the occurrences of the exterior powers of the reflection representation in the cohomology of Springer fibers. The actual theorem proved is a slight extension of the original…
On the example of the quantized spinor field, interacting with arbitrary external electromagnetic field, the commutation function is studied. It is shown that a proper time representation is available in any dimensions. Using it, all the…
The covering of the affine symmetry group, a semidirect product of translations and special linear transformations, in $D \geq 3$ dimensional spacetime is considered. Infinite dimensional spinorial representations on states and fields are…
In this paper we show that the nontrivial fundamental group $\pi_1 SO(3) ={\Bbb Z}_2$ for the group SO(3) of global proper rotations of a four-dimensional Euclidian space (when a spin structure is introduced preliminarily in that space)…
Application of the so-called refined algebraic quantization scheme for constrained systems to the relativistic particle provides an inner product that defines a unique Fock representation for a scalar field in curved space-time. The…
Let $\psi$ and $F$ be positive definite forms with integral coefficients of equal degree. Using the circle method, we establish an asymptotic formula for the number of identical representations of $\psi$ by $F$, provided $\psi$ is…
There are two well-known ways of describing elements of the rotation group SO$(m)$. First, according to the Cartan-Dieudonn\'e theorem, every rotation matrix can be written as an even number of reflections. And second, they can also be…
The exactly and quasi-exactly solvable problems for spin one-half in one dimension on the basis of a hidden dynamical symmetry algebra of Hamiltonian are discussed. We take the supergroup, $OSP(2|1)$, as such a symmetry. A number of exactly…
Given a natural number $n \geq 1$, the odometer semigroup $O_n$, also known as the adding machine or the Baumslag-Solitar monoid with two generators, is a well-known object in group theory. This paper examines the odometer semigroup in…
We study the representation of a finite group acting on the cohomology of a non-degenerate, invariant hypersurface of a projective toric variety. We deduce an explicit description of the representation when the toric variety has at worst…
We consider a bounded domain $\Omega \subseteq \mathbb C^d$ which is a $G$-space for a finite complex reflection group $G$. For each one-dimensional representation of the group $G,$ the relative invariant subspace of the weighted Bergman…
This is the second of a pair of papers on extended geometrically finite (EGF) representations, which were originally posted as a single article under the title "An extended definition of Anosov representation for relatively hyperbolic…
We develop a theory of Anosov representation of geometrically finite Fuchsian groups in SL(d,R) and show that cusped Hitchin representations are Borel Anosov in this sense. We establish analogues of many properties of traditional Anosov…
Some new representations of the supersymmetric transformations are derived, and the supermultiplets are introduced. Based on these representations, various formulations (equations, commutation relations, propagators, Jacobi identities,…