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Article overview
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Universal Cartan-Lie algebroid of an anchored bundle with connection and compatible geometries | | Alexei Kotov
; Thomas Strobl
; | | Date: |
11 Apr 2019 | | Abstract: | Consider an anchored bundle $(E,
ho)$, i.e. a vector bundle $E o M$
equipped with a bundle map $
ho colon E o 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 $FR(E)supset E$. We adapt his construction to the case of an
anchored bundle equipped with an arbitrary connection, $(E,
abla)$, and show
that it gives rise to a unique connection $ ilde
abla$ on $FR(E)$ which is
compatible with its Lie algebroid structure, thus turning $(FR(E), ilde
abla)$ into a Cartan-Lie algebroid. Moreover, this construction is universal:
any connection-preserving vector bundle morphism from $(E,
abla)$ to a
Cartan-Lie Algebroid $(A,ar
abla)$ factors through a unique Cartan-Lie
algebroid morphism from $(FR(E), ilde
abla)$ to $(A,ar
abla)$.
Suppose that, in addition, $M$ is equipped with a geometrical structure
defined by some tensor field $t$ which is compatible with $(E,
ho,
abla)$ 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 $(E,
ho)$, this condition implies that $M$ carries
a Riemannian foliation. %In general, the compatibility of a tensor $t$ with
$(E,
ho,
abla)$ implies its adequate invariance transversal to $
ho(E)$. 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 $(FR(E), ilde
abla)$. | | Source: | arXiv, 1904.5809 | | Services: | Forum | Review | PDF | Favorites | |
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