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Clifford algebras and Lie theory. (English) Zbl 1267.15021
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 58. Berlin: Springer (ISBN 978-3-642-36215-6/hbk; 978-3-642-36216-3/ebook). xx, 321 p. (2013).
The present book constitutes an in depth presentation of the relation between Clifford algebras and Lie groups (and Lie algebras). Given a vector space \(V\) equipped with a symmetric bilinear form \(B\), the Clifford algebra \(\text{Cl}(V; B)\) is the associative algebra generated by the elements of \(V\) with relations \(v_1v_2 +v_2v_1=2B(v_1, v_2)\) \((v_1, v_2\in V)\). If \(B=0\), this is just the exterior algebra \(\wedge(V)\), and for general \(B\) there is an isomorphism (quantization map) \(q:\wedge (V)\to\text{Cl}(V; B)\). Clifford algebras are linked to the theory of Lie groups and Lie algebras in various forms. For instance, they are used to define the spin groups \(\text{Spin}(n)\), the simply connected coverings of \(\text{SO}(n)\) for \(n\geq 3\). Well known “accidental” isomorphisms of Lie groups such as \(\text{Spin}(6)\cong \text{SU}(4)\) find natural explanations using spin representations. Furthermore, there are explicit constructions of the exceptional Lie groups \(E_6, E_7, E_8, F_4, G_2\). The book also presents further relations between Lie groups and Clifford algebras from the theory of Dirac operators on homogeneous spaces and relations to representation theory (especially of non compact Lie groups). A central goal is the presentation of the cubic Dirac operator and the Hopf-Koszul-Samelson theorem which identifies the space of invariants in the exterior algebra \((\wedge\mathfrak{g})^{\mathfrak{g}}\) with the exterior algebra over its subspace of primitive elements.
Many of the topics in the book play a role in theoretical physics and this is close to the reviewer’s personal flavour. It is directed, however, to a slightly specialized audience. The titles of the chapters are the following: Symmetric bilinear forms, Clifford algebras, The spin representations, Covariant and contravariant spinors; Enveloping algebras; Weil algebras; Quantum Weil algebras; Applications to reductive Lie algebras; The relative Dirac operator as a geometric Dirac operator; The Hopf-Koszul-Samelson theorem; The Clifford algebra of a reductive Lie algebra. There are also three appendices: Graded and filtered super spaces; Reductive Lie algebras; Background on Lie groups.

MSC:
15A66 Clifford algebras, spinors
17B20 Simple, semisimple, reductive (super)algebras
17B35 Universal enveloping (super)algebras
17B55 Homological methods in Lie (super)algebras
15-02 Research exposition (monographs, survey articles) pertaining to linear algebra
15A75 Exterior algebra, Grassmann algebras
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras
22-02 Research exposition (monographs, survey articles) pertaining to topological groups
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