Lie Groups and Algebras with Applications to Physics, by D.H. Sattinger, O.L. Weaver

By D.H. Sattinger, O.L. Weaver

This e-book is meant as an introductory textual content near to Lie teams and algebras and their function in a variety of fields of arithmetic and physics. it really is written through and for researchers who're essentially analysts or physicists, no longer algebraists or geometers. now not that we have got eschewed the algebraic and geo­ metric advancements. yet we would have liked to provide them in a concrete approach and to teach how the topic interacted with physics, geometry, and mechanics. those interactions are, after all, manifold; we've got mentioned lots of them here-in specific, Riemannian geometry, ordinary particle physics, sym­ metries of differential equations, thoroughly integrable Hamiltonian platforms, and spontaneous symmetry breaking. a lot ofthe fabric now we have handled is commonplace and largely on hand; yet we have now attempted to guide a path among the descriptive procedure similar to present in Gilmore and Wybourne, and the summary mathematical strategy of Helgason or Jacobson. Gilmore and Wybourne tackle themselves to the physics neighborhood while Helgason and Jacobson handle themselves to the mathematical group. This booklet is an try to synthesize the 2 issues of view and tackle either audiences concurrently. we would have liked to provide the topic in a fashion that's right now intuitive, geometric, functions­ orientated, mathematically rigorous, and available to scholars and researchers with no an in depth heritage in physics, algebra, or geometry.

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Later, in Chapter 12 we construct representations of some semi-simple Lie algebras from this point of view. The advantage of this approach, we feel, is that it lends insight into the structure and representation of semi-simple Lie algebras by linking them to a bona fide physical model. Indeed, thinking in terms of elementary particles enhances ones understanding of the semi-simple Lie algebras. Let us turn to the construction of representations of su(2) from the calculus of two boson operators aI' a2 , aT, a!

The proof of this simple lemma is left to the reader. An immediate consequence is that the matrix ezA is defined for any matrix A and any complex number z. A second consequence is that log B is uniquely defined for all B such that II I - B II < 1. This follows from the fact that log z = log [1 + (z - 1)] = L( 00 n=l It (z - _l)n+l-'------'_ n converges for Iz - 11 < 1. Furthermore, since e10gz = z is an identity in Iz - 11 < 1, e10gB = B is an identity on liB - I II < 1. 3. The mapping B ~ log B is an analytic mapping of the neighborhood U = {Bill B-1 II < I} of the identity in GL(n, C) into the algebra gl(n, C).

The operator Q is multiplication by x; thus (Qf)(x) = xf(x). The commutation relations are [P,QJ=~. Introducing P+Q a=--, j2 Q-P a*=-- j2 we have [a, a*] = ~. The Lie algebra {P, Q, ~} is called the Heisenberg algebra and plays a fundamental role in quantum mechanics (see Chapter 4). 26 2. Lie Algebras 9. Some Important Concepts and Examples in Lie Algebras (A) We say that 5 is a subalgebra of 9 if 5 is closed under commutation, that is, if [Sl' S2] E 5 whenever Sl and S2 belong to 5. We often abbreviate this by writing [5,5] c 5.

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