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R. Campoamor-Stursberg

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

31 records · Page 2Linked to original sources
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↗
arXiv2007

Non-solvable contractions of semisimple Lie algebras in low dimension

The problem of non-solvable contractions of Lie algebras is analyzed. By means of a stability theorem, the problem is shown to be deeply related to the embeddings among semisimple Lie algebras and the resulting branching rules for representations. With this procedure, we determine all deformations of indecomposable Lie algebras having a nontrivial Levi decomposition onto semisimple Lie algebras of dimension $n\leq 8$, and obtain the non-solvable contractions of the latter class of algebras.

hep-th↗
arXiv2007

Contractions and deformations of quasi-classical Lie algebras preserving a non-degenerate quadratic Casimir operator

By means of contractions of Lie algebras, we obtain new classes of indecomposable quasi-classical Lie algebras that satisfy the Yang-Baxter equations in its reformulation in terms of triple products. These algebras are shown to arise naturally from non-compact real simple algebras with non-simple complexification, where we impose that a non-degenerate quadratic Casimir operator is preserved by the limiting process. We further consider the converse problem, and obtain sufficient conditions on integrable cocycles of quasi-classical Lie algebras in order to preserve non-degenerate quadratic Casimir operators by the associated linear deformations.

hep-th↗
arXiv2006

Quasi-classical Lie algebras and their contractions

After classifying indecomposable quasi-classical Lie algebras in low dimension, and showing the existence of non-reductive stable quasi-classical Lie algebras, we focus on the problem of obtaining sufficient conditions for a quasi-classical Lie algebras to be the contraction of another quasi-classical algebra. It is illustrated how this allows to recover the Yang-Mills equations of a contraction by a limiting process, and how the contractions of an algebra may generate a parameterized families of Lagrangians for pairwise non-isomorphic Lie algebras.

hep-th↗
arXiv2006

Symplectic forms on six dimensional real solvable Lie algebras I

We analyze symplectic forms on six dimensional real solvable and non-nilpotent Lie algebras. More precisely, we obtain all those algebras endowed with a symplectic form that decompose as the direct sum of two ideals or are indecomposable solvable algebras with a four dimensional nilradical.

math.DG↗
arXiv2006

Determinantal formulae for the Casimir operators of inhomogeneous Lie algebras

Contractions of Lie algebras are combined with the classical matrix method of Gel'fand to obtain matrix formulae for the Casimir operators of inhomogeneous Lie algebras. The method is presented for the inhomogeneous pseudo-unitary Lie algebras $I\frak{u}(p,q)$. This procedure is extended to contractions of $I\frak{u}(p,q)$ isomorphic to an extension by a derivation of the inhomogeneous special pseudo-unitary Lie algebras $I\frak{su}(p-1,q)$, providing an additional analytical method to obtain their invariants. Further, matrix formulae for the invariants of other inhomogeneous Lie algebras are presented.

hep-th↗
arXiv2005

Solvable real rigid Lie algebras are not necessarily completely solvable [Les algèbres de Lie résolubles rigides réelles ne sont pas nécessairement complètement résolubles]

We show that a solvable real rigid Lie algebra is not completelt rigid, by constructing an example of minimal dimension where the external torus is not spanned by $ad$-semisimple derivations over $\mathbb{R}$. We analyze the real forms of nilradicals of solvable rigid Lie algebras in dimensions $n\leq 7$ and give the real classification for dimension 8.

math.RT↗
arXiv2005

Invariant Tensors Formulae via Chord Diagrams

We provide an explicit algorithm to calculate invariant tensors for the adjoint representation of the simple Lie algebra $sl(n)$, as well as arbitrary representation in terms of roots. We also obtain explicit formulae for the adjoint representations of the orthogonal and symplectic Lie algebras $so(n)$ and $sp(n)$.

math.GT↗