Related papers: Integrating morphisms of Lie 2-algebras
A study is made of real Lie algebras admitting compatible complex and product structures, including numerous 4-dimensional examples. If g is a Lie algebra with such a structure then its complexification has a hypercomplex structure. It is…
Given a coalgebra C over a cooperad, and an algebra A over an operad, it is often possible to define a natural homotopy Lie algebra structure on hom(C,A), the space of linear maps between them, called the convolution algebra of C and A. In…
Like categories, small 2-categories have well-understood classifying spaces. In this paper, we deal with homotopy types represented by 2-diagrams of 2-categories. Our results extend to homotopy colimits of 2-functors lower categorical…
The main goal of this article is to develop integration theory for absolute partition $L_\infty$-algebras, which are point-set models for the (spectral) partition Lie algebras of Brantner-Mathew where infinite sums of operations are…
A bi-invariant differential 2-form on a Lie group G is a highly constrained object, being determined by purely linear data: an Ad-invariant alternating bilinear form on the Lie algebra of G. On a compact connected Lie group these have an…
Bakalov, Kac and Voronov introduced Leibniz conformal algebras (and their cohomology) as a non-commutative analogue of Lie conformal algebras. Leibniz conformal algebras are closely related to field algebras which are non-skew-symmetric…
We show that the category of Lie triple systems is equivalent to the category of Lie algebras graded by Z/(2Z) such that the odd component generates the algbera and the second graded cohomology group coefficients in any trivial module is…
We show that the Morse complex of a compact Lie monoid can be given the structure of an $f$-bialgebra, a chain-level version of bialgebras introduced in [CHM24]; and that this assignment defines an $\infty$-functor. As a consequence, we…
We construct Quillen equivalent semi-model structures on the categories of dg-Lie algebroids and $L_\infty$-algebroids over a commutative dg-algebra in characteristic zero. This allows one to apply the usual methods of homotopical algebra…
We compare three different ways of defining group cohomology with coefficients in a crossed-module: 1) explicit approach via cocycles; 2) geometric approach via gerbes; 3) group theoretic approach via butterflies. We discuss the case where…
We introduce a remarkable subset "the stem" of the set of positive roots of a reduced root system. The stem determines several interesting decompositions of the corresponding reductive Lie algebra. It gives also a nice simple three…
This is an old paper put here for archeological purposes. We derive a general formula expressing the second homology of a Lie algebra of the form L\otimes A with coefficients in the trivial module through homology of $L$, cyclic homology of…
We categorify the theory of Lie algebras beginning with a new notion of categorified vector space, or `2-vector space', which we define as an internal category in Vect, the category of vector spaces. We then define a `semistrict Lie…
In this paper, we introduce the notions of crossed module of associative conformal algebras, 2-term strongly homotopy associative conformal algebras, and discuss the relationship between them and the 3-th Hochschild cohomology of…
This paper studies differential graded modules and representations up to homotopy of Lie $n$-algebroids, for general $n\in\mathbb{N}$. The adjoint and coadjoint modules are described, and the corresponding split versions of the adjoint and…
We build free, bigraded bidifferential algebra models for the forms on a complex manifold, with respect to a strong notion of quasi-isomorphism and compatible with the conjugation symmetry. This answers a question of Sullivan. The resulting…
Given a foliation, there is a well-known notion of holonomy, which can be understood as an action that differentiates to the Bott connection on the normal bundle. We present an analogous notion for Lie subalgebroids, consisting of an…
We consider a simple instance of action up to homotopy. More precisely, we consider strict actions of DGLAs in degrees -1 and 0 on degree 1 NQ-manifolds. In a more conventional language this means: strict actions of Lie algebra crossed…
Quantum $L_\infty$ algebras are higher loop generalizations of cyclic $L_\infty$ algebras. Motivated by the problem of defining morphisms between such algebras, we construct a linear category of $(-1)$-shifted symplectic vector spaces and…
Lie algebras are an important class of algebras which arise throughout mathematics and physics. We report on the formalisation of Lie algebras in Lean's Mathlib library. Although basic knowledge of Lie theory will benefit the reader, none…