# What is the tangent Lie algebroid to a Lie groupoid?

How do you define the tangent Lie algebroid to a Lie groupoid?

In this online note Lie Algebroids, Lie Groupoids and Poisson Geometry by Sébastien Racanière, it states that if $t\colon G_1\to G_0$ is the target map from arrows to objects of the Lie groupoid, then since it is a submersion, its kernel is a vector bundle $T^tG_1\to G_1$. But the target map $t$ is just a smooth submersion, and its domain. Does he mean kernel of the differential of the target map $dt\colon TG_1\to TG_0$?

Also, once we have a definition of the tangent Lie algebroid, how do we induce a homomorphisms of Lie algebroids, starting from a homomorphism of Lie groupoids?

• Does he mean kernel of the differential of the target map $𝑑𝑡: TG_1 \to TG_0$? -- Yes, he does. – Max Nov 29 '19 at 19:11
• "2. Left invariant vector fields, with commutators of vector fields as Lie bracket." -- this is what he is generalizing, it's in the text. – Max Nov 29 '19 at 19:13