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Lie Algebra
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Lie Theory Summer School and Workshop - From June 15–27, 2009, the University of Ottawa will host a two week Fields Institute summer school focusing on geometric representation theory and extended affine Lie algebras. The summer school will be followed by a workshop (June 28 - July 3) featuring research talks on recent developments in these areas. Organizers: Erhard Neher and Alistair Savage.
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The workshop will feature talks on recent advances in the theory of Lie algebras and related subjects. In particular, there will be a focus on current developments in geometric representation theory. During the workshop Shrawan Kumar (University of North Carolina) will present a CRM-University of Ottawa Distinguished Lecture.
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A Lie algebra g is called semi-simple if the only solvable ideal of g is trivial. Equivalently, g is semi-simple if and only if the Killing form K(u,v) = tr(ad(u)ad(v)) is non-degenerate; here tr denotes the trace operator. When the field F is of characteristic zero, g is semi-simple if and only if every representation is completely reducible, that is for every invariant subspace of the representation there is an invariant complement (Weyl's theorem).
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Here L is a semisimple Lie algebra, and w a dominant weight. This function returns the dimension of the highest-weight module over L with highest weight w. The algorithm uses Weyl's dimension formula.
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A Lie algebra is simple if it contains no nonzero invariant Lie subalgebras A Lie algebra is semisimple if it contains no nonzero, nontrivial *commutative*, invariant Lie subalgebras. As the words might suggest, simplicity implies semisimplicity.
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In many ways, the classes of semisimple and solvable Lie algebras are at the opposite ends of the full spectrum of the Lie algebras. The Levi decomposition expresses an arbitrary Lie algebra as a semidirect product of its solvable radical and a semisimple Lie algebra, almost in a canonical way. Semisimple Lie algebras over an algebraically closed field have been completely classified through their root systems. The classification of solvable Lie algebras is a 'wild' problem, and cannot be accomplished in general.
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Lie Algebra