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Andrianopoli, L., Merino, N., Nadal, F., & Trigiante, M. (2013). General properties of the expansion methods of Lie algebras. J. Phys. A, 46(36), 365204–33pp.
Abstract: The study of the relation between Lie algebras and groups, and especially the derivation of new algebras from them, is a problem of great interest in mathematics and physics, because finding a new Lie group from an already known one also means that a new physical theory can be obtained from a known one. One of the procedures that allow us to do so is called expansion of Lie algebras, and has been recently used in different physical applications-particularly in gauge theories of gravity. Here we report on further developments of this method, required to understand in a deeper way their consequences in physical theories. We have found theorems related to the preservation of some properties of the algebras under expansions that can be used as criteria and, more specifically, as necessary conditions to know if two arbitrary Lie algebras can be related by some expansion mechanism. Formal aspects, such as the Cartan decomposition of the expanded algebras, are also discussed. Finally, an instructive example that allows us to check explicitly all our theoretical results is also provided.
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Caroca, R., Kondrashuk, I., Merino, N., & Nadal, F. (2013). Bianchi spaces and their three-dimensional isometries as S-expansions of two-dimensional isometries. J. Phys. A, 46(22), 225201–24pp.
Abstract: In this paper we show that certain three-dimensional isometry algebras, specifically those of type I, II, III and V (according to Bianchi's classification), can be obtained as expansions of the isometries in two dimensions. In particular, we use the so-called S-expansionmethod, which makes use of the finite Abelian semigroups, because it is the most general procedure known until now. Also, it is explicitly shown why it is impossible to obtain the algebras of type IV, VI-IX as expansions from the isometry algebras in two dimensions. All the results are checked with computer programs. This procedure shows that the problem of how to relate, by an expansion, two Lie algebras of different dimensions can be entirely solved. In particular, the procedure can be generalized to higher dimensions, which could be useful for diverse physical applications, as we discuss in our conclusions.
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Cervantes, D., Fioresi, R., Lledo, M. A., & Nadal, F. A. (2016). Quantum Twistors. P-Adic Num., 8(1), 2–30.
Abstract: We compute explicitly a star product on the Minkowski space whose Poisson bracket is quadratic. This star product corresponds to a deformation of the conformal spacetime, whose big cell is the Minkowski spacetime. The description of Minkowski space is made in the twistor formalism and the quantization follows by substituting the classical conformal group by a quantum group.
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Cervantes, D., Fioresi, R., Lledo, M. A., & Nadal, F. A. (2012). Quadratic deformation of Minkowski space. Fortschritte Phys.-Prog. Phys., 60(9-10), 970–976.
Abstract: We present a deformation of the Minkowski space as embedded into the conformal space (in the formalism of twistors) based in the quantum versions of the corresponding kinematic groups. We compute explicitly the star product, whose Poisson bracket is quadratic. We show that the star product although defined on the polynomials can be extended differentiably. Finally we compute the Eucliden and Minkowskian real forms of the deformation.
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Fioresi, R., Latini, E., Lledo, M. A., & Nadal, F. A. (2018). The Segre embedding of the quantum conformal superspace. Adv. Theor. Math. Phys., 22(8), 1939–2000.
Abstract: In this paper we study the quantum deformation of the superflag Fl(2 vertical bar 0, 2 vertical bar 1, 4 vertical bar 1), and its big cell, describing the complex conformal and Minkowski superspaces respectively. In particular, we realize their projective embedding via a generalization to the super world of the Segre map and we use it to construct a quantum deformation of the super line bundle realizing this embedding. This strategy allows us to obtain a description of the quantum coordinate superring of the superflag that is then naturally equipped with a coaction of the quantum complex conformal supergroup SLq (4 vertical bar 1).
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