Diagrams of Lie algebras

Gerstenhaber, Murray; Giaquinto, Anthony; Schack, Samuel D.

doi:10.1016/j.jpaa.2004.08.022

Cited by 10 publications

(9 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Extensions of Lie and associative algebras by ideals are a classical subject [4,21], which have been recast in many forms and generalized extensively [18,19], in terms of diagrams of algebras. Deformation theory of associative algebras is still an active subject of research [1].…”

Section: Preliminariesmentioning

confidence: 99%

Extensions of associative algebras

Fialowski

Penkava

2014

Commun. Contemp. Math.

View full text Add to dashboard Cite

In this paper, we translate the problem of extending an associative algebra by another associative algebra into the language of codifferentials. The authors have been constructing moduli spaces of algebras and studying their structure by constructing their versal deformations. The codifferential language is very useful for this purpose. The goal of this paper is to express the classical results about extensions in a form which leads to a simpler construction of moduli spaces of low-dimensional algebras.

show abstract

Section: Preliminariesmentioning

confidence: 99%

Extensions of associative algebras

Fialowski

Penkava

2014

Commun. Contemp. Math.

View full text Add to dashboard Cite

show abstract

“…. is a contravariant functor A from C to the category of unital associative algebras, i.e., a presheaf of algebras over C. (One can make the same definition for diagrams of other kinds of algebras; for the Lie case cf [9].) For example, the sets in an open covering U (closed under taking intersections) of a complex manifold M may be viewed as the objects of a category in which the morphisms are the inclusion maps.…”

Section: Diagram Cohomologymentioning

confidence: 99%

Deformations associated with rigid algebras

Gerstenhaber

Giaquinto

2013

J. Homotopy Relat. Struct.

View full text Add to dashboard Cite

The deformations of an infinite dimensional algebra may be controlled not just by its own cohomology but by that of an associated diagram of algebras, since an infinite dimensional algebra may be absolutely rigid in the classical deformation theory for single algebras while depending essentially on some parameters. Two examples studied here, the function field of a sphere with four marked points and the first Weyl algebra, show, however, that the existence of these parameters may be made evident by the cohomology of a diagram (presheaf) of algebras constructed from the original. The Cohomology Comparison Theorem asserts, on the other hand, that the cohomology and deformation theory of a diagram of algebras is always the same as that of a single, but generally rather large, algebra constructed from the diagram.

show abstract

“…Also, instead of associative algebras, one may consider any other type of algebras for which a convenient cohomology is known (e.g. Lie algebras, [5]). In this paper we stick to associative algebras, but we believe that other types can be handled in a similar way.…”

Section: Gerstenhaber-schack Diagram Cohomologymentioning

confidence: 99%