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Finite dimensional irreducible modules of the two-parameter quantum enveloping algebra $U_{r,s}(\mathfrak{sl}_n)$ are explicitly constructed using the fusion procedure when $rs^{-1}$ is generic. This provides an alternative and…
We introduce a $Sim(2)$ invariant dimensional regularization of loop integrals. Then we compute the one loop quantum corrections to the photon self energy, electron self energy and vertex in the Electrodynamics sector of the Very Special…
It was established long ago that SO(2) electric-magnetic duality is an {\em off-shell} symmetry of the free Maxwell theory, i.e., that it leaves invariant the action and not just the equations of motion. We review here that analysis and…
Semilinear maps are a generalization of linear maps between vector spaces where we allow the scalar action to be twisted by a ring homomorphism such as complex conjugation. In particular, this generalization unifies the concepts of linear…
An infinite family of ND spaces endowed with sl(2)-coalgebra symmetry is introduced. For all these spaces the geodesic flow is superintegrable, and the explicit form of their common set of integrals is obtained from the underlying…
We study factorizations of rational matrix functions with simple poles on the Riemann sphere. For the quadratic case (two poles) we show, using multiplicative representations of such matrix functions, that a good coordinate system on this…
We study integral transforms mapping a function on the Euclidean space to the family of its integration on some hypersurfaces, that is, a function of hypersurfaces. The hypersurfaces are given by the graphs of functions with fixed axes of…
We obtain a decomposition formula of a representation of Sp(p,q) and SO^\ast(2n) unitarily induced from a derived functor module, which enables us to reduce the problem of irreducible decompositions to the study of derived functor modules.…
Given a finite connected graph $\Lambda$, the space of $SU(2)$ lattice gauge-fields on $\Lambda$, modulo gauge transformations, is a Lagrangian submanifold -- with mild singularities -- of the $SU(2)$ character variety (= phase-space of…
We consider the partition function of beta-gamma systems in curved space of the type discussed by Nekrasov and Witten. We show how the Koszul resolution theorem can be applied to the computation of the partition functions and to characters…
We recall the structure of the indecomposable sl(2) modules in the Bernstein-Gelfand-Gelfand category O. We show that all these modules can arise as quantized phase spaces of physical models. In particular, we demonstrate in a path integral…
We develop uniform approximations for the trace formula for non-integrable systems in which SU(2) symmetry is broken by a non-linear term of the Hamiltonian. As specific examples, we investigate H\'enon-Heiles type potentials. Our formalism…
The consequences of the attractive, short-range nucleon-nucleon (NN) interaction on the wave functions of the Elliott SU(3) and the proxy-SU(3) symmetry are discussed. The NN interaction favors the most symmetric spatial SU(3) irreducible…
In this paper we revisit the isomorphism $SU(2)\otimes SU(2)\cong SO(4)$ to apply to some subjects in Quantum Computation and Mathematical Physics. The unitary matrix $Q$ by Makhlin giving the isomorphism as an adjoint action is studied and…
Using connections to random matrix theory and orthogonal polynomials, we develop a framework for obtaining explicit closed-form formulae for the number, $\mathscr{N}_{g}(2\nu,j)$, of connected $2\nu$-valent labeled graphs with $j$ vertices…
In our previous papers we repeatedly emphasized the special role in Quaternionic Analysis of the conformal group SU(2,2) and other real forms of its complexification SL(4,C). In particular, the natural product map of the left and right…
We represent a version of multidimensional quasilinear partial differential equation (PDE) together with large manifold of particular solutions given in an integral form. The dimensionality of constructed PDE can be arbitrary. We call it…
We recall the classification of the irreducible representations of $SL(2)_q$, and then give fusion rules for these representations. We also consider the problem of $\cR$-matrices, intertwiners of the differently ordered tensor products of…
The partition function of the SO(4)- or Spin(4)-symmetric Euclidean Barrett-Crane model can be understood as a sum over all quantized geometries of a given triangulation of a four-manifold. In the original formulation, the variables of the…
We continue to study the matrix model of the $N_f =2$ $SU(2)$ case that represents the irregular conformal block. What provides us with the Painlev\'{e} system is not the instanton partition function per se but rather a finite analog of its…