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

By Andreas Arvanitogeorgos

It truly is amazing that rather a lot approximately Lie teams will be packed into this small ebook. yet after examining it, scholars can be well-prepared to proceed with extra complicated, graduate-level subject matters in differential geometry or the speculation of Lie groups.

The idea of Lie teams consists of many components of arithmetic: algebra, differential geometry, algebraic geometry, research, and differential equations. during this e-book, Arvanitoyeorgos outlines adequate of the must haves to get the reader begun. He then chooses a course via this wealthy and various idea that goals for an realizing of the geometry of Lie teams and homogeneous areas. during this means, he avoids the additional aspect wanted for an intensive dialogue of illustration theory.

Lie teams and homogeneous areas are specially invaluable to review in geometry, as they supply very good examples the place amounts (such as curvature) are more straightforward to compute. an outstanding knowing of them offers lasting instinct, specially in differential geometry.

The writer offers numerous examples and computations. themes mentioned comprise the type of compact and attached Lie teams, Lie algebras, geometrical features of compact Lie teams and reductive homogeneous areas, and significant periods of homogeneous areas, similar to symmetric areas and flag manifolds. purposes to extra complex subject matters also are integrated, equivalent to homogeneous Einstein metrics, Hamiltonian platforms, and homogeneous geodesics in homogeneous spaces.

The e-book is appropriate for complicated undergraduates, graduate scholars, and examine mathematicians drawn to differential geometry and neighboring fields, resembling topology, harmonic research, and mathematical physics.

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

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

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Recall that a representation of G defines an action on the vector space V and viceversa). Similarly, we can define the standard representations of the groups GLn C, SU(n) and U(n) on C. A representation is called trivial if each group element acts as the identity transformation. It is denoted by 1. Maximal Tori and the Classification Theorem 32 There is some notation used for the standard representations: denote by A,,, the standard representation of GL,,,IIS, and by A,,, the standard representation of 80(n).

1) We compute the Killing form of SU(2). zu(2) (in this example at the diagonal elements). su(2) used in example (1) of the previous section, and we compute that ad 0 (iO -Ze 0 0 0 0 o 0 -2e 0 20 0 Then a simple calculation using this basis gives that if X = io and Y = 0 (iO o 0 -Ze then B(X, Y) = tr(ad(X)ad(Y)) = -880 = 4 tr XY. (2) The Killing form of U(2). 0 0 i i 0 then 4(0101+0202)-2(01+ 02)(01 + 02) = 4trXY - 2trXtrY. Notice that if 01 = 02 = 01 = c2 = 1, then B(X, Y) = 0, so U(2) is not semisimple.

Ei(en-1-en)) that 5. Classification of compact Lie groups 39 maps T onto the maximal torus in U (n - 1). Maximality is shown as before. 15 can be seen easily in this case: A standard result of linear algebra says that any A E SU(n) can be diagonalized, that is, there is a U E SU(n) with UAU-1 E T. (3) Let rot B = cos B sin B . Then a maximal torus in - sin B cos B SO(2n + 1) is the set of "block matrices" {diag(rot B1, ... , rot On, 1)}. The rank of SO (2n + 1) is n. (4) A maximal torus in SO(2n) is the set {diag(rot B1, ...

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