Universal Cartan-Lie algebroid of an anchored bundle with connection and compatible geometries

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F62690094%3A18470%2F19%3A50015565" target="_blank" >RIV/62690094:18470/19:50015565 - isvavai.cz</a>

Výsledek na webu

<a href="https://www.sciencedirect.com/science/article/pii/S0393044018305734?via%3Dihub" target="_blank" >https://www.sciencedirect.com/science/article/pii/S0393044018305734?via%3Dihub</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.geomphys.2018.09.004" target="_blank" >10.1016/j.geomphys.2018.09.004</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Universal Cartan-Lie algebroid of an anchored bundle with connection and compatible geometries

Popis výsledku v původním jazyce

Consider an anchored bundle (E, rho), i.e. a vector bundle E -&gt; M equipped with a bundle map rho: E -&gt; TM covering the identity. M. Kapranov showed in the context of Lie-Rinehard algebras that there exists an extension of this anchored bundle to an infinite rank universal free Lie algebroid FR(E) superset of E. We adapt his construction to the case of an anchored bundle equipped with an arbitrary connection, (E, del), and show that it gives rise to a unique connection, (del) over tilde on FR(E) which is compatible with its Lie algebroid structure, thus turning (FR(E), (del) over tilde) into a Cartan-Lie algebroid. Moreover, this construction is universal: any connection-preserving vector bundle morphism from (E, del) to a Cartan-Lie Algebroid (A, (del) over bar) factors through a unique Cartan-Lie algebroid morphism from (FR(E), (del) over tilde) to (A, (del) over bar). Suppose that, in addition, M is equipped with a geometrical structure defined by some tensor field t which is compatible with (E, rho, del) in the sense of being annihilated by a natural E-connection that one can associate to these data. For example, for a Riemannian base (M, g) of an involutive anchored bundle (E, rho), this condition implies that M carries a Riemannian foliation. It is shown that every E-compatible tensor field t becomes invariant with respect to the Lie algebroid representation associated canonically to the Cartan-Lie algebroid (FR(E), (del) over tilde).

Název v anglickém jazyce

Universal Cartan-Lie algebroid of an anchored bundle with connection and compatible geometries

Popis výsledku anglicky

Consider an anchored bundle (E, rho), i.e. a vector bundle E -&gt; M equipped with a bundle map rho: E -&gt; TM covering the identity. M. Kapranov showed in the context of Lie-Rinehard algebras that there exists an extension of this anchored bundle to an infinite rank universal free Lie algebroid FR(E) superset of E. We adapt his construction to the case of an anchored bundle equipped with an arbitrary connection, (E, del), and show that it gives rise to a unique connection, (del) over tilde on FR(E) which is compatible with its Lie algebroid structure, thus turning (FR(E), (del) over tilde) into a Cartan-Lie algebroid. Moreover, this construction is universal: any connection-preserving vector bundle morphism from (E, del) to a Cartan-Lie Algebroid (A, (del) over bar) factors through a unique Cartan-Lie algebroid morphism from (FR(E), (del) over tilde) to (A, (del) over bar). Suppose that, in addition, M is equipped with a geometrical structure defined by some tensor field t which is compatible with (E, rho, del) in the sense of being annihilated by a natural E-connection that one can associate to these data. For example, for a Riemannian base (M, g) of an involutive anchored bundle (E, rho), this condition implies that M carries a Riemannian foliation. It is shown that every E-compatible tensor field t becomes invariant with respect to the Lie algebroid representation associated canonically to the Cartan-Lie algebroid (FR(E), (del) over tilde).

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GA18-00496S" target="_blank" >GA18-00496S: Singulární prostory ze speciální holonomie a foliací</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2019

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Journal of geometry and physics

ISSN

0393-0440

e-ISSN

—

Svazek periodika

135

Číslo periodika v rámci svazku

1

Stát vydavatele periodika

NL - Nizozemsko

Počet stran výsledku

6

Strana od-do

1-6

Kód UT WoS článku

000453339800001

EID výsledku v databázi Scopus

2-s2.0-85054093958