Cohomology and deformation of Leibniz pairs

Cohomology and deformation theories are developed for Poisson algebras starting with the more general concept of a Leibniz pair, namely of an associative algebraA together with a Lie algebraL mapped into the derivations ofA. A bicomplex (with both Hochschild and Chevalley-Eilenberg cohomologies) is essential.     

Graded quasi-Lie algebras

This paper is concerned with a new class of graded algebras naturally uniting within a single framework various deformations of the Witt, Virasoro and other Lie algebras based on twisted and deformed derivations, as well as color Lie algebras and Lie superalgebras. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005. Supported by the Liegrits network Supported by the Crafoord foundation     

Quasigraded deformations of loop algebras and classical integrable systems

We construct a family of infinite-dimensional quasigraded Lie algebras, that could be viewed as deformation of the graded loop algebras. Using them we obtain new series of integrable Hamiltonian systems on semisimple Lie algebras and their extensions. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.     

Higher-Order Simple Lie Algebras

It is shown that the non-trivial cocycles on simple Lie algebras may be used to introduce antisymmetric multibrackets which lead to higher-order Lie algebras, the definition of which is given. Their generalised Jacobi identities turn out to be satisfied by the antisymmetric tensors (or higher-order “structure constants”) which characterise the Lie algebra cocycles. This analysis allows us to present a classification of the higher-order simple Lie algebras as well as a constructive procedure for them. Our results are synthesised by the introduction of a single, complete BRST operator associated with each simple algebra. Received: 3 June 1996 / Accepted: 8 November 1996     

Modular Classes of Regular Twisted Poisson Structures on Lie Algebroids

We derive a formula for the modular class of a Lie algebroid with a regular twisted Poisson structure in terms of a canonical Lie algebroid representation of the image of the Poisson map. We use this formula to compute the modular classes of Lie algebras with a twisted triangular r-matrix. The special case of r-matrices associated to Frobenius Lie algebras is also studied.      

Graded contractions of Lie algebras and central extensions

I explain how the concept ofgrading of Lie algebras can be used to investigate the appearance of central charges during a contraction. I illustrate the method with the kine-matical algebras of spacetime. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

Global Geometric Deformations of the Virasoro Algebra,Current and Affine Algebras by Krichever–Novikov Type Algebras

In two earlier articles we constructed algebraic-geometric families of genus one (i.e. elliptic) Lie algebras of Krichever–Novikov type. The considered algebras are vector fields, current and affine Lie algebras. These families deform the Witt algebra, the Virasoro algebra, the classical current, and the affine Kac–Moody Lie algebras respectively. The constructed families are not equivalent (not even locally) to the trivial families, despite the fact that the classical algebras are formally rigid. This effect is due to the fact that the algebras are infinite dimensional. In this article the results are reviewed and developed further. The constructions are induced by the geometric process of degenerating the elliptic curves to singular cubics. The algebras are of relevance in the global operator approach to the Wess–Zumino–Witten–Novikov models appearing in the quantization of Conformal Field Theory.     

Strongly Homotopy Lie Bialgebras and Lie Quasi-bialgebras

Structures of Lie algebras, Lie coalgebras, Lie bialgebras and Lie quasibialgebras are presented as solutions of Maurer–Cartan equations on corresponding governing differential graded Lie algebras using the big bracket construction of Kosmann–Schwarzbach. This approach provides a definition of an L _∞-(quasi)bialgebra (strongly homotopy Lie (quasi)bialgebra). We recover an L _∞-algebra structure as a particular case of our construction. The formal geometry interpretation leads to a definition of an L _∞ (quasi)bialgebra structure on V as a differential operator Q on V, self-commuting with respect to the big bracket. Finally, we establish an L _∞-version of a Manin (quasi) triple and get a correspondence theorem with L _∞-(quasi)bialgebras. This paper is dedicated to Jean-Louis Loday on the occasion of his 60th birthday with admiration and gratitude.     

Induced Lie Algebras of a Six-Dimensional Matrix Lie Algebra

By using a six-dimensional matrix Lie algebra [Y.F. Zhang and Y. Wang, Phys. Lett. A 360 (2006) 92], three induced Lie algebras are constructed. One of them is obtained by extending Lie bracket, the others are higher-dimensional complex Lie algebras constructed by using linear transformations. The equivalent Lie algebras of the later two with multi-component forms are obtained as well. As their applications, we derive an integrable coupling and quasi-Hamiltonian structure of the modified TC hierarchy of soliton equations.     

Loop Homotopy Algebras in Closed String Field Theory

Barton Zwiebach constructed [20] “string products” on the Hilbert space of a combined conformal field theory of matter and ghosts, satisfying the “main identity”. It has been well known that the “tree level” of the theory gives an example of a strongly homotopy Lie algebra (though, as we will see later, this is not the whole truth). Strongly homotopy Lie algebras are now well-understood objects. On the one hand, strongly homotopy Lie algebra is given by a square zero coderivation on the cofree cocommutative connected coalgebra [13, 14]; on the other hand, strongly homotopy Lie algebras are algebras over the cobar dual of the operad &?om for commutative algebras [9]. As far as we know, no such characterization of the structure of string products for arbitrary genera has been available, though there are two series of papers directly pointing towards the requisite characterization. As far as the characterization in terms of (co)derivations is concerned, we need the concept of higher order (co)derivations, which has been developed, for example, in[2, 3]. These higher order derivations were used in the analysis of the ”master identity“. For our characterization we need to understand the behavior of these higher (co)derivations on (co)free (co)algebras. The necessary machinery for the operadic approach is that of modular operads, anticipated in [5] and introduced in [8]. We believe that the modular operad structure on the compactified moduli space of Riemann surfaces of arbitrary genera implies the existence of the structure we are interested in the same manner as was explained for the tree level in [11]. We also indicate how to adapt the loop homotopy structure to the case of open string field theory [19]. Received: 10 November 1999 / Accepted: 29 March 2001     

Noncommutative *-product continual Lie algebras and solvable models

Noncommutative counterparts of exactly solvable models are proposed on the basis of *-product continual Lie algebras. Examples of noncommutative Liouville and sine/sh-Gordon equations are discussed.     

Graded contractions of lie algebras and central extensions

We generalize the methods of graded contractions in order to determine, using grading arguments only, the existence of central charges within the limit Lie algebras. As an illustration we show how this formalism allows one to recover the u(n)-bosons limits of the classical Lie algebras. Presented by Marc de Montigny at the DI-CRM Workshop held in Prague, 18–21 June 2000.     

Existence and equivalence of twisted products on a symplectic manifold

The twisted products play an important role in Quantum Mechanics [1, 2]. We introduce here a distinction between Vey *_ν-products and strong Vey *_ν-products and prove that each *_ν-product is equivalent to a Vey *_ν-product. If b ₃(W)=0, the symplectic manifold (W, F) admits strong Vey *_ν-products. If b ₂(W)=0, all *_ν-products are equivalent as well as the Vey Lie algebras. In the general case, we characterize the formal Lie algebras which are generated by a *_ν-product and we prove that the existence of a *_ν-product is equivalent to the existence of a formal Lie algebra infinitesimally equivalent to a Vey Lie algebra at the first order.     

and as Vertex Operator Extensions¶of Dual Affine Algebras

We discover a realisation of the affine Lie superalgebra and of the exceptional affine superalgebra as vertex operator extensions of two algebras with “dual” levels (and an auxiliary level-1 algebra). The duality relation between the levels is . We construct the representation of on a sum of tensor products of , , and modules and decompose it into a direct sum over the spectral flow orbit. This decomposition gives rise to character identities, which we also derive. The extension of the construction to is traced to the properties of embeddings into and their relation with the dual pairs. Conversely, we show how the representations are constructed from representations. Received: 29 July 1999 / Accepted: 6 February 2000     

Integro-differential non-linear equations and continual Lie algebras

The integrability problem of integro-differential equations with, generally speaking, singular kernels is discussed after an example of new continual analogs of the two-dimensional Toda lattices. These equations are associated with new infinite-dimensional Lie algebras via zero curvature type representation. The structural constants of these algebras are distributions. A formal solution of the Goursát problem is obtained. For the case with the kernel of the integral operator being _±-distribution an explicit expression in quadratures for the solutions is given.     

Constructing lie algebras of first-order differential operators

Formulas for calculating vector fields — generators of groups of transformations to a uniform space — from specified structural constants are obtained. The problem of vector-field continuation — the construction of Lie algebras of inhomogeneous first-order differential operators — is considered. It is also shown that the existence of a nontrivial continuation is closely associated with the structure of the isotopic subalgebra and, in particular, that no nontrivial continuation exists for semisimple algebras. Omsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 25–32, June, 1997.     

Lie algebras of nonstandard relativity groups

It is shown that representations of Lie algebras of the possible nonstandard spacetime symmetry groups may be derived from the representations of the Poincaré group. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.     

Homotopy Approach to Quantum Gravity

Gravity may be a quantum-space-time effect. General relativity is quantized by small generic changes in its commutation relations that make its Lie algebras simple on all levels, positing extra variables frozen by self-organization as needed. This quantizes space-time coordinates as well as fields and eliminates physical singularities. Fermi statistics and sl (nℝ) Lie algebras are assumed for all levels. Spin 1/2 is taken to be anomalous, arising from vacuum organization; the spin-statistics relation is incorporated. The gravitational field is quartic in Fermi variables. Einstein’s non-commutativity of parallel transport emerges as a vestige of Heisenberg’s quantum non-commutativity near the classical limit.     

There is No “Theory of Everything” Inside E8

We analyze certain subgroups of real and complex forms of the Lie group E₈, and deduce that any “Theory of Everything” obtained by embedding the gauge groups of gravity and the Standard Model into a real or complex form of E₈ lacks certain representation-theoretic properties required by physical reality. The arguments themselves amount to representation theory of Lie algebras in the spirit of Dynkin’s classic papers and are written for mathematicians.