By Andreas Arvanitogeorgos

ISBN-10: 0821827782

ISBN-13: 9780821827789

It really is extraordinary that lots approximately Lie teams can be packed into this small ebook. yet after interpreting it, scholars can be well-prepared to proceed with extra complex, graduate-level issues in differential geometry or the idea of Lie groups.

The idea of Lie teams comprises many parts of arithmetic: algebra, differential geometry, algebraic geometry, research, and differential equations. during this e-book, Arvanitoyeorgos outlines adequate of the necessities to get the reader all started. He then chooses a direction via this wealthy and various thought that goals for an realizing of the geometry of Lie teams and homogeneous areas. during this approach, he avoids the additional aspect wanted for an intensive dialogue of illustration theory.

Lie teams and homogeneous areas are particularly valuable to review in geometry, as they supply very good examples the place amounts (such as curvature) are more uncomplicated to compute. a superb figuring out of them offers lasting instinct, specially in differential geometry.

The writer offers numerous examples and computations. issues mentioned comprise the type of compact and attached Lie teams, Lie algebras, geometrical elements of compact Lie teams and reductive homogeneous areas, and significant sessions of homogeneous areas, comparable to symmetric areas and flag manifolds. purposes to extra complicated themes also are integrated, corresponding to homogeneous Einstein metrics, Hamiltonian structures, and homogeneous geodesics in homogeneous spaces.

The booklet is appropriate for complex undergraduates, graduate scholars, and examine mathematicians attracted to differential geometry and neighboring fields, reminiscent of topology, harmonic research, and mathematical physics.

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

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**Additional info for An Introduction to Lie Groups and the Geometry of Homogeneous Spaces**

**Sample text**

That is, Aej is left fixed by Tj, so Aej = Ajej for some complex number Aj of modulus 1 (as A E U(n)), therefore ) = e4j. Since this is true for each j, this means that A = diag(ez01, ... , so A E T Hence T is maximal. (2) A maximal torus of S U (n) is the set T = {diag(e6',. +On =0}, hence the rank of SU(n) is n - 1. This is a torus because of the isomorphism diag(eiel , ... , eiOn) f--* diag(ez(e1 -en) , .. , ei(en-1-en)) that 5. Classification of compact Lie groups 39 maps T onto the maximal torus in U (n - 1).

5) A maximal torus in Sp(n) is the set {diag(e9',.. , ezan) }. Its rank is also n. 15 and the examples above justify the simplifications in the computations for the Killing form that we did in the previous section. 5. The classification of compact and connected Lie groups All groups in this section are assumed to be compact and connected. We will present the classification theorem for such Lie groups. For a detailed presentation we refer to [ Definition. A Lie group is called simple if it is non-abelian and it does not contain any proper normal Lie subgroups.

Then the map X9 F (g, dL9-1(X9)) is the desired isomorphism. 0 The following special class of vector fields on a Lie group will play an important role from now on. Definition. A vector field X on a Lie group G is left-invariant if X o La = dLa (X) for all a E G, or more explicitly Xag = (dLa)g(Xg) for all a, g e G. A left-invariant vector field has the important property that it is determined by its value at the identity element e of the Lie group, since Xa =dLa (Xe) for all a E G. Also, since multiplication in G is smooth, so is a left-invariant vector field.

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

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