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Lectures on exceptional Lie groups. Ed. by Zafer Mahmud and Mamoru Mimura. (English) Zbl 0866.22008

Chicago Lectures in Mathematics. Chicago, IL: University of Chicago Press. xiv, 122 p. (1996).
The book came out of a set of notes of various lectures given by Adams on Exceptional Lie groups. They were edited by Mahmud and Mimura. Although most of the material can be read without any prior knowledge, some parts of J. F. Adams’ [Lectures on Lie Groups, Math. Lect. Note Series, New York-Amsterdam 1969; Zbl 0206.31604] are used every now and then. The book begins with a synoptical chapter on (mostly reductive) complex and real Lie groups and their representations. Next, the construction of Spin groups is carried out via the Clifford algebra. The Spin group of an (complex or real) orthogonal space \(V,(\cdot,\cdot)\) is defined as the Lie group of all elements in the Clifford algebra which have values 1 under a certain norm function and which ”conjugate” the subspace \(V\) into itself. It is shown (and used as a motivation) that the homogeneous 2 part of the Clifford algebra is the Lie algebra of \(Spin(V)\). This, and some properties (e.g., characters) as well as some applications make up Chapters 2-5. The applications in Chapter 5 comprise exceptional isomorphisms between low rank simple Lie groups, and the construction of \(G_2\) as the stabilizer in \(Spin(7)\) of a vector in the spin representation. Next, the Lie group \(E_8\) is constructed by use of the centralizer of an involution of type \(D_8\). The restriction of the adjoint representation of \(E_8\) to this half spin group \(HSpin(16)\cong Spin(16)/{\mathbb{Z}}_2\) decomposes into a 120-dimensional Lie algebra and a 128-dimensional half spin representation. But these two irreducible representations for \(HSpin(16)\) have been analysed in previous chapters, and so starting with this sum of two irreducibles, a Lie algebra is manufactured, which is proven to be of type \(E_8\). It is noted that similar constructions can be done with centralizers of involutions in \(F_4\), \(E_6\), and \(E_7\), of types \(B_4\), \(D_5T_1\), and \(D_6T_1\), respectively (all three involve spin groups). But rather than repeating the direct construction, Adams finds the three remaining exceptional groups from the commuting pairs \((G_2,F_4)\), \((A_2,E_6)\), and \((A_1,A_7)\). Thus, for instance \(E_7\) is found as the connected centralizer of an \(A_1\) diagonally embedded in the \(A_1A_1\) factor of the product \(D_6A_1A_1\) of subgroups in the subgroup \(HSpin(16)\) (of type \(D_8\)) of \(E_8\). (For \(F_4\) the construction above for \(G_2\) is needed.) One of the advantages of the constructions given is an explicit calculational machinery for the maximal torus coming from Clifford algebra. In Chapter 11, the smallest faithful representations of \(E_7\) and \(E_6\) (of degrees 56 and 27, respectively) are studied, especially their restrictions to the subgroups of type \(A_7\) and \(A_2A_2A_2\), respectively. Then, in Chapters 12 and 13, this point of view is used to construct \(E_7\) and \(E_6\), respectively, in their smallest dimensional faithful representations. The ”trick” here, besides the usual quartic and cubic form in the respective cases, is to keep the Lie algebra at hand in the guise of endomorphisms of the representation space. In the final Chapters 14-16, the usual Jordan algebra and Cayley algebra for \(F_4\) and \(G_2\) are found from the Lie algebra for \(E_8\), by use of the embeddings of these groups in \(E_8\), and subsequently direct constructions are given. The book is concisely written, very explicit in its constructions, and very direct. It often uses the fact that the characteristic is zero (e.g., compact forms, sin and cos, \(\sqrt{-1}\) appear at various places), and so it will not be entirely straightforward to carry all arguments over to arbitrary characteristic. Additional information, e.g., on the root systems, and on real forms, is given along the way.

MSC:

22E15 General properties and structure of real Lie groups
22E10 General properties and structure of complex Lie groups
15A66 Clifford algebras, spinors
17B25 Exceptional (super)algebras
22-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to topological groups

Citations:

Zbl 0206.31604
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