Irreducible Lie algebra extensions of the Poincaré algebra

We use cohomology of Lie algebras to analyse the abelian extensions of the Poincaré algebraP. We study particularly the irreducible and truly irreducible extensions: some irreducibility criteria are proved and applied to obtain a classification of types of irreducible abelian extensions ofP. We give a characterization of the minimal essential extensions in terms of truly irreducible extensions.     

Irreducible Lie algebra extensions of the Poincaré algebra

We analyse the extensions of the Poincaré algebraP with arbitrary kernels. The main tool is a reduction theorem which generalizes the Hochschild-Serre theorem forn=2. This reduction theorem is proved and used to investigate the structure of the Lie algebras obtained by extension.We look particularly for the irreducible and -irreducible extensions ofP and we classify the types of irreducible extensions with arbitrary kernels.     

Relativity symmetries and Lie algebra contractions

We revisit the notion of possible relativity or kinematic symmetries mutually connected through Lie algebra contractions under a new perspective on what constitutes a relativity symmetry. Contractions of an SO(m,n)$SO(m,n)$ symmetry as an isometry on an m+n$m+n$ dimensional geometric arena which generalizes the notion of spacetime are discussed systematically. One of the key results is five different contractions of a Galilean-type symmetry G(m,n)$G(m,n)$ preserving a symmetry of the same type at dimension m+n−1$m+n-1$, e.g.   a G(m,n−1)$G(m,n-1)$, together with the coset space representations that correspond to the usual physical picture. Most of the results are explicitly illustrated through the example of symmetries obtained from the contraction of SO(2,4)$SO(2,4)$, which is the particular case for our interest on the physics side as the proposed relativity symmetry for “quantum spacetime”. The contractions from G(1,3)$G(1,3)$ may be relevant to real physics.     

Existence of a Lie bialgebra structure on every Lie algebra

We prove that every non-Abelian, finite-dimensional Lie algebra admits an exact bialgebra structure.     

The graded Lie algebra structure of Lie superalgebra deformation theory

We show how Lie superalgebra deformation theory can be treated by graded Lie algebras formalism. Rigidity and integrability theorems are obtained.     

Lie algebra cohomology and the fusion rules

We prove a vanishing theorem for Lie algebra cohomology which constitutes a loop group analogue of Kostant's Lie algebra version of the Borel-Weil-Bott theorem. Consider a complex semi-simple Lie algebra and an integrable, irreducible, negative energy representation of. Givenn distinct pointsz _k in , with a finite-dimensional irreducible representationV _k of assigned to each, the Lie algebra of-valued polynomials acts on eachV _k , via evaluation atz _k . Then, the relative Lie algebra cohomologyH ^* is concentrated in one degree. As an application, based on an idea of G. Segal's, we prove that a certain homolorphic induction map from representations ofG to representations ofLG at a given level takes the ordinary tensor product into the fusion product. This result had been conjectured by R. Bott.     

Continual Lie algebra bicomplexes and integrable models

We introduce bicomplex structures associated with Saveliev-Vershik continual Lie algebras, and derive non-linear dynamical systems resulting from the bicomplex conditions. Examples related to classes of continual Lie algebras, including contact Lie, Poisson bracket, and Hilbert-Cartan ones are discussed. Using the bicomplex linearization problem, we derive corresponding conservation laws. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

Quantum Observables,Lie Algebra Homology and TQFT

Let us consider a Lie (super)algebra G spanned by T where T are quantum observables in BV formalism. It is proved that for every tensor c^... that determines a homology class of the Lie algebra G the expression c^...T...T is again a quantum observable. This theorem is used to construct quantum observables in the BV sigma model. We apply this construction to explain Kontsevich's results about the relation between homology of the Lie algebra of Hamiltonian vector fields and topological invariants of manifolds.     

Lie algebra contractions on two-dimensional hyperboloid

The Inönü-Wigner contraction from the SO(2, 1) group to the Euclidean E(2) and E(1, 1) group is used to relate the separation of variables in Laplace-Beltrami (Helmholtz) equations for the four corresponding two-dimensional homogeneous spaces: two-dimensional hyperboloids and two-dimensional Euclidean and pseudo-Euclidean spaces. We show how the nine systems of coordinates on the two-dimensional hyperboloids contracted to the four systems of coordinates on E ₂ and eight on E _1,1. The text was submitted by the authors in English.     

On the quantum super Virasoro algebra

The quantum super-algebra structure on the deformed super Virasoro algebra is investigated. More specifically we established the possibility of defining a nontrivial Hopf super-algebra on both one and two-parameters deformed super Virasoro algebras.     

The integrability of a Lie algebra representation

The paper concerns the so-called integration problem for the representation of a Lie algebra by operators (not necessarily bounded) acting in a Banach space. Some general assumptions have been admitted about resolvents of these operators.     

Internally connected graphs and the Kashiwara-Vergne Lie algebra

It is conjectured that the Kashiwara-Vergne Lie algebra \(\widehat{\mathfrak {krv}}_2\) is isomorphic to the direct sum of the Grothendieck-Teichmüller Lie algebra \(\mathfrak {grt}_1\) and a one-dimensional Lie algebra. In this paper, we use the graph complex of internally connected graphs to define a nested sequence of Lie subalgebras of \(\widehat{\mathfrak {krv}}_2\) whose intersection is \(\mathfrak {grt}_1\), thus giving a way to interpolate between these two Lie algebras.     

Lie superalgebraic approach to super Toda lattice and generalized super KdV equations

We propose a super Lax type equation based on a certain class of Lie superalgebra as a supersymmetric extension of generalized (modified) KdV hierarchy. We are able to construct an infinite set of conservation laws and the consistent time evolution generators for generalized modified super KdV equations. Thefirst few of the conserved currents, the (modified) super KdV equation and the super Miura transformation are worked out explicitly in the case of twisted affine Lie superalgebraOSp(2/2)⁽²⁾.Partially supported by the Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture (#01540246 and #01790203).RIFP will be known as Yukawa Institute for Theoretical Physics from June 8, 1990     

Multi-black holes from nilpotent Lie algebra orbits

For N 3 2{\mathcal{N}\ge 2} supergravities, BPS black hole solutions preserving four supersymmetries can be superposed linearly, leading to well defined solutions containing an arbitrary number of such BPS black holes at arbitrary positions. Being stationary, these solutions can be understood via associated non-linear sigma models over pseudo-Riemannian spaces coupled to Euclidean gravity in three spatial dimensions. As the main result of this paper, we show that whenever this pseudo-Riemannian space is an irreducible symmetric space \mathfrakG/\mathfrakH^*{\mathfrak{G}/\mathfrak{H}^*}, the most general solutions of this type can be entirely characterised and derived from the nilpotent orbits of the associated Lie algebra \mathfrakg{\mathfrak{g}}. This technique also permits the explicit computation of non-supersymmetric extremal solutions which cannot be obtained by truncation to N=2{\mathcal{N}=2} supergravity theories. For maximal supergravity, we not only recover the known BPS solutions depending on 32 independent harmonic functions, but in addition find a set of non-BPS solutions depending on 29 harmonic functions. While the BPS solutions can be understood within the appropriate N=2{\mathcal{N}=2} truncation of N=8{\mathcal{N}=8} supergravity, the general non-BPS solutions require the whole field content of the theory.     

Poisson structures due to Lie algebra representations

Using a unitary solution of the classical Yang-Baxter equation on a Lie algebraG we describe a particular way of constructing homogeneous quadratic Poisson structures on the dual of aG-moduleV and study some local features of the symplectic foliation due to the involutive distribution of the Hamiltonian vector fields. We also give some examples where the symplectic leaves are explicitly calculated.