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We present a classification and an explicit form of all constant solutions of the Yang–Mills equations with SU (2) gauge symmetry for an arbitrary constant non-Abelian current in pseudo-Euclidean space ℝ p , q of arbitrary finite dimension n = p + q. Using hyperbolic singular value decomposition and two-sheeted covering of orthogonal group by spin group, we solve the nontrivial system for constant solutions of the Yang–Mills equations of 3 n cubic equations with 3 n unknowns and 3 n parameters in the general case. We present a new symmetry of this system of equations. All solutions in terms of the potential, strength and invariant of the Yang–Mills field are presented. Nonconstant solutions of the Yang–Mills equations can be considered in the form of series of perturbation theory using all constant solutions as a zeroth approximation. [ABSTRACT FROM AUTHOR]

We consider systems where dynamical variables are the generators of the SU(2) group. A subset of these Hamiltonians is exactly solvable using the Bethe ansatz techniques. We show that Bethe ansatz equations are equivalent to polynomial relationships between the operator invariants, or equivalently, between eigenvalues of those invariants. [ABSTRACT FROM AUTHOR]

The knot coloring polynomial defined by Eisermann for a finite pointed group is generalized to an infinite pointed group as the longitudinal mapping invariant of a knot. In turn, this can be thought of as a generalization of the quandle 2-cocycle invariant for finite quandles. If the group is a topological group, then this invariant can be thought of as a topological generalization of the 2-cocycle invariant. The longitudinal mapping invariant is based on a meridian–longitude pair in the knot group. We also give an interpretation of the invariant in terms of quandle colorings of a 1-tangle for generalized Alexander quandles without use of a meridian–longitude pair in the knot group. The invariant values are concretely evaluated for the torus knots T (2 , n) , their mirror images, and the figure eight knot for the group SU(2). [ABSTRACT FROM AUTHOR]

We exhibit the traceless SU(2) character variety of a 6-punctured 2-sphere as a 2-fold branched cover of , branched over the singular Kummer surface, with the branch locus in corresponding to the binary dihedral representations. This follows from an analysis of the map induced on SU(2) character varieties by the 2-fold branched cover branched over points, combined with the theorem of Narasimhan-Ramanan which identifies with . The singular points of correspond to abelian representations, and we prove that each has a neighborhood in homeomorphic to a cone on . [ABSTRACT FROM AUTHOR]

Explicit, analytic and closed expressions for boson realizations of the (m+3)-parameter nonlinearly deformed angular momentum algebra {R}{r, s}(m) with its highest power m of polynomial function being arbitrary, which combines and generalizes Witten's two deformation schemes, are investigated in terms of the single boson and the single inversion boson, respectively. For each kind, the unitary Holstein-Primakoff-like realization, the non-unitary Dyson-Maléev-like realization and their connections are respectively discussed. Using these realizations, the corresponding representations of {R}{r, s}(m) as well as their respective acting spaces in the Fock space are obtained. [ABSTRACT FROM AUTHOR]

Starting with the zero-square "zeon algebra," the regular representation gives rise to a Boolean lattice representation of sl(2). We detail the su(2) content of the Boolean lattice, providing the irreducible representations carried by the algebra generated by the subsets of an n-set. The group elements are found, exhibiting the "special functions" in this context. The corresponding Leibniz rule and group law are shown. Krawtchouk polynomials, the Hamming and the Johnson schemes appear naturally. Applications to the Boolean poset and the structure of Hadamard-Sylvester matrices are shown as well. [ABSTRACT FROM AUTHOR]