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Results tagged with lie-algebroids
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user 7031
In differential geometry, Lie algebroids generalize on one hand Lie algebras, on the other hand the tangent bundle of a manifold: they are vector bundles equipped with an anchor map, i.e. a vector bundle morphism to the tangent bundle, and a Lie algebra structure on the space of sections subject to certain Leibniz rules. The integrated version of a Lie algebroid is a Lie groupoid. A purely algebraic version is a Lie-Rinehart algebra.
7
votes
For which algebras does \{Differential Operators\} satisfy a PBW-like theorem?
I think a sufficient condition is:
the $A$-module $Der_k(A)$ of $k$-derivations
of $A$ is projective.
A hint for the proof you may find in an old paper of G.S. Rinehart: Differential Forms on …
- 8,385
6
votes
Examples of Lie Algebroids
Here is an exotic Lie algebroid structure. Consider an inclusion of algebraic varieties $i:X\hookrightarrow Y$. Then we have a short exact sequence in $\mathcal O_X$-mod:
$$
0\to T_X\to i^*T_Y\to N\t …
- 8,385
5
votes
Accepted
What's an example of a commutative algebra over $\mathbb Q$ that fails to satisfy this versi...
Let me try to give a naive answer. Consider the following symmetric triderivation on $A=k[x]/x^2$:
$$
x\partial_x\otimes\partial_x\otimes\partial_x:(x,x,x)\mapsto x
$$
How could it be in the image of …
- 8,385