I'm interested in the crossover of Lie groups/differential geometry and I'm a bit confused about the relation of Lie algebras with symmetric spaces.(adsbygoogle = window.adsbygoogle || []).push({});

Take for instance the Lie group G=SL(2,R), we take the quotient by K=SO(2) as isotropic group(maximal compact subgroup) and get the symmetric space G/K= H2(hyperbolic plane).

How is it then the tangent vector space of the hyperbolic plane exactly related to the Lie algebra sl(2,R) of G (if at all)?

Thanks in advance, I would also be interested in references on introductory textbooks that treat Lie groups from the geometrical side rather than the purely abstract algebraic one.

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# Riemannian symmetric spaces and Lie algebras

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