process

188 P. Carrillo Rouse3.1 The tangent groupoidDefinition 3.8 (Tangent groupoid). Let G G .0/ be a Lie groupoid. The tangentgroupoid associated to G is the groupoid that has D G as the set of arrows andG .0/G .0/ Œ0; 1 as the units with• s T .x;;0/D .x; 0/ and r T .x;;0/D .x; 0/ at t D 0,• s T .; t/ D .s./; t/ and r T .; t/ D .r./; t/ at t ¤ 0,• the product is given by m T ..x; ; 0/; .x; ; 0// D .x; C ;0/ andm T ..; t/; .ˇ; t// D .m.; ˇ/; t/ if t ¤ 0 and if r.ˇ/ D s./,• the unit map u T W G .0/ ! G T is given by u T .x; 0/ D .x; 0/ and u T .x; t/ D.u.x/; t/ for t ¤ 0.We denote G T WD D G G .0/ .As we have seen above G T can be considered as a C 1 manifold with border. As aconsequence of the functoriality of the DNC construction we can show that the tangentgroupoid is in fact a Lie groupoid. Indeed, it is easy to check that if we identify in acanonical way D G .2/with .G T / .2/ , thenG .0/m T D D.m/; s T D D.s/; r T D D.r/; u T D D.u/where we are considering the following pair morphisms:mW ..G / .2/ ; G .0/ / ! .G ; G .0/ /;s; r W .G ; G .0/ / ! .G .0/ ; G .0/ /;uW .G .0/ ; G .0/ / ! .G ; G .0/ /:Finally, if f x g is a smooth Haar system on G , then, setting• .x;0/ WD x at .G T / .x;0/ D T x G x , and• .x;t/ WD t q x at .G T / .x;t/ D G x for t ¤ 0, where q D dim G x ,one obtains a smooth Haar system for the tangent groupoid (details may be found in[22]).We finish this section with some interesting examples of groupoids and their tangentgroupoids.Examples 3.9. 1. The tangent groupoid of a group. Let G be a Lie group consideredas a Lie groupoid, G WD G feg. In this case the normal bundle to the inclusionfeg ,! G is of course identified with the Lie algebra of the group. Hence, the tangentgroupoid is a deformation of the group in its Lie algebra:G T D g f0g tG .0; 1: