Cartan algebras and involutions

Algebras and Involutions

• Vectorspaces over division rings • Matrices, opposite rings • Semi-simple modules and rings • Semi-simple algebras • Reduced trace and norm • Other criteria for simplicity • Involutions • Brauer group of a field • Tensor products of fields • Crossed product construction of simple algebras • Cyclic algebra construction of simple algebras • Quaternion algebras • Examples • Unramified extensions...

Hyperbolic Monodromy Groups for the Hypergeometric Equation and Cartan Involutions

We give a criterion which ensures that a group generated by Cartan involutions in the automorph group of a rational quadratic form of signature (n−1, 1) is “thin”, namely it is of infinite index in the latter. It is based on a graph defined on the integral Cartan root vectors, as well as Vinberg’s theory of hyperbolic reflection groups. The criterion is shown to be robust for showing that many ...

Cartan Subalgebras in C*-Algebras

According to J. Feldman and C. Moore’s wellknown theorem on Cartan subalgebras, a variant of the group measure space construction gives an equivalence of categories between twisted countable standard measured equivalence relations and Cartan pairs, i.e., a von Neumann algebra (on a separable Hilbert space) together with a Cartan subalgebra. A. Kumjian gave a C∗-algebraic analogue of this theore...

Involutions of reductive Lie algebras

Let G be a reductive group over a field of characteristic 6= 2, let g = Lie(G), let θ be an involutive automorphism of G and let g = k⊕p be the associated symmetric space decomposition. For k = C, Kostant and Rallis studied [17] properties of orbits, centralizers, and invariants related to the (−1) eigenspace p. In this paper, we generalise [17] to the case of good positive characteristic. Amon...

Involutions on graded matrix algebras