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R. Campoamor-Stursberg

Publications and source records attributed to R. Campoamor-Stursberg.

At least 19 recordsLinked to original sources
arXiv2024

Lie-Hamilton systems associated with the symplectic Lie algebra $\mathfrak{sp}(6, \mathbb{R})$

New classes of Lie-Hamilton systems are obtained from the six-dimensional fundamental representation of the symplectic Lie algebra $\mathfrak{sp}(6,\mathbb{R})$. The ansatz is based on a recently proposed procedure for constructing higher-dimensional Lie-Hamilton systems through the representation theory of Lie algebras. As applications of the procedure, we study a time-dependent electromagnetic field and several types of coupled oscillators. The irreducible embedding of the special unitary Lie algebra $\mathfrak{su}(3)$ into $\mathfrak{sp}(6, \mathbb{R})$ is also considered, yielding Lie-Hamilton systems arising from the sum of the quark and antiquark three-dimensional representations of $\mathfrak{su}(3)$, which are applied in the construction of t-dependent coupled systems. In addition, t-independent constants of the motion are obtained explicitly for all these Lie-Hamilton systems, which allows the derivation of a nonlinear superposition rule

math-ph↗
arXiv2013

Higher-order superintegrability of a Holt related potential

In a recent paper, Post and Winternitz studied the properties of two-dimensional Euclidean potentials that are linear in one of the two Cartesian variables. In particular, they proved the existence of a potential endowed with an integral of third-order and an integral of fourth-order. In this paper we show that these results can be obtained in a more simple and direct way by noting that this potential is directly related with the Holt potential. It is proved that the existence of a potential with higher order superintegrability is a direct consequence of the integrability of the family of Holt type potentials.

math-ph↗
arXiv2011

Projective Representations of the Inhomogeneous Hamilton Group: Noninertial Symmetry in Quantum Mechanics

Symmetries in quantum mechanics are realized by the projective representations of the Lie group as physical states are defined only up to a phase. A cornerstone theorem shows that these representations are equivalent to the unitary representations of the central extension of the group. The formulation of the inertial states of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group, and its nonrelativistic limit in terms of the Galilei group, are fundamental examples. Interestingly, neither of these symmetries includes the Weyl-Heisenberg group; the hermitian representations of its algebra are the Heisenberg commutation relations that are a foundation of quantum mechanics. The Weyl-Heisenberg group is a one dimensional central extension of the abelian group and its unitary representations are therefore a particular projective representation of the abelian group of translations on phase space. A theorem involving the automorphism group shows that the maximal symmetry that leaves invariant the Heisenberg commutation relations are essentially projective representations of the inhomogeneous symplectic group. In the nonrelativistic domain, we must also have invariance of Newtonian time. This reduces the symmetry group to the inhomogeneous Hamilton group that is a local noninertial symmetry of Hamilton's equations. The projective representations of these groups are calculated using the Mackey theorems for the general case of a nonabelian normal subgroup.

quant-ph↗
arXiv2009

Geometrical foundations of plasticity yield criteria: A unified theory

A new model for elucidating the mathematical foundation of plasticity yield criteria is proposed. The proposed ansatz uses differential geometry and group theory concepts in addition to elementary hypotheses based on well-established experimental evidence. Its theoretical development involves the analysis of tensor functions and provides a series expansion which allows the functional stress-dependence of plasticity yield criteria to be predicted. The theoretical framework for the model includes a series of spatial coefficients that provide a more flexible theory for in-depth examination of symmetry and anisotropy in compact solid materials. It describes the classical yield criteria (like those of Tresca, Von Mises, Hosford, Hill, etc) and accurately describes the anomalous behaviour of metals such as aluminium, which was elucidated by Hill (1979). Further, absolutely new instances of stress-dependence are predicted; this makes it highly useful for fitting experimental data with a view to studying the phenomena behind plasticity.

cond-mat.mtrl-sci↗
arXiv2009

Composition algebras and the two faces of $G_{2}$

We consider composition and division algebras over the real numbers: We note two rôles for the group $G_{2}$: as automorphism group of the octonions and as the isotropy group of a generic 3-form in 7 dimensions. We show why they are equivalent, by means of a regular metric. We express in some diagrams the relation between some pertinent groups, most of them related to the octonions. Some applications to physics are also discussed.

math-ph↗
arXiv2009

Parafermions for higher order extensions of the Poincaré algebra and their associated superspace

Parafermions of order two and three are shown to be the fundamental tool to construct superspaces related to cubic and quartic extensions of the Poincaré algebra. The corresponding superfields are constructed, and some of their main properties analyzed in detail. In this context, the existence problem of operators acting like covariant derivatives is analyzed, and the associated operators are explicitly constructed.

hep-th↗
arXiv2009

Commutativity of missing label operators in terms of Berezin brackets

We obtain a criterion on the commutativity of polynomials in the enveloping algebra of a Lie algebra in terms of an involution condition with respect to the Berezin bracket. As an application, it is shown that the commutativity requirement of missing label operators for reduction chains in the missing label problem can be solved analytically.

math-ph↗
arXiv2008

Color Lie algebras and Lie algebras of order F

The notion of color algebras is generalized to the class of F-ary algebras, and corresponding decoloration theorems are established. This is used to give a construction of colored structures by means of tensor products with Clifford-like algebras. It is moreover shown that color algebras admit realisations as q=0 quon algebras.

math-ph↗
arXiv2008

Obtainment of internal labelling operators as broken Casimir operators by means of contractions related to reduction chains in semisimple Lie algebras

We show that the Inönü-Wigner contraction naturally associated to a reduction chain $\frak{s}\supset \frak{s}^{\prime}$ of semisimple Lie algebras induces a decomposition of the Casimir operators into homogeneous polynomials, the terms of which can be used to obtain additional mutually commuting missing label operators for this reduction. The adjunction of these scalars that are no more invariants of the contraction allow to solve the missing label problem for those reductions where the contraction provides an insufficient number of labelling operators.

math-ph↗
arXiv2008

Casimir operators induced by Maurer-Cartan equations

It is shown that for inhomogeneous Lie algebras $\frak{g}=\frak{s}\overrightarrow{\oplus}_Λ(\dim Λ)L_{1}$ satisfying the condition $\mathcal{N}(\frak{g})=1$, the only Casimir operator can be explicitly constructed from the Maurer-Cartan equations by means of wedge products. It is shown that this constraint imposes sharp bounds for the dimension of the representation $R$. The procedure is generalized to compute also the rational invariant of some Lie algebras.

hep-th↗
arXiv2008

Virtual copies of semisimple Lie algebras in enveloping algebras of semidirect products and Casimir operators

Given a semidirect product $\frak{g}=\frak{s}\uplus\frak{r}$ of semisimple Lie algebras $\frak{s}$ and solvable algebras $\frak{r}$, we construct polynomial operators in the enveloping algebra $\mathcal{U}(\frak{g})$ of $\frak{g}$ that commute with $\frak{r}$ and transform like the generators of $\frak{s}$, up to a functional factor that turns out to be a Casimir operator of $\frak{r}$. Such operators are said to generate a virtual copy of $\frak{s}$ in $\mathcal{U}(\frak{g})$, and allow to compute the Casimir operators of $\frak{g}$ in closed form, using the classical formulae for the invariants of $\frak{s}$. The behavior of virtual copies with respect to contractions of Lie algebras is analyzed. Applications to the class of Hamilton algebras and their inhomogeneous extensions are given.

math-ph↗
arXiv2008

Embeddings of Lie algebras, contractions and the state labelling problem

Many relevant applications of group theoretical methods to physical problems are related, in some manner, to classification schemes by means of symmetry groups. In these schemes, irreducible representations of a Lie group have to be decomposed according to some subgroup, thus providing a labeling of states in terms of the subgroup. This is the typical situation in nuclear physics, where Hamiltonians are described as functions of the Casimir operators of the groups involved in some reduction chain. This further gives an effective method to deduce the corresponding energy formulae. It is not however infrequent that the embedding of a Lie group as subgroup of a larger symmetry group is not canonical, hence leading to different branching rules of representations and to an insufficient number of labeling operators. The objective of this work is to review the actual developments of the contraction method, including also reduction chains with respect to abelian (Cartan) subalgebras. The decomposition of the invariants of the coadjoint representation provide natural classes of labeling operators for the corresponding embeddings, which are of interest for various specific problems ranging from nuclear physics to the group theoretical foundations of the periodic table of chemical elements.

hep-th↗
arXiv2008

Kinematical superalgebras and Lie algebras of order 3

We study and classify kinematical algebras which appear in the framework of Lie superalgebras or Lie algebras of order three. All these algebras are related through generalised Inonü-Wigner contractions from either the orthosymplectic superalgebra or the de Sitter Lie algebra of order three.

hep-th↗
arXiv2007

Internal labelling operators and contractions of Lie algebras

We analyze under which conditions the missing label problem associated to a reduction chain $\frak{s}^{\prime}\subset \frak{s}$ of (simple) Lie algebras can be completely solved by means of an Inönü-Wigner contraction $\frak{g}$ naturally related to the embedding. This provides a new interpretation of the missing label operators in terms of the Casimir operators of the contracted algebra, and shows that the available labeling operators are not completely equivalent. Further, the procedure is used to obtain upper bounds for the number of invariants of affine Lie algebras arising as contractions of semisimple algebras.

hep-th↗