Semi-Simple Lie Algebras and Their Representations
Designed to acquaint students of particle physics already familiar with SU(2) and SU(3) with techniques applicable to all simple Lie algebras, this text is especially suited to the study of grand unification theories. Author Robert N. Cahn, who is affiliated with the Lawrence Berkeley National Laboratory in Berkeley, California, has provided a new preface for this edition. Subjects include the killing form, the structure of simple Lie algebras and their representations, simple roots and the Cartan matrix, the classical Lie algebras, and the exceptional Lie algebras. Additional topics include Casimir operators and Freudenthal's formula, the Weyl group, Weyl's dimension formula, reducing product representations, subalgebras, and branching rules. 1984 edition.
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A1 and A2 adjoint representation An_1 anti-symmetric product bilinear invariant branching rules calculate Cartan matrix Cartan subalgebra Casimir operator Chapter commutation relations conjugacy classes contains corresponding deﬁne deﬁnition deleting determine diagonal dimension dual space Dynkin coefﬁcients Dynkin diagram eigenvalue eigenvector eight dimensional elements of H embedding example exp(M extended diagram ﬁnd ﬁnding ﬁrst formula full algebra integers invariant bilinear form irreducible component irreducible representation JACOBSON Killing form linearly independent lowering operators mapping maximal semi-simple subalgebras maximal subalgebra non-negative obtained orthogonal positive roots product representation reducible reﬂection regular subalgebra representation of G representation of SU(2 representation whose highest representation with highest root space root vectors scalar product semi-simple Lie algebras simple Lie algebra simple roots speciﬁc SU(n subspace sufﬁces Suppose three dimensional representation three-by-three traceless vector space weight space weight vector Weyl group zero