Weyl Groups of some Hyperbolic Kac-Moody Algebras

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Group Theory, Rings and Algebras, Representation Theory


We use the theory of Clifford algebras and Vahlen groups to study Weyl groups of hyperbolic Kac-Moody algebras T ++ n , obtained by a process of double extension from a Cartan matrix of finite type Tn, whose corresponding generalized Cartan matrices are symmetric.



  • Introduction
  • Generalities on orthogonal geometries
  • Clifford algebras, Pin and Spin groups
  • The Clifford group
  • Abstract Pin and Spin groups
  • Pin and Spin groups
  • Lorentzian geometry over R
  • Change of fields
  • Vahlen groups
  • Vahlen groups
  • Change of fields
  • Generalized Cartan matrices, system of simple roots and Weyl groups
  • A useful normalization
  • Canonical Lorentzian extensions
  • Change of fields
  • Weyl groups of the hyperbolic canonical Lorentzian extensions T ++ n with symmetric Cartan matrices
  • Spinor norm of outer autormorphisms
  • Connections with previous descriptions of the Weyl group
  • Paravectors
  • Vahlen groups for paravectors
  • Hermitian matrices
  • References