March 7, 2017

An Introduction to Lie Groups and the Geometry of by Andreas Arvanitogeorgos

By Andreas Arvanitogeorgos

It's awesome that rather a lot approximately Lie teams can be packed into this small e-book. yet after studying it, scholars may be well-prepared to proceed with extra complicated, graduate-level subject matters in differential geometry or the idea of Lie groups.

The concept of Lie teams contains many components of arithmetic: algebra, differential geometry, algebraic geometry, research, and differential equations. during this publication, Arvanitoyeorgos outlines sufficient of the necessities to get the reader all started. He then chooses a direction via this wealthy and various conception that goals for an figuring out of the geometry of Lie teams and homogeneous areas. during this method, he avoids the additional element wanted for an intensive dialogue of illustration theory.

Lie teams and homogeneous areas are in particular invaluable to check in geometry, as they supply first-class examples the place amounts (such as curvature) are more uncomplicated to compute. a superb knowing of them presents lasting instinct, in particular in differential geometry.

The writer offers numerous examples and computations. subject matters mentioned comprise the category of compact and hooked up Lie teams, Lie algebras, geometrical points of compact Lie teams and reductive homogeneous areas, and demanding periods of homogeneous areas, similar to symmetric areas and flag manifolds. purposes to extra complex issues also are integrated, resembling homogeneous Einstein metrics, Hamiltonian structures, and homogeneous geodesics in homogeneous spaces.

The publication is acceptable for complicated undergraduates, graduate scholars, and learn mathematicians attracted to differential geometry and neighboring fields, reminiscent of topology, harmonic research, and mathematical physics.

Readership: complicated undergraduates, graduate scholars, and examine mathematicians attracted to differential geometry, topology, harmonic research, and mathematical physics

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Extra resources for An Introduction to Lie Groups and the Geometry of Homogeneous Spaces

Example text

3) If T is a connected Lie subgroup of a compact Lie group G whose Lie algebra is a maximal abelian subalgebra of g, then T is a maximal torus in G. 15. Let G be a compact and connected Lie group. Then: (a) Any element in G is contained in some maximal torus. (b) Any two maximal Lori are conjugate. That is, if T1, T2 are maximal Lori in G, then there exists an element g c G such that gT1 g -1 = T2. From (b) in the above theorem, we see that every maximal torus has the same dimension. So this is an invariant for a compact and connected Lie group.

Let G be a compact and connected Lie group. Then: (a) Any element in G is contained in some maximal torus. (b) Any two maximal Lori are conjugate. That is, if T1, T2 are maximal Lori in G, then there exists an element g c G such that gT1 g -1 = T2. From (b) in the above theorem, we see that every maximal torus has the same dimension. So this is an invariant for a compact and connected Lie group. Hence we can define: Definition. The rank of a compact and connected Lie group is the dimension of a maximal torus.

I. Representation theory: elementary concepts 25 Definition. A representation is called irreducible if the only invariant subspaces are {0} and V Definition. , there exists a linear isomorphism A : V1 -* V2 such that A(01(g) (v)) = 02 (9)(A (v)), for all g E G and v E V1. In shorthand A01 = 02A. This means that the following diagram is commutative: Given two representations (G, V) and (G, W) of G, we can define the following representations on the corresponding vector spaces with the obvious notations: (a) Dual space: V* with (v, g v') = (g-1 v, v') for all v E V, v'EV* (b) Direct sum: V ® W with g (x, y) = (gx, gy).

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