Two-step solvable Lie algebras and weight graphs

By using the concept of weight graph associated to nonsplit complex nilpotent Lie algebras \mathfrakg\mathfrak{g}, we find necessary and sufficient conditions for a semidirect product \mathfrakg?^? T_i\mathfrak{g}\overrightarrow{\oplus } T_{i} to be two-step solvable, where $T_{i}TT over \mathfrakg\mathfrak{g} which induces a decomposition of \mathfrakg\mathfrak{g} into one-dimensional weight spaces without zero weights. In particular we show that the semidirect product of such a Lie algebra with a maximal torus of derivations cannot be itself two-step solvable. We also obtain some applications to rigid Lie algebras, as a geometrical proof of the nonexistence of two-step nonsplit solvable rigid Lie algebras in dimensions n\geqslant 3n\geqslant 3.     

Representations of Loop Kac-Moody Lie Algebras

We study representations of the Loop Kac-Moody Lie algebra 𝔤 ?A, where 𝔤 is any Kac-Moody algebra and A is a ring of Laurent polynomials in n commuting variables. In particular, we study representations with finite dimensional weight spaces and their graded versions. When we specialize 𝔤 to be a finite dimensional or affine Lie algebra we obtain modules for toroidal Lie algebras.     

A Lie algebra attached to a projective variety

Abstract. Each choice of a K?hler class on a compact complex manifold defines an action of the Lie algebra sl(2) on its total complex cohomology. If a nonempty set of such K?hler classes is given, then we prove that the corresponding sl(2)-copies generate a semisimple Lie algebra. We investigate the formal properties of the resulting representation and we work things out explicitly in the case of complex tori, hyperk?hler manifolds and flag varieties. We pay special attention to the cases where this leads to a Jordan algebra structure or a graded Frobenius algebra. Oblatum 21-V-1996 & 15-X-1996     

Lie algebras admitting a unique quadratic structure

We prove that any Lie algebra g over a field K of characteristic zero admitting a unique up to a constant quadratic structure is necessarily a simple Lie algebra. If the field K is algebraically closed, such condition is also sufficient.

Further, a real Lie algebra g admits a unique quadratic structure if and only if its complexification g^C is a simple Lie algebra over C     

Lie ideals of graded associative algebras

We study the Lie structure of graded associative algebras. Essentially, we analyze the relation between Lie and associative graded ideals, and between Lie and associative graded derivations. Gathering together results on both directions, we compute maximal graded algebras of quotients of graded Lie algebras that arise from associative algebras. We also show that the Lie algebra Der _gr (A) of graded derivations of a graded semiprime associative algebra is strongly non-degenerate (modulo a certain ideal containing the center of Der _gr (A)).     

Toroidal Lie algebras and vertex representations

The paper describes the theory of the toroidal Lie algebra, i.e. the Lie algebra of polynomial maps of a complex torus ^××^× into a finite-dimensional simple Lie algebra g. We describe the universal central extension t of this algebra and give an abstract presentation for it in terms of generators and relations involving the extended Cartan matrix of g. Using this presentation and vertex operators we obtain a large class of integrable indecomposable representations of t in the case that g is of type A, D, or E. The submodule structure of these indecomposable modules is described in terms of the ideal structure of a suitable commutative associative algebra.To Professor J. Tits for his sixtieth birthday     

Test rank of an abelian product of a free Lie algebra and a free abelian Lie algebra

Let F be a free Lie algebra of rank n ≥ 2 and A be a free abelian Lie algebra of rank m ≥ 2. We prove that the test rank of the abelian product F ×A is m. Morever we compute the test rank of the algebra F/g_k( F) ^￠F/\gamma _{k}\left( F\right) ^{^{\prime }}.     

Classification of Lie bialgebras over current algebras

In this paper, we give a classification of Lie bialgebra structures on Lie algebras of type \mathfrak g{\mathfrak {g}} [[x]] and \mathfrak g[x]{\mathfrak g[x]}, where \mathfrak g{\mathfrak g} is a simple complex finite dimensional Lie algebra.     

Lie algebra of formal vector fields which are extended by formal g-valued functions

In this paper, we consider the infinite-dimensional Lie algebra W_n⋉g⊗O _n of formal vector fields on the n-dimensional plane which is extended by formal g-valued functions of n variables. Here g is an arbitrary Lie algebra. We show that the cochain complex of this Lie algebra is quasi-isomorphic to the quotient of the Weyl algebra of (gl _n ⊕ g) by the (2n+1)st term of the standard filtration. We consider separately the case of a reductive Lie algebra g. We show how one can use the methods of formal geometry to construct characteristic classes of bundles. For every G-bundle on an n-dimensional complex manifold, we construct a natural homomorphism from the ring A of relative cohomologies of the Lie algebra W_n⋉g⊗O _n to the ring of cohomologies of the manifold. We show that generators of the ring A are mapped under this homomorphism to characteristic classes of tangent and G-bundles. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 335, 2006, pp. 205–230.     

Weyl symbolic calculus on any Lie group

For any Lie groupG we extend the Weyl symbolic calculus to analytic symbol classes on the dual g^* of the Lie algebra g ofG.     

On Strongly Inert Subalgebras of an Infinite-Dimensional Lie Algebra

We study infinite-dimensional Lie algebras L over an arbitrary field that contain a subalgebra A such that dim(A + [A, L])/A < . We prove that if an algebra L is locally finite, then the subalgebra A is contained in a certain ideal I of the Lie algebra L such that dimI/A <. We show that the condition of local finiteness of L is essential in this statement.     

Determination of 7-Dimensional Indecomposable Nilpotent Complex Lie Algebras by Adjoining a Derivation to 6-Dimensional Lie Algebras

For any complex 6-dimensional nilpotent Lie algebra \mathfrakg,\mathfrak{g}, we compute the strain of all indecomposable 7-dimensional nilpotent Lie algebras which contain \mathfrakg\mathfrak{g} by the adjoining a derivation method. We get a new determination of all 7-dimensional complex nilpotent Lie algebras, allowing to check earlier results (some contain errors), along with a cross table intertwining nilpotent 6- and 7-dimensional Lie algebras.     

Application of a semi-direct sum of Lie algebras to the TD hierarchy

We construct a Lie algebra G by using a semi-direct sum of Lie algebra G₁ with Lie algebra G₂. A direct application to the TD hierarchy leads to a novel hierarchy of integrable couplings of the TD hierarchy. Furthermore, the generalized variational identity is applied to Lie algebra G to obtain quasi-Hamiltonian structures of the associated integrable couplings.     

A Selberg integral for the Lie algebra A<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

A new q-binomial theorem for Macdonald polynomials is employed to prove an A _n analogue of the celebrated Selberg integral. This confirms the \mathfrakg = A_n \mathfrak{g} ={\rm{A}}_{n} case of a conjecture by Mukhin and Varchenko concerning the existence of a Selberg integral for every simple Lie algebra \mathfrakg \mathfrak{g} .     

A construction for coisotropic subalgebras of Lie bialgebras

Given a Lie bialgebra (g,g^∗), we present an explicit procedure to construct coisotropic subalgebras, i.e. Lie subalgebras of g whose annihilator is a Lie subalgebra of g^∗. We write down families of examples for the case that g is a classical complex simple Lie algebra.     

Semi-centre de l'algèbre enveloppante d'une sous-algèbre parabolique d'une algèbre de Lie semi-simple

Our aim is to describe the semicentre of the enveloping algebra of a parabolic subalgebra p of a semisimple finite dimensional complex Lie algebra g. Whilst [F. Fauquant-Millet, A. Joseph, Transformation Groups 6 (2) (2001) 125-142] describes explicitly the semicentre of the quantized enveloping algebra associated to p, specialization at q=1 gives only part of the required classical semicentre, even when p is a Borel. Similarly the graded of a polynomial subalgebra of the Hopf dual of the enveloping algebra of g, associated to the Kostant filtration, gives a lower bound on the required semicentre. Then by a method developed from [A. Joseph, Amer. J. Math. 99 (6) (1977) 1151-1165; J. Algebra 48 (1977) 241-289] we obtain an upper bound. Finally when g is a product of simple Lie algebras of type An or Cn we show that these bounds coincide and conclude that in this case the semicentre of the enveloping algebra of p is a polynomial algebra.     

Differential Invariants of the 2D Conformal Lie Algebra Action

We consider the Lie algebra that corresponds to the Lie pseudogroup of all conformal transformations on the plane. This conformal Lie algebra is canonically represented as the Lie algebra of holomorphic vector fields in ℝ²≃ℂ. We describe all representations of \mathfrakg\mathfrak{g} via vector fields in J ⁰ℝ²=ℝ³(x,y,u), which project to the canonical representation, and find their algebra of scalar differential invariants.     

Etingof-Kazhdan quantization of Lie superbialgebras

For every semi-simple Lie algebra g one can construct the Drinfeld-Jimbo algebra . This algebra is a deformation Hopf algebra defined by generators and relations. To study the representation theory of , Drinfeld used the KZ-equations to construct a quasi-Hopf algebra Ag. He proved that particular categories of modules over the algebras and Ag are tensor equivalent. Analogous constructions of the algebras and Ag exist in the case when g is a Lie superalgebra of type A-G. However, Drinfeld's proof of the above equivalence of categories does not generalize to Lie superalgebras. In this paper, we will discuss an alternate proof for Lie superalgebras of type A-G. Our proof utilizes the Etingof-Kazhdan quantization of Lie (super)bialgebras. It should be mentioned that the above equivalence is very useful. For example, it has been used in knot theory to relate quantum group invariants and the Kontsevich integral.     

Multiplicative Semiprimeness of Skew Lie Algebras

Abstract

Let A be a semiprime associative algebra with an involution over a field of characteristic not 2, let K be the Lie algebra of all skew elements of A, and let Z _{[K, K]} denote the annihilator of the Lie algebra [K, K]. We will prove that the multiplication algebra of the semiprime Lie algebra [K, K]/Z _{[K, K]} is also semiprime. As a consequence, the multiplication algebra of [K, K]/Z _{[K, K]} is prime, whenever [K, K]/Z _{[K, K]} is prime. We will obtain similar results for the Lie algebra K/Z _K whenever the base field has characteristic zero.