ترغب بنشر مسار تعليمي؟ اضغط هنا

Realization of Lie algebras and classifying spaces of crossed modules

77   0   0.0 ( 0 )
 نشر من قبل Daniel Tanr\\'e
 تاريخ النشر 2021
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

The category of complete differential graded Lie algebras provides nice algebraic models for the rational homotopy types of non-simply connected spaces. In particular, there is a realization functor, $langle -rangle$, of any complete differential graded Lie algebra as a simplicial set. In a previous article, we considered the particular case of a complete graded Lie algebra, $L_{0}$, concentrated in degree 0 and proved that $langle L_{0}rangle$ is isomorphic to the usual bar construction on the Malcev group associated to $L_{0}$. Here we consider the case of a complete differential graded Lie algebra, $L=L_{0}oplus L_{1}$, concentrated in degrees 0 and 1. We establish that this category is equivalent to explicit subcategories of crossed modules and Lie algebra crossed modules, extending the equivalence between pronilpotent Lie algebras and Malcev groups. In particular, there is a crossed module $mathcal{C}(L)$ associated to $L$. We prove that $mathcal{C}(L)$ is isomorphic to the Whitehead crossed module associated to the simplicial pair $(langle Lrangle, langle L_{0}rangle)$. Our main result is the identification of $langle Lrangle$ with the classifying space of $mathcal{C}(L)$.



قيم البحث

اقرأ أيضاً

We introduce a notion of $n$-Lie Rinehart algebras as a generalization of Lie Rinehart algebras to $n$-ary case. This notion is also an algebraic analogue of $n$-Lie algebroids. We develop representation theory and describe a cohomology complex of $n $-Lie Rinehart algebras. Furthermore, we investigate extension theory of $n$-Lie Rinehart algebras by means of $2$-cocycles. Finally, we introduce crossed modules of $n$-Lie Rinehart algebras to gain a better understanding of their third dimensional cohomology groups.
81 - Charles Rezk 2016
A 1-truncated compact Lie group is any extension of a finite group by a torus. In this note we compute the homotopy types of $Map_*(BG,BH)$, $Map(BG,BH)$, and $Map(EG, B_GH)^G$ for compact Lie groups $G$ and $H$ with $H$ 1-truncated, showing that the y are computed entirely in terms of spaces of homomorphisms from $G$ to $H$. These results generalize the well-known case when $H$ is finite, and the case of $H$ compact abelian due to Lashof, May, and Segal.
We study Lie bialgebroid crossed modules which are pairs of Lie algebroid crossed modules in duality that canonically give rise to Lie bialgebroids. A one-one correspondence between such Lie bialgebroid crossed modules and co-quadratic Manin triples $(K,P,Q)$ is established, where $K$ is a co-quadratic Lie algebroid and $(P,Q)$ is a pair of transverse Dirac structures in $K$.
We prove that the set of concordance classes of sections of an infinity-sheaf on a manifold is representable, extending a theorem of Madsen and Weiss. This is reminiscent of an h-principle in which the role of isotopy is played by concordance. As an application, we offer an answer to the question: what does the classifying space of a Segal space classify?
Let G be a compact Lie group. By work of Chataur and Menichi, the homology of the space of free loops in the classifying space of G is known to be the value on the circle in a homological conformal field theory. This means in particular that it admit s operations parameterized by homology classes of classifying spaces of diffeomorphism groups of surfaces. Here we present a radical extension of this result, giving a new construction in which diffeomorphisms are replaced with homotopy equivalences, and surfaces with boundary are replaced with arbitrary spaces homotopy equivalent to finite graphs. The result is a novel kind of field theory which is related to both the diffeomorphism groups of surfaces and the automorphism groups of free groups with boundaries. Our work shows that the algebraic structures in string topology of classifying spaces can be brought into line with, and in fact far exceed, those available in string topology of manifolds. For simplicity, we restrict to the characteristic 2 case. The generalization to arbitrary characteristic will be addressed in a subsequent paper.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا