arXiv subjects

Xiongwei Cai

Publications and source records attributed to Xiongwei Cai.

5 recordsLinked to original sources

arXiv2022

The standard cohomology of regular Courant algebroids

For any regular Courant algebroid $E$ over a smooth manifold $M$ with characteristic distribution $F$ and ample Lie algebroid $A_E$, we prove that there exists a canonical homological vector field on the graded manifold $A_E[1] \oplus (TM/F)^\ast[2]$ such that the resulting dg manifold $\mathcal{M}_E$, which we call the minimal model of the Courant algebroid $E$, encodes all cohomological information of $E$. Indeed, the standard cohomology of $E$ can be identified with the cohomology of the function space on $\mathcal{M}_E$, which can be computed by a Hodge-to-de Rham type spectral sequence. We apply this result to generalized exact Courant algebroids and those arising from regular Lie algebroids.

math.DG↗

arXiv2020

Cohomology of hemistrict Lie 2-algebras

We study representations of hemistrict Lie 2-algebras and give a functorial construction of their cohomology. We prove that both the cohomology of an injective hemistrict Lie 2-algebra $L$ and the cohomology of the semistrict Lie 2-algebra obtained from skew-symmetrization of $L$ are isomorphic to the Chevalley-Eilenberg cohomology of the induced Lie algebra $L_{\operatorname{Lie}}$.

math.RA↗

arXiv2020

Transgression maps for crossed modules of groupoids

Given a crossed module of groupoids $N\rightarrow G$, we construct (1) a natural homomorphism from the product groupoid $\mathbb{Z}\times(N\rtimes G)\rightrightarrows N$ to the crossed product groupoid $N\rtimes G\rightrightarrows N$ and (2) a transgression map from the singular cohomology $H^\ast(G_\bullet,\mathbb{Z})$ of the nerve of the groupoid $G$ to the singular cohomology $H^{\ast-1}\big((N\rtimes G)_\bullet,\mathbb{Z}\big)$ of the nerve of the crossed product groupoid $N\rtimes G$. The latter turns out to be identical to the transgression map obtained by Tu--Xu in their study of equivariant $K$-theory.

math.AT↗

arXiv2016

Derived brackets for fat Leibniz algebras

Given a Leibniz algebra L with left center Z, we work on C(L,Z,S(Z)), the Z-standard complex of L with coefficients in S(Z). We construct the derived bracket for a fat Leibniz algebra in terms of a certain 3-cocycle and a Poisson algebra structure on the space of so-called "representable cochains".

math.RA↗

arXiv2016

H-standard cohomology for Courant-Dorfman algebras and Leibniz algebras

We introduce the notion of H-standard cohomology for Courant-Dorfman algebras and Leibniz algebras, by generalizing Roytenberg's construction. Then we generalize a theorem of Ginot-Grutzmann on transitive Courant algebroids, which was conjectured by Stienon-Xu. The relation between H-standard complexes of a Leibniz algebra and the associated crossed product is also discussed.

math.RA↗