By Andreas Arvanitogeorgos
It's notable that lots approximately Lie teams should be packed into this small e-book. yet after examining it, scholars might be well-prepared to proceed with extra complex, graduate-level themes in differential geometry or the idea of Lie groups.
The conception of Lie teams consists of many components of arithmetic: algebra, differential geometry, algebraic geometry, research, and differential equations. during this e-book, Arvanitoyeorgos outlines sufficient of the must haves to get the reader began. He then chooses a direction via this wealthy and numerous thought that goals for an knowing of the geometry of Lie teams and homogeneous areas. during this approach, he avoids the additional aspect wanted for a radical dialogue of illustration theory.
Lie teams and homogeneous areas are specially necessary to review in geometry, as they supply very good examples the place amounts (such as curvature) are more uncomplicated to compute. a superb realizing of them offers lasting instinct, particularly in differential geometry.
The writer presents a number of examples and computations. issues mentioned contain the category of compact and attached Lie teams, Lie algebras, geometrical features of compact Lie teams and reductive homogeneous areas, and demanding periods of homogeneous areas, corresponding to symmetric areas and flag manifolds. purposes to extra complicated issues also are integrated, equivalent to homogeneous Einstein metrics, Hamiltonian platforms, and homogeneous geodesics in homogeneous spaces.
The e-book is acceptable for complex undergraduates, graduate scholars, and examine mathematicians drawn to differential geometry and neighboring fields, equivalent to topology, harmonic research, and mathematical physics.
Readership: complicated undergraduates, graduate scholars, and study mathematicians drawn to differential geometry, topology, harmonic research, and mathematical physics
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Extra info for An Introduction to Lie Groups and the Geometry of Homogeneous Spaces
2. The adjoint representation An automorphism of a Lie group G is a map 0: G -* G that is a diffeomorphism and a group isomorphism. Let G be a Lie group and x E G. Then the map I,;: G -* G sending each g to xgx-1 is a homomorphism and, since I,; = R,;- o L, is a diffeomorphism, it is called an inner automorphism of G. Definition. The adjoint representation of G is the homomorphism Ad: G --+ Aut(g) given by Ad(g) = (dIg)e. This is a homomorphism since Icy = I,; o Iy implies that Ad,;y = Ad,, o Ady (we take differentials).
We refer to [Jac], [Wan] for more details on these. (4) Concerning the idea of the classification, here is how it goes. We know that a simply connected Lie group is determined by its Lie algebra, so the compact semisimple Lie algebras are in one-toone correspondence (up to isomorphism) with compact Lie groups. By complexifying these Lie algebras, we obtain a one-to-one correspondence between these, and the complex semisimple Lie algebras. However the complex semisimple Lie algebras are classified by their, still to come, root systems, and the root systems are classified by their bases.
3. If G is abelian, then any complex irreducible representation is one-dimensional. Proof. Let 0: G - Aut(V) be a complex irreducible representation. Since G is abelian, the map q5(g) is a G-equivariant self-map of V; therefore, q5(g) = c(g) Id for some complex scalar c(g). Since g is an arbitrary element of G, Im 0 C C* Id, so any subspace of V is G-invariant; therefore, it can be irreducible only when dim V = 1. 1. Representation theory: elementary concepts 27 Irreducible representations are the building blocks of any representation.