The Koszul formulae for graded Lie-Cartan pairs (super BRS operator)

This note is a sequel to the paper [1] codifying the /2-graded (i.e. fermionic) differential calculus within the frame of «graded Lie-Cartan pairså. We here present in this frame the Koszul formulae for the generalized exterior derivative, resp. the generalized covariant exterior derivative. The latter yields in particular the cohomological aspect of the «super BRS operatorå.     

Three carbon pairs in Si

Carbon impurities in Si are common in floating-zone and cast-Si materials. The simplest and most discussed carbon complex is the interstitial–substitutional C_iC_s pair, which readily forms when self-interstitials are present in the material. This pair has three possible configurations, each of which is electrically active. The less common C_sC_s pair has been studied in irradiated material but has also recently been seen in as-grown C-rich cast-Si, which is commonly used to fabricate solar cells. The third pair consists of two interstitial C atoms: C_iC_i. Although its formation probability is low for several reasons, the C_iC_i pair is very stable and electrically inactive. In this contribution, we report preliminary results of first-principles calculations of these three C pairs in Si. The structures, binding energies, vibrational spectra, and electrical activity are predicted.     

The Weil-B.R.S. algebra of a Lie algebra and the anomalous terms in gauge theory

In this paper, we explain the computation we made in collaboration with M. Talon and C.M. Viallet of anomalous terms in gauge theory [1], [2], [3]. We relate our constructions to standard mathematical constructions. The paper is self-contained in the sense that all mathematical concepts and results we use are explained.     

The action of the group of bundle-automorphisms on the space of connections and the geometry of gauge theories

Some aspects of the geometry of gauge theories are sketched in this review. We deal essentially with Yang-Mills theory, discussing the structure of the space of gauge orbits and the geometrical interpretation of ghosts and anomalies. Occasionally we deal also with classical gauge theories of gravitation and in particular we study the action of the group of diffeomorphisms on the space of linear connections. Finally we comment on the mathematical interpretation of anomalies in field theories.     

Anomalies, gauge field topology, and the lattice

Motivated by the connection between gauge field topology and the axial anomaly in fermion currents, I suggest that the fourth power of the naive Dirac operator can provide a natural method to define a local lattice measure of topological charge. For smooth gauge fields this reduces to the usual topological density. For typical gauge field configurations in a numerical simulation, however, quantum fluctuations dominate, and the sum of this density over the system does not generally give an integer winding. On cooling with respect to the Wilson gauge action, instanton like structures do emerge. As cooling proceeds, these objects tend shrink and finally “fall through the lattice.” Modifying the action can block the shrinking at the expense of a loss of reflection positivity. The cooling procedure is highly sensitive to the details of the initial steps, suggesting that quantum fluctuations induce a small but fundamental ambiguity in the definition of topological susceptibility.     

Composition of Lie Group Elements from Basis Lie Algebra Elements

It is shown explicitly how one can obtain elements of Lie groups as compositions of products of other elements based on the commutator properties of associated Lie algebras. Problems of this kind can arise naturally in control theory. Suppose an apparatus has mechanisms for moving in a limited number of ways with other movements generated by compositions of allowed motions. Two concrete examples are: (1) the restricted parallel parking problem where the commutator of translations in y and rotations in the xy-plane yields translations in x. Here the control problem involves a vehicle that can only perform a series of translations in y and rotations with the aim of efficiently obtaining a pure translation in x; (2) involves an apparatus that can only perform rotations about two axes with the aim of performing rotations about a third axis. Both examples involve three-dimensional Lie algebras. In particular, the composition problem is solved for the nine three- and four-dimensional Lie algebras with non-trivial solutions. Three different solution methods are presented. Two of these methods depend on operator and matrix representations of a Lie algebra. The other method is a differential equation method that depends solely on the commutator properties of a Lie algebra. Remarkably, for these distinguished Lie algebras the solutions involve arbitrary functions and can be expressed in terms of elementary functions.     

A decomposition theorem for SU(n) and its application to CP-violation through quark mass diagonalisation

It is proved that the groupG=SU(n) has a decompositionG=FCF whereF is a maximal abelian subgroup andC is an (n − 1)² parameter subset of matrices. The result is applied to the problem of absorbing the maximum possible number of phases in the mass-diagonalising matrix of the charged weak current into the quark fields; i.e., of determining the exact number of CP-violating phases for arbitrary number of generations. The inadequacies of the usual way of solving this problem are discussed. Then=3 case is worked out in detail as an example of the constructive procedure furnished by the proof of the decomposition theorem.     

A Crossed Module Representing the Godbillon–Vey Cocycle

We exhibit crossed modules of Lie algebras (i.e. certain four-term exact sequences) corresponding to the Godbillon–Vey generator for the Lie algebras of formal vector fields in one variable, of vector fields on the circle, of differentiable vector fields of holomorphic type and of holomorphic vector fields on a punctured Riemann surface.     

Infinitesimal deformations of filiform Lie algebras of order 3

The Lie algebras of order$F$ have important applications for the fractional supersymmetry, and on the other hand the filiform Lie (super)algebras have very important properties into the Lie Theory. Thus, the aim of this work is to study filiform Lie algebras of order$F$ which were introduced in Navarro (2014). In this work we obtain new families of filiform Lie algebras of order 3, in which the complexity of the problem rises considerably respecting to the cases considered in Navarro (2014).     

Projective geometry from Poisson algebras

By analogy with the Poisson algebra of quadratic forms on the symplectic plane and with the concept of duality in the projective plane introduced by Arnold (2005) [1], where the concurrence of the triangle altitudes is deduced from the Jacobi identity, we consider the Poisson algebras of the first degree harmonics on the sphere, on the pseudo-sphere and on the hyperboloid, to obtain analogous duality concepts and similar results for spherical, pseudo-spherical and hyperbolic geometry. Such algebras, including the algebra of quadratic forms, are isomorphic either to the Lie algebra of the vectors in R³${\mathbb{R}}^3$, with the vector product, or to algebra s_l2(R)$s{l}_2\left(\mathbb{R}\right)$. The Tomihisa identity, introduced in (Tomihisa, 2009) [3] for the algebra of quadratic forms, holds for all these Poisson algebras and has a geometrical interpretation. The relationships between the different definitions of duality in projective geometry inherited by these structures are shown here.     

Quasigraded Lie Algebras and Hierarchies of Integrable Equations

Using special quasigraded Lie algebras we obtain new hierarchies of integrable equations in partial derivatives admitting zero-curvature representations. In particular, we obtain new type of so(3) anisotropic chiral-field equation along with its higher rank generalization.     

Iwasawa nilpotency degree of non compact symmetric cosets in \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ \mathcal{N} $\end{document}‐extended supergravity

We analyze the polynomial part of the Iwasawa realization of the coset representative of non compact symmetric Riemannian spaces. We start by studying the role of Kostant's principal SU(2)_P subalgebra of simple Lie algebras, and how it determines the structure of the nilpotent subalgebras. This allows us to compute the maximal degree of the polynomials for all faithful representations of Lie algebras. In particular the metric coefficients are related to the scalar kinetic terms while the representation of electric and magnetic charges is related to the coupling of scalars to vector field strengths as they appear in the Lagrangian. We consider symmetric scalar manifolds in ��‐extended supergravity in various space‐time dimensions, elucidating various relations with the underlying Jordan algebras and normed Hurwitz algebras. For magic supergravity theories, our results are consistent with the Tits‐Satake projection of symmetric spaces and the nilpotency degree turns out to depend only on the space‐time dimension of the theory. These results should be helpful within a deeper investigation of the corresponding supergravity theory, e.g. in studying ultraviolet properties of maximal supergravity in various dimensions.     

Homogeneous structures on three-dimensional Lorentzian manifolds

We prove that any non-symmetric three-dimensional homogeneous Lorentzian manifold is isometric to a Lie group equipped with a left-invariant Lorentzian metric. We then classify all three-dimensional homogeneous Lorentzian manifolds.     

Fine Gradings on Non-simple Lie Algebras: Example of o(4,C)

On any Lie algebra L, it is of significant convenience to have at one's disposal all the possible fine gradings of L, since they reflect the basic structural properties of the Lie algebra. They also provide useful bases of the representations of the algebra -- namely such bases that are preserved by the commutator.We list all the six fine gradings on the non-simple Lie algebra o(4,C) and we explain their relation to the fine gradings of the Lie algebra sl(2,C) where relevant. The existence of such relation is not surprising, since o(4,C) is in fact a product of two specimen of sl(2,C). The example of o(4,C) is especially important due to the fact that one of its fine gradings is not generated by any MAD-group. This proves that, unlike in the case of classical simple Lie algebras over C, on the non-simple classical Lie algebras over C there can exist a fine grading that is not generated by any MAD-group on the Lie algebra.     

Noether Symmetries Versus Killing Vectors and Isometries of Spacetimes

Symmetries of spacetime manifolds which are given by Killing vectors are compared with the symmetries of the Lagrangians of the respective spacetimes. We find the point generators of the one parameter Lie groups of transformations that leave invariant the action integral corresponding to the Lagrangian (Noether symmetries). In the examples considered, it is shown that the Noether symmetries obtained by considering the Larangians provide additional symmetries which are not provided by the Killing vectors. It is conjectured that these symmetries would always provide a larger Lie algebra of which the KV symmetres will form a subalgebra. PACS: 04.25.-g, 02.20.Sv, 11.30.-j     

Classification of Lie algebras with naturally graded quasi-filiform nilradicals

The whole class of complex Lie algebras g$\mathfrak{g}$ having a naturally graded nilradical with characteristic sequence c(g)=(dimg−2,1,1)$c\left(\mathfrak{g}\right)=(dim\mathfrak{g}-2,1,1)$ is classified. It is shown that up to one exception, such Lie algebras are solvable.     

Deformation Quantization of Poisson Manifolds

I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven (the Formality conjecture), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not explicitly use the language of functional integrals.