A non-abelian tensor product of precrossed modules in lie algebras

Abstract

In this paper, we introduce the non-abelian tensor square of precrossed modules in Lie algebras and investigate some of its properties. In particular, for an arbitrary Lie algebra L, we study the relation of the second homology of a precrossed L-module and the non-abelian exterior square. Also, we show how this non-abelian tensor product is related to the universal central extensions (with respect to the subcategory of crossed modules) of a precrossed module.     

Compatible actions of Lie algebras

Abstract

We study compatible actions (introduced by Brown and Loday in their work on the non-abelian tensor product of groups) in the category of Lie algebras over a fixed ring. We describe the Peiffer product via a new diagrammatic approach, which specializes to the known definitions both in the case of groups and of Lie algebras. We then use this approach to transfer a result linking compatible actions and pairs of crossed modules over a common base object L from groups to Lie algebras. Finally, we show that the Peiffer product, naturally endowed with a crossed module structure, has the universal property of the coproduct in XMod_L(Lie_R).     

Universal Strict General Actors and Actors in Categories of Interest

For any category of interest ℂ we define a general category of groups with operations \mathbbC_G, \mathbbC\hookrightarrow\mathbbC_G\mathbb{C_G}, \mathbb{C}\hookrightarrow\mathbb{C_G}, and a universal strict general actor USGA(A) of an object A in ℂ, which is an object of \mathbbC_G\mathbb{C_G}. The notion of actor is equivalent to the one of split extension classifier defined for an object in more general settings of semi-abelian categories. It is proved that there exists an actor of A in ℂ if and only if the semidirect product \textUSGA(A)\ltimes A{\text{USGA}}(A)\ltimes A is an object of ℂ and if it is the case, then USGA(A) is an actor of A. We give a construction of a universal strict general actor for any A ∈ ℂ, which helps to detect more properties of this object. The cases of groups, Lie, Leibniz, associative, commutative associative, alternative algebras, crossed and precrossed modules are considered. The examples of algebras are given, for which always exist actors.     

Representability of actions in the category of (Pre)crossed modules in Leibniz algebras

We investigate the representability of actions in the category of (pre)crossed modules in Leibniz algebras. For this, we construct an actor of a (pre)cat¹-Leibniz algebra and then by using the natural equivalence of the categories of (pre)cat¹-Leibniz algebras and that of (pre)crossed modules, we construct the split extension classifier of the corresponding (pre)crossed module.     

Actor of a Precrossed Module

Applying the equivalence of the category of precrossed modules with the category of groups with two additional unary operations satisfying the corresponding conditions, the construction of an actor is given in terms of Whitehead group of generalized regular derivations, defined in the article, and the automorphism group of a precrossed module. The analogous approach to this problem in the case of crossed modules leads to the well-known construction given in the works of Lue and Norrie.     

Capability of Crossed Modules of Lie Algebras

In this paper, we introduce the concept of capability for crossed modules of Lie algebras, which is a generalization of capability in Lie algebras and groups. By using a special central ideal of a crossed module, we give a sufficient condition for the capability of a crossed module of Lie algebras. Also, we will extend the five-term exact sequence on homology of crossed modules of Lie algebras one term further and study the connection between the capability of crossed modules and this sequence. Finally, we study the relation between the capability and the center of a cover of a crossed module.     

From Lie algebra crossed modules to tensor hierarchies

The present paper, though inspired by the use of tensor hierarchies in theoretical physics, establishes their mathematical credentials, especially as genetically related to Lie algebra crossed modules. Gauging procedures in supergravity rely on a pairing – the embedding tensor – between a Leibniz algebra and a Lie algebra. Two such algebras, together with their embedding tensor, form a triple called a Lie-Leibniz triple, of which Lie algebra crossed modules are particular cases. This paper is devoted to showing that any Lie-Leibniz triple induces a differential graded Lie algebra – its associated tensor hierarchy – whose restriction to the category of Lie algebra crossed modules is the canonical assignment associating to any Lie algebra crossed module its corresponding unique 2-term differential graded Lie algebra. This shows that Lie-Leibniz triples form natural generalizations of Lie algebra crossed modules and that their associated tensor hierarchies can be considered as some kind of ‘lie-ization’ of the former. We deem the present construction of such tensor hierarchies clearer and more straightforward than previous derivations. We stress that such a construction suggests the existence of further well-defined Leibniz gauge theories.     

Baer Invariants and Cohomology of Precrossed Modules

In this paper we study Baer invariants of precrossed modules relative to the subcategory of crossed modules, following Fröhlich and Furtado-Coelho’s general theory on Baer invariants in varieties of Ω-groups and Modi’s theory on higher dimensional Baer invariants. Several homological invariants of precrossed and crossed modules were defined in the last two decades. We show how to use Baer invariants in order to connect these various homology theories. First, we express the low-dimensional Baer invariants of precrossed modules in terms of a new non-abelian tensor product of a precrossed module. This expression is used to analyze the connection between the Baer invariants and the homological invariants of precrossed modules defined by Conduché and Ellis. Specifically we prove that the second homological invariant of Conduché and Ellis is in general a quotient of the first component of the Baer invariant we consider. The definition of classical Baer invariants is generalized using homological methods. These generalized Baer invariants of precrossed modules are applied to the construction of five term exact sequences connecting the generalized Baer invariants with the cohomology theory of crossed modules considered by Carrasco, Cegarra and R.-Grandjeán and the cohomology theory of precrossed modules.     

Restricted Lie 2-algebras

In this article, we introduce the notions of restricted Lie 2-algebras and crossed modules of restricted Lie algebras, and give a series of examples of restricted Lie 2-algebras. We also construct restricted Lie 2-algebras from A(m)-algebras, restricted Leibniz algebras, restricted right-symmetric algebras. Finally, we prove that there is a one-to-one correspondence between strict restricted Lie 2-algebras and crossed modules of restricted Lie algebras.     

Universal central extension and the second invariant of homology of crossed modules in lie algebras

In this paper we show that the kernel of the universal central extension of a crossed module in Lie algebras is the second invariant of this crossed module. As a consequence of this result we obtain a recognition criterion for universal central extensions and a vanishing situation of two invariants associated to a crossed module in Lie algebras.     

Strongly Flat and PO-Flat S-Posets

Abstract

The main purpose of this paper is to study Lie algebras L such that if a subalgebra U of L has a maximal subalgebra of dimension one then every maximal subalgebra of U has dimension one. Such an L is called lm(0)-algebra. This class of Lie algebras emerges when it is imposed on the lattice of subalgebras of a Lie algebra the condition that every atom is lower modular. We see that the effect of that condition is highly sensitive to the ground field F. If F is algebraically closed, then every Lie algebra is lm(0). By contrast, for every algebraically non-closed field there exist simple Lie algebras which are not lm(0). For the real field, the semisimple lm(0)-algebras are just the Lie algebras whose Killing form is negative-definite. Also, we study when the simple Lie algebras having a maximal subalgebra of codimension one are lm(0), provided that char(F) ≠ 2. Moreover, lm(0)-algebras lead us to consider certain other classes of Lie algebras and the largest ideal of an arbitrary Lie algebra L on which the action of every element of L is split, which might have some interest by themselves.     

A categorical approach to Weyl modules

Global and local Weyl modules were introduced via generators and relations in the context of affine Lie algebras in [CP2] and were motivated by representations of quantum affine algebras. In [FL] a more general case was considered by replacing the polynomial ring with the coordinate ring of an algebraic variety and partial results analogous to those in [CP2] were obtained. In this paper we show that there is a natural definition of the local and global Weyl modules via homological properties. This characterization allows us to define the Weyl functor from the category of left modules of a commutative algebra to the category of modules for a simple Lie algebra. As an application we are able to understand the relationships of these functors to tensor products, generalizing results in [CP2] and [FL]. We also analyze the fundamental Weyl modules and show that, unlike the case of the affine Lie algebras, the Weyl functors need not be left exact.     

A Program for Computing the Cohomology of Lie Superalgebras of Vector Fields

The cohomology of Lie (super)algebras has many important applications in mathematics and physics. At present, since the required algebraic computations are very tedious, the cohomology is explicitly computed only in a few cases for different classes of Lie (super)algebras. That is why application of computer algebra methods is important for this problem. We describe an algorithm (and its C implementation) for computing the cohomology of Lie algebras and superalgebras. In elaborating the algorithm, we focused mainly on the cohomology with coefficients in trivial, adjoint, and coadjoint modules for Lie (super)algebras of the formal vector fields. These algebras have many applications to modern supersymmetric models of theoretical and mathematical physics. As an example, we consider the cohomology of the Poisson algebra Po(2) with coefficients in the trivial module and present 3- and 5-cocycles found by a computer. Bibliography: 6 titles.     

A Push Forward Construction and the Comprehensive Factorization for Internal Crossed Modules

In a semi-abelian category, we give a categorical construction of the push forward of an internal pre-crossed module, generalizing the pushout of a short exact sequence in abelian categories. The main properties of the push forward are discussed. A simplified version is given for action accessible categories, providing examples in the categories of rings and Lie algebras. We show that push forwards can be used to obtain the crossed module version of the comprehensive factorization for internal groupoids.     

Depth Generated Simple Lie Algebras

A Lie module algebra for a Lie algebra L is an algebra and L-module A such that L acts on A by derivations. The depth Lie algebra of a Lie algebra L with Lie module algebra A acts on a corresponding depth Lie module algebra . This determines a depth functor from the category of Lie module algebra pairs to itself. Remarkably, this functor preserves central simplicity. It follows that the Lie algebras corresponding to faithful central simple Lie module algebra pairs (A,L) with A commutative are simple. Upon iteration at such (A,L), the Lie algebras are simple for all i ∈ ω. In particular, the (i ∈ ω) corresponding to central simple Jordan Lie algops (A,L) are simple Lie algebras. Presented by Don Passman.     

On derived categories of modules over 2-vertex basic algebras

We investigate finite dimensional 2-vertex basic algebras of finite global dimension and the derived categories of modules over such algebras. We prove that any superrigid object in the derived category of modules over a “loop-kind” two-vertex algebra is a pure module up to the action of Serre functor and translation. All superrigid objects in the derived categories of modules over two-vertex algebras of global dimension 2 are described. Also we obtain a complete classification of two-vertex basic algebras possessing a full exceptional pair in the derived category of modules.     

Projective functors and the restriction of a Verma module to a subalgebra of Levi type

We observe that the restriction of a Verma module over a semi-simple Lie algebra to a subalgebra of Levi type may be viewed as a projective functor. By simple arguments we prove that this restriction can be decomposed into a direct sum of standard indecomposables in the category O. For the restriction problem from sl(n+1) to gl(n) we describe the complete answer. We study the properties of the modules with Verma flag also and prove that any module with Verma flag is a submodule of some projective.