By Luc Frappat;Antonino Sciarrino;Paul Sorba
This e-book is a close reference on Lie algebras and Lie superalgebras provided within the type of a dictionary. it truly is meant to be worthy to mathematical and theoretical physicists, from the extent of the graduate scholar upwards. The Dictionary will function the reference of selection for practitioners and scholars alike. Key gains: * Compiles and offers fabric presently scattered all through quite a few textbooks and professional magazine articles * Dictionary structure presents a simple to exploit reference at the crucial issues touching on Lie algebras and Lie superalgebras * Covers the constitution of Lie algebras and Lie superalgebras and their finite dimensional illustration thought * comprises a number of tables of the homes of person Lie algebras and Lie superalgebras
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Additional info for Dictionary on Lie Algebras and Superalgebras
An analogous analysis for SU (2) would show that SU (2) is simply connected, or that its homotopy group reduces to the identity. Moreover, it appears that SU (2) has one and only one (if we exclude the identity) invariant discrete subgroup Z2 = f;I Ig and that the quotient group SU (2)=Z2 is isomorphic to SO(3). Actually, this relationship between homotopy groups and discrete invariant subgroups (or between algebra and topology) in the theory of Lie groups is more general and allows to relate the di erent Lie groups with the same Lie algebra.
47), one knows that any Lie algebra can be seen as the semi-direct sum of a solvable Lie algebra by a semi-simple one. Let us remember that a semi-simple Lie algebra is a direct sum of simple Lie algebras. Owing to the work of Cartan one knows that any simple nite dimensional Lie algebra is one of the following types: Lie Algebras 21 AN ;1 or su(N ), where N 2 is an integer, is the Lie algebra of the group of unitary N N matrices of determinant 1. Its rank is N ; 1 and its dimension is N 2 ; 1.
Z n;r=2p 1 C C C C C C C 1A 0 1 0 } De nition The set H generated by the generators (H1 : : : Hr ) is called the Cartan subalgebra of G . The dimension of the Cartan subalgebra is the rank of the Lie algebra G : rank G = dim H Property The Cartan subalgebra H is unique up to conjugation of G . It is maximal as Abelian subalgebra of G , that is any Abelian subalgebra of G is in H (up to a conjugation). Because of its uniqueness, one can say that H is \the" maximal Abelian subalgebra of G . e. for any X 2 G , one can choose H such that X 2 H).