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

By Andreas Arvanitogeorgos

It really is extraordinary that quite a bit approximately Lie teams will be packed into this small booklet. yet after interpreting it, scholars might 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 parts of arithmetic: algebra, differential geometry, algebraic geometry, research, and differential equations. during this ebook, Arvanitoyeorgos outlines adequate of the necessities to get the reader began. He then chooses a course via this wealthy and various thought that goals for an knowing of the geometry of Lie teams and homogeneous areas. during this manner, he avoids the additional element wanted for a radical dialogue of illustration theory.

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

The writer presents numerous examples and computations. themes mentioned comprise the class of compact and attached Lie teams, Lie algebras, geometrical points of compact Lie teams and reductive homogeneous areas, and critical periods of homogeneous areas, akin to symmetric areas and flag manifolds. functions to extra complicated subject matters also are incorporated, resembling homogeneous Einstein metrics, Hamiltonian platforms, and homogeneous geodesics in homogeneous spaces.

The ebook is appropriate for complex undergraduates, graduate scholars, and study mathematicians attracted to differential geometry and neighboring fields, akin to topology, harmonic research, and mathematical physics.

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

Show description

Read or Download An Introduction to Lie Groups and the Geometry of Homogeneous Spaces PDF

Similar linear books

Linear Models: An Integrated Approach

Linear versions: An built-in process goals to supply a transparent and deep figuring out of the overall linear version utilizing easy statistical principles. based geometric arguments also are invoked as wanted and a evaluation of vector areas and matrices is equipped to make the therapy self-contained. advanced, matrix-algebraic equipment, reminiscent of these utilized in the rank-deficient case, are changed through statistical proofs which are extra obvious and that express the parallels with the straightforward linear version.

Graphs and Matrices

While it's a moot aspect among researchers, linear algebra is a crucial part within the learn of graphs. This booklet illustrates the splendor and gear of matrix innovations within the research of graphs by way of numerous effects, either classical and up to date. The emphasis on matrix thoughts is bigger than different usual references on algebraic graph idea, and the $64000 matrices linked to graphs equivalent to prevalence, adjacency and Laplacian matrices are handled intimately.

Theory of Operator Algebras II

To the Encyclopaedia Subseries on Operator Algebras and Non-Commutative Geometry the idea of von Neumann algebras used to be initiated in a chain of papers via Murray and von Neumann within the 1930's and 1940's. A von Neumann algebra is a self-adjoint unital subalgebra M of the algebra of bounded operators of a Hilbert house that's closed within the susceptible operator topology.

Clifford algebras and spinor structures : a special volume dedicated to the memory of Albert Crumeyrolle (1919-1992)

This quantity is devoted to the reminiscence of Albert Crumeyrolle, who died on June 17, 1992. In organizing the amount we gave precedence to: articles summarizing Crumeyrolle's personal paintings in differential geometry, normal relativity and spinors, articles which offer the reader an idea of the intensity and breadth of Crumeyrolle's study pursuits and impact within the box, articles of excessive medical caliber which might be of common curiosity.

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

Sample text

If the Lie groups G1, G2 are simply connected (i. e. their fundamental groups are trivial), then G1 is isomorphic to G2. 22 Lie Groups Remarks. 14 is a consequence of Ado's theorem which states that any finite-dimensional (abstract) real Lie algebra is isomorphic to a Lie subalgebra of the Lie algebra GL,zR for sufficiently large n. (b) Concerning part (2), if j : H - G is the immersion of H in G, then the required subalgebra is dje (1)) . Conversely, if [3 is a Lie subalgebra of g, then the required subgroup is the one generated by {exp tX : X E [3 J.

We will see soon that, for familiar groups, the adjoint representation is a very familiar representation. By taking the derivative of Ad we obtain a representation of g. Definition. The adjoint representation of g is the homomorphism ad: g -* End(g) given by ad(X) = (dAd)6(X). 2. The adjoint representation 29 LetZ(G)={gEG:gh=hgforallhEG}andZ(g)={XE g : [X, Y] = 0 for all Y E g} denote the centers of G and g respectively. 7. Let G be a connected Lie group. Then ker Ad = Z(G) and ker ad = Z(g). 8.

Definition. Let (G, V) be a representation. A subspace U of V is called invariant or G-invariant if g U C U for all g E G. A representation (G, V) has always at least two invariant subspaces, namely {0} and V The first is called the trivial subspace. 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.

Download PDF sample

Rated 4.11 of 5 – based on 33 votes