$$ \mathcal{N} $$ = 2 extended MacDowell-Mansouri supergravity

We construct a gauge theory based in the supergroup G = SU(2, 2|2) that generalizes MacDowell-Mansouri supergravity. This is done introducing an extended notion of Hodge operator in the form of an outer automorphism of su(2, 2|2)-valued 2-form tensors. The model closely resembles a Yang-Mills theory — including the action principle, equations of motion and gauge transformations — which avoids the use of the otherwise complicated component formalism. The theory enjoys H = SO(3, 1) × ℝ × U(1) × SU(2) off-shell symmetry whilst the broken symmetries G/H, translation-type symmetries and supersymmetry, can be recovered on surface of integrability conditions of the equations of motion, for which it suffices the Rarita-Schwinger equation and torsion-like constraints to hold. Using the matter ansatz —projecting the 1 ⊗ 1/2 reducible representation into the spin-1/2 irreducible sector — we obtain (chiral) fermion models with gauge and gravity interactions.

Rigidity at infinity for lattices in rank-one Lie groups

Let [Formula: see text] be a non-uniform lattice in [Formula: see text] without torsion and with [Formula: see text]. By following the approach developed in [S. Francaviglia and B. Klaff, Maximal volume representations are Fuchsian, Geom. Dedicata 117 (2006) 111–124], we introduce the notion of volume for a representation [Formula: see text] where [Formula: see text]. We use this notion to generalize the Mostow–Prasad rigidity theorem. More precisely, we show that given a sequence of representations [Formula: see text] such that [Formula: see text], then there must exist a sequence of elements [Formula: see text] such that the representations [Formula: see text] converge to a reducible representation [Formula: see text] which preserves a totally geodesic copy of [Formula: see text] and whose [Formula: see text]-component is conjugated to the standard lattice embedding [Formula: see text]. Additionally, we show that the same definitions and results can be adapted when [Formula: see text] is a non-uniform lattice in [Formula: see text] without torsion and for representations [Formula: see text], still maintaining the hypothesis [Formula: see text].

Irreducible representations of Leavitt path algebras

We construct some irreducible representations of the Leavitt path algebra of an arbitrary quiver. The constructed representations are associated to certain algebraic branching systems. For a row-finite quiver, we classify algebraic branching systems, to which irreducible representations of the Leavitt path algebra are associated. For a certain quiver, we obtain a faithful completely reducible representation of the Leavitt path algebra. The twisted representations of the constructed ones under the scaling action are studied.

METHOD OF CONSTRUCTING BRAID GROUP REPRESENTATION AND ENTANGLEMENT IN A 9 × 9 YANG–BAXTER SYSTEM

In this paper, we present reducible representation of the n2 braid group representation which is constructed on the tensor product of n-dimensional spaces. Specifically, it is shown that via a combining method, we can construct more n2 dimensional braiding S-matrices which satisfy the braid relations. By Yang–Baxterization approach, we derive a 9 × 9 unitary [Formula: see text]-matrix according to a 9 × 9 braiding S-matrix we have constructed. The entanglement properties of [Formula: see text]-matrix is investigated, and the arbitrary degree of entanglement for two-qutrit entangled states can be generated via [Formula: see text]-matrix acting on the standard basis.

GAUSSIAN FLUCTUATIONS OF REPRESENTATIONS OF WREATH PRODUCTS

We study the asymptotics of the reducible representations of the wreath products G≀Sq = Gq ⋊ Sq for large q, where G is a fixed finite group and Sq is the symmetric group in q elements; in particular for G = ℤ/2ℤ we recover the hyperoctahedral groups. We decompose such a reducible representation of G≀Sq as a sum of irreducible components (or, equivalently, as a collection of tuples of Young diagrams) and we ask what is the character of a randomly chosen component (or, what are the shapes of Young diagrams in a randomly chosen tuple). Our main result is that for a large class of representations, the fluctuations of characters (and fluctuations of the shape of the Young diagrams) are asymptotically Gaussian. The considered class consists of the representations for which the characters asymptotically almost factorize and it includes, among others, the left regular representation therefore we prove the analogue of Kerov's central limit theorem for wreath products.

CLASSICAL AND QUANTUM FERMIONS LINKED BY AN ALGEBRAIC DEFORMATION

We study the regular representation ρζ of the single-fermion algebra [Formula: see text], i.e., c2 = c+2 = 0, cc+ + c+c = ζ1, for ζ ∈ [0,1]. We show that ρ0 is a four-dimensional nonunitary representation of [Formula: see text] which is faithfully irreducible (it does not admit a proper faithful subrepresentation). Moreover, ρ0 is the minimal faithfully irreducible representation of [Formula: see text] in the sense that every faithful representation of [Formula: see text] has a subrepresentation that is equivalent to ρ0. We therefore identify a classical fermion with ρ0 and view its quantization as the deformation: ζ : 0 → 1 of ρζ. The latter has the effect of mapping ρ0 into the four-dimensional, unitary, (faithfully) reducible representation ρ1 of [Formula: see text] that is reminiscent of a Dirac fermion.

CANONICAL QUANTIZATION OF NONLOCAL FIELD EQUATIONS

We consistently quantize a class of relativistic nonlocal field equations characterized by a nonlocal kinetic term in the Lagrangian. We solve the classical nonlocal equations of motion for a scalar field and evaluate the on-shell Hamiltonian. The quantization is realized by imposing Heisenberg’s equation, which leads to the commutator algebra obeyed by the Fourier components of the field. We show that the field operator carries, in general, a reducible representation of the Poincaré group. We also consider the Gupta-Bleuler quantization of a nonlocal gauge theory and analyze the propagators and the physical modes of the gauge field.

Yang–Mills theory and nonirreducible representations of the Lorentz group

We start from an O(4) = SU(2) × SU(2) Yang–Mills theory and argue that the O(4) indices can, in fact, be space-time indices. The resulting theory is that of a tensor, Cμαβ = −Cμβα, which is in a reducible representation of the Lorentz group.fj This is a special case of the extended Yang–Mills formalism of Gabrielli. Some special solutions of the classical field equations are found.