• Corpus ID: 219635883

Algebraic exponentiation for Lie algebras.

@article{GarciaMartinez2020AlgebraicEF,
  title={Algebraic exponentiation for Lie algebras.},
  author={Xabier Garc'ia-Mart'inez and James Richard Andrew Gray},
  journal={arXiv: Category Theory},
  year={2020}
}
It is known that the category of Lie algebras over a ring admits algebraic exponents. The aim of this paper is to show that the same is true for the category of internal Lie algebras in an additive, cocomplete, symmetric, closed, monoidal category. In this way, we add some new examples to the brief list of known locally algebraically cartesian closed categories, including the categories of Lie superalgebras and differentially graded Lie algebras amongst others. 
View PDF on arXiv
3 Citations
Background Citations
2

3 Citations

Citation Type

Citation Type

ON SOME PROPERTIES OF Lie-CENTROIDS OF LEIBNIZ ALGEBRAS
We study some properties on Lie-centroids related to central Liederivations, generalized Lie-derivations and almost inner Lie-derivations. We also determine the Lie-centroid of the tensor product of…
On some properties of $\mathsf{Lie}$-centroids of Leibniz algebras
We study some properties on Lie-centroids related to central Liederivations, generalized Lie-derivations and almost inner Lie-derivations. We also determine the Lie-centroid of the tensor product of…
Algebras with representable representations
Abstract Just like group actions are represented by group automorphisms, Lie algebra actions are represented by derivations: up to isomorphism, a split extension of a Lie algebra $B$ by a Lie algebra…

References

SHOWING 1-10 OF 27 REFERENCES
A characterisation of Lie algebras via algebraic exponentiation
A Note on the Categorification of Lie Algebras
In this short note we study Lie algebras in the framework of symmetric monoidal categories. After a brief review of the existing work in this field and a presentation of earlier studied and new…
Algebraic exponentiation for categories of Lie algebras
A characterisation of Lie algebras amongst anti-commutative algebras
Abelian extensions and crossed modules of Hom-Lie algebras
Algebraic Exponentiation in General Categories
  • J. Gray
  • Mathematics
    Appl. Categorical Struct.
  • 2012
TLDR
A categorical-algebraic concept of exponentiation, namely, right adjoints for the pullback functors between D. Bourn’s categories of points, is studied, in particular, for semi-abelian, protomodular, (weakly) Mal’tsev, ( weakly) unital, and more general categories.
The cohomological comparison arising from the associated abelian object
We make explicit some conditions on a semi-abelian category D such that, for any abelian group A in D and any object Y in D, the cohomology group homomorphisms with coefficients in A, induced by the…
Aspects of algebraic exponentiation
We analyse some aspects of the notion of algebraic exponentiation introduced by the second author [16] and satisfied by the category of groups. We show how this notion provides a new approach to the…
Lie monads and dualities
Semi-abelian categories
...
...