Clifford algebras and their applications to Lie groups and spinors

Abstract

In these lectures, we discuss some well-known facts about Clifford algebras: matrix representations, Cartan's periodicity of 8, double coverings of orthogonal groups by spin groups, Dirac equation in different formalisms, spinors in $n$ dimensions, etc. We also present our point of view on some problems. Namely, we discuss the generalization of the Pauli theorem, the basic ideas of the method of averaging in Clifford algebras, the notion of quaternion type of Clifford algebra elements, the classification of Lie subalgebras of specific type in Clifford algebra, etc.

arXiv comment44 pages

References in corpus (13)

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• Unitary spaces on Clifford algebras
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• Quaternion typification of Clifford algebra elements
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• A classification of Lie algebras of pseudounitary groups in the techniques of Clifford algebras
• Development of the method of quaternion typification of Clifford algebra elements
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• Cambridge Lectures on Supersymmetry and Extra Dimensions
• On some Lie groups containing spin group in Clifford algebra
• Constant solutions of Yang-Mills equations and generalized Proca equations
• Method of averaging in Clifford algebras
• Quaternion types of Clifford algebra elements, basis-free approach

Cited by in corpus (10)

• Calculation of elements of spin groups using method of averaging in Clifford's geometric algebra
• On inner automorphisms preserving subspaces of Clifford algebras
• On generalization of Lipschitz groups and spin groups
• Basis-free solution to Sylvester equation in Clifford algebra of arbitrary dimension
• Generalization of Dirac conjugation in the superalgebraic theory of spinors
• On SVD and Polar Decomposition in Real and Complexified Clifford Algebras
• A Note on Centralizers and Twisted Centralizers in Clifford Algebras
• Generalized Degenerate Clifford and Lipschitz Groups in Geometric Algebras
• On Rank of Multivectors in Geometric Algebras
• On Lie Groups Preserving Subspaces of Degenerate Clifford Algebras