Actions in the Category of Precrossed Modules in Lie Algebras

For any object L in the category of precrossed modules in Lie algebras PXLie, we construct the object Act(L), which we call the actor of this object. From this construction, we derive the notions of action, center, semidirect product, derivation, commutator, and abelian precrossed module in PXLie. We show that the notion of action is equivalent to the one given in semi-abelian categories, and Act(L) is the split extension classifier for L. In the case of a crossed module in Lie algebras we show how to recover its actor in the category of crossed modules from its actor in the category 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.     

The construction of 3-Lie 2-algebras

We construct a 3-Lie 2-algebra from a 3-Leibniz algebra and a Rota-Baxter 3-Lie algebra. Moreover, we give some examples of 3-Leibniz algebras.     

Galois Theory and a New Homotopy Double Groupoid of a Map of Spaces

The authors have used generalised Galois Theory to construct a homotopy double groupoid of a surjective fibration of Kan simplicial sets. Here we apply this to construct a new homotopy double groupoid of a map of spaces, which includes constructions by others of a 2-groupoid, cat¹-group or crossed module. An advantage of our construction is that the double groupoid can give an algebraic model of a foliated bundle.     

Irreducible weight modules for the Virasoro-like algebra and its q-analog

In this paper, using Larsson’s functor with irreducible 𝔰𝔩₂-modules V, we construct a class of ?²-graded modules for the Virasoro-like algebra and its q-analogs. We determine the irreducibility of these modules for finite-dimensional or infinite-dimensional V using a unified method. In particular, these modules provide new irreducible weight modules with infinite-dimensional weight spaces for the corresponding algebras.     

Multiplicity computation of modules overk[x1,…,xn] and an application to weyl algebras

Let A = k[x₁,…,x_n] be the polynomial algebra over a field kof characteristic 0Ian ideal of A, M = A/Iand ^αHP _I the (affine) Hilbert polynomial of M. By further exploring the algorithmic procedure given in [CLO'] for deriving the existence of ^αHP _I , we compute the leading coefficient of ^αHP _I by looking at the leading monomials of a Grobner basis of Iwithout computing ^αHP _I . Using this result and the filtered-graded transfer of Grobner basis obtained in [LW] for (noncommutative) solvable polynomial algebras (in the sense of [K-RW]), we are able to compute the multiplicity of a cyclic module over the Weyl algebra A _n (k) without computing the Hilbert polynomial of that module, and consequently to give a quite easy algorithmic characterization of the “smallest“ modules over Weyl algebras. Using the same methods as before, we also prove that the tensor product of two cyclic modules over the Weyl algebras has the multiplicity which is equal to the product of the multiplicities of both modules. The last result enables us to construct examples of “smallest“ irreducible modules over Weyl algebras.     

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.     

Cocycles on a finite dimensional quotient of uq (sl(2))

We prove that certain algebra quotients of Hopf algebras are twisted Hopf algebras. On the other handu_q (sl(2)) is a crossed product of a central subalgebra with a quotient [Ubar], when q is a root of 1. Using the cocycle involved in this crossed product we construct non-trivial complex cocycles τ and we find the isomorphism classes of the corresponding twisted Hopf algebras _τ [Ubar]. These provide complex projective representations of [Ubar] which are not ordinary representations.     

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.     

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.     

More on the Hochschild and Cyclic Homologies of Crossed Modules of Algebras

We investigate the Hochschild and cyclic homologies of crossed modules of algebras in some special cases. We prove that the cotriple cyclic homology of a crossed module of algebras (I, A, ρ) is isomorphic to HC _*(ρ): HC _*(I) → HC _*(A), provided I is H-unital and the ground ring is a field with characteristic zero. We also calculate the Hochschild and cyclic homologies of a crossed module of algebras (R, 0, 0) for each algebra R with trivial multiplication. At the end, we give some applications proving a new five term exact sequence.     

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).     

On corepresentations of multiplier Hopf algebras

We present a general construction producing a unitary corepresentation of a multiplier Hopf algebra in itself and study RR-corepresentations of twisted tensor coproduct multiplier Hopf algebra. Then we investigate some properties of functors, integrals and morphisms related to the categories of the crossed modules and covariant modules over multiplier Hopf algebras. By doing so, we can apply our theory to the case of group-cograded multiplier Hopf algebras, in particular, the case of Hopf group-coalgebras.     

Annihilators, Multipliers and Crossed Modules

Using multiplication algebras we introduce actor crossed modules of commutative algebras and use it to generalise some aspects from commutative algebras to crossed modules of commutative algebras. This is applied to the Peiffer pairings in the Moore complex of a simplicial commutative algebra.     

DG Poisson algebra and its universal enveloping algebra

We introduce the notions of differential graded (DG) Poisson algebra and DG Poisson module. Let A be any DG Poisson algebra. We construct the universal enveloping algebra of A explicitly, which is denoted by A^ue. We show that A^ue has a natural DG algebra structure and it satisfies certain universal property. As a consequence of the universal property, it is proved that the category of DG Poisson modules over A is isomorphic to the category of DG modules over A^ue. Furthermore, we prove that the notion of universal enveloping algebra A^ue is well-behaved under opposite algebra and tensor product of DG Poisson algebras. Practical examples of DG Poisson algebras are given throughout the paper including those arising from differential geometry and homological algebra.     

Four dimensional regular algebras with point scheme,anonsingular quadric in P3

In [22], a class of four-dimensional, quadratic, Artin-Schelter regular algebras was introduced, whose point scheme is the graph of an automorphism of a nonsingular quadric in P³. These algebras are the first examples of quadratic Artin-Schelter regular algebras whose defining relations are not determined by the point scheme and, hence, not determined by the algebraic data obtained from the point modules. In this paper, we study these algebras via their line modules. In particular, the set of lines in P³ that correspond to left line modules is not the set of lines in P³ that correspond to right line modules. Our analysis focuses on a distinguished member R _λ of this class of algebras, where R _λ is a twist by a twisting system of the other algebras. We prove that R _λ is a finite module over its center and that its central Proj is a smooth quadric inP⁴.     

Smoothness and Locality for Nonunital Spectral Triples

To deal with technical issues in noncommutative geometry for nonunital algebras, we introduce a useful class of algebras and their modules. These algebras and modules allow us to extend all of the smoothness results for spectral triples to the nonunital case. In addition, we show that smooth spectral triples are closed under the C functional calculus of self-adjoint elements. In the final section we show that our algebras allow the formulation of Poincaré Duality and that the algebras of smooth spectral triples are H-unital.     

n-representation infinite algebras

From the viewpoint of higher dimensional Auslander–Reiten theory, we introduce a new class of finite dimensional algebras of global dimension n, which we call n-representation infinite. They are a certain analog of representation infinite hereditary algebras, and we study three important classes of modules: n-preprojective, n-preinjective and n  -regular modules. We observe that their homological behaviour is quite interesting. For instance they provide first examples of algebras having infinite Ext¹${\mathrm{Ext}}^1$-orthogonal families of modules. Moreover we give general constructions of n-representation infinite algebras.