The Schur Lie-multiplier of Leibniz algebras

Abstract

For a free presentation 0 → τ → → → 0 of a Leibniz algebra , the Baer invariant is called the Schur multiplier of relative to the Liezation functor or Schur Lie-multiplier. For a two-sided ideal of a Leibniz algebra , we construct a four-term exact sequence relating the Schur Lie-multipliers of and /, which is applied to study and characterize Lie-nilpotency, Lie-stem covers and Lie-capability of Leibniz algebras.     

Lie algebra computations

In the context of prolongation theory, introduced by Wahlquist and Estabrook, computations of a lot of Jacobi identities in (infinite-dimensional) Lie algebras are necessary. These computations can be done (automatically) using symbolic computations. A package written in REDUCE is demonstrated to give an idea of the chosen approach.     

Computation by computer of Lie superalgebra homology and cohomology

In this paper, we introduce a package to compute homology and cohomology spaces of Lie superalgebras. We describe most of its features and the implementation in REDUCE.     

Bosonic and fermionic representations of Lie algebra central extensions

Given any representation of an arbitrary Lie algebra g over a field K of characteristic 0, we construct representations of a central extension of g on bosonic and fermionic Fock space. The method gives an explicit formula for a (sometimes trivial) 2-cocycle in H²(g;K). We illustrate these techniques with several concrete examples.     

On the Lie algebra of skew-symmetric elements of an enveloping algebra

Let L be a restricted Lie algebra over a field of characteristic p>2 and denote by u(L) its restricted enveloping algebra. We establish when the Lie algebra of skew-symmetric elements of u(L) under the principal involution is solvable, nilpotent, or satisfies an Engel condition.     

Derived brackets and sh Leibniz algebras

We develop a general framework for the construction of various derived brackets. We show that suitably deforming the differential of a graded Leibniz algebra extends the derived bracket construction and leads to the notion of strong homotopy (sh) Leibniz algebra. We discuss the connections among homotopy algebra theory, deformation theory and derived brackets. We prove that the derived bracket construction induces a map from suitably defined deformation theory equivalence classes to the isomorphism classes of sh Leibniz algebras.     

The adjoint representation inside the exterior algebra of a simple Lie algebra

For a simple complex Lie algebra g$\mathfrak{g}$ we study the space of invariants A=(?g^?⊗g^?)^g$A={(?{\mathfrak{g}}^{?}\otimes {\mathfrak{g}}^{?})}^{\mathfrak{g}}$, which describes the isotypic component of type g$\mathfrak{g}$ in ?g^?$?{\mathfrak{g}}^{?}$, as a module over the algebra of invariants (?g^?)^g${(?{\mathfrak{g}}^{?})}^{\mathfrak{g}}$. As main result we prove that A   is a free module, of rank twice the rank of g$\mathfrak{g}$, over the exterior algebra generated by all primitive invariants in (?g^?)^g${(?{\mathfrak{g}}^{?})}^{\mathfrak{g}}$, with the exception of the one of highest degree.     

Lie triple derivation of the Lie algebra of strictly upper triangular matrix over a commutative ring

Let N(n,R) be the nilpotent Lie algebra consisting of all strictly upper triangular n×n matrices over a 2-torsionfree commutative ring R with identity 1. In this paper, we prove that any Lie triple derivation of N(n,R) can be uniquely decomposited as a sum of an inner triple derivation, diagonal triple derivation, central triple derivation and extremal triple derivation for n≧6. In the cases 1≦n≦5, the results are trivial.     

Quillen-Barr-Beck (co-) homology for restricted Lie algebras

In this paper we use Quillen-Barr-Beck's theory of (co-) homology of algebras in order to define (co-) homology for the category RLie of restricted Lie algebras over a field k of characteristic p≠0. In contrast with the cases of groups, associative algebras and Lie algebras we do not obtain Hochschild (co-) homology shifted by 1.Precisely, we determine for L∈RLie the category of Beck L-modules and the group of Beck derivations of g∈RLie/L to a Beck L-module M. Moreover, we prove a classification theorem which gives a one-to-one correspondence between the one cohomology and the set of equivalent classes of p-extensions. Finally, a universal coefficient theorem is proved, relating the homology to the Hochschild homology via a short exact sequence. This shows that the new homology determines the Hochschild homology.     

Quasifinite modules of a Lie algebra related to Block type

Let B be the universal central extension of a graded Lie algebra of Block type. In this paper, it is proved that any quasifinite irreducible B-module is either highest weight, lowest weight or uniformly bounded. Furthermore, the quasifinite irreducible highest weight B-modules are classified, and the intermediate series B-modules are classified and constructed.     

Leibniz cohomology and the calculus of variations

A geometric interpretation of the Leibniz coboundary is given in terms of the calculus of variations. For a differentiable manifold M, Leibniz cohomology generalizes de Rham cohomology by including all tensors as cochains. When applied to two-tensors, the conditions for the vanishing of a Leibniz cochain are related to the necessary conditions to achieve an extreme value of the integral of the tensor over an immersed surface. A local formula for the coboundary of any tensor is given in terms of a coordinate chart, and the Leibniz coboundary of the Riemann curvature tensor is computed in terms of the derivative of sectional curvature.     

Separated Lie models and the homotopy Lie algebra

A simply connected topological space X has homotopy Lie algebra π_∗(ΩX)⊗Q. Following Quillen, there is a connected differential graded free Lie algebra (dgL) called a Lie model, which determines the rational homotopy type of X, and whose homology is isomorphic to the homotopy Lie algebra. We show that such a Lie model can be replaced with one that has a special property that we call being separated. The homology of a separated dgL has a particular form which lends itself to calculations.     

Trilinear products and comtrans algebra representations

Comtrans algebras are modules over a commutative ring R equipped with two trilinear operations: a left alternative commutator and a translator satisfying the Jacobi identity, the commutator and translator being connected by the so-called comtrans identity. The standard construction of a comtrans algebra uses the ternary commutator and translator of a trilinear product. If 6 is invertible in R, then each comtrans algebra arises in this standard way from the so-called bogus product.Consider a vector space E of dimension n over a field R. While the dimension of the space of all trilinear products on E is n⁴, the dimension of the space of all comtrans algebras on E is less, namely . The paper determines which trilinear products may be represented as linear combinations of the commutator and translator of a comtrans algebra. For R not of characteristic 3, the necessary and sufficient condition for such a representation is the strong alternativityxxy+xyx+yxx=0 of the trilinear product xyz. For R also not of characteristic 2, it is shown that the representation may be given by the bogus product. A suitable representation for the characteristic 2 case is also obtained.     

Representations of the Lie algebra of vector fields on a sphere

For an affine algebraic variety X we study a category of modules that admit compatible actions of both the algebra A of functions on X and the Lie algebra of vector fields on X. In particular, for the case when X is the sphere ${\mathbb{S}}^2$, we construct a set of simple modules that are finitely generated over A. In addition, we prove that the monoidal category that these modules generate is equivalent to the category of finite-dimensional rational ${\mathrm{GL}}_2$-modules.     

Infinitesimal deformations of restricted simple Lie algebras II

We compute the infinitesimal deformations of two families of restricted simple modular Lie algebras of Cartan-type: the Contact and the Hamiltonian Lie algebras.