Classification of involutions on graded-division simple real algebras

Involutions on graded matrix algebras

Graded Algebras, Anti-involutions, Simple Groups and Symmetric Spaces

Decomposition of Involutions on Inertially Split Division Algebras

Let F be a Henselian valued field with char(F ) 6= 2, and let S be an inertially split F -central division algebra with involution σ∗ that is trivial on an inertial lift in S of the field Z(S). We prove necessary and sufficient conditions for S to contain a σ∗stable quaternion F -subalgebra, and for (S, σ∗) to decompose into a tensor product of quaternion algebras. These conditions are in terms...

Correspondences between Valued Division Algebras and Graded Division Algebras

If D is a tame central division algebra over a Henselian valued field F , then the valuation on D yields an associated graded ring GD which is a graded division ring and is also central and graded simple over GF . After proving some properties of graded central simple algebras over a graded field (including a cohomological characterization of its graded Brauer group), it is proved that the map ...

Simple Graded Commutative Algebras

We study the notion of Γ-graded commutative algebra for an arbitrary abelian group Γ. The main examples are the Clifford algebras already treated in [2]. We prove that the Clifford algebras are the only simple finitedimensional associative graded commutative algebras over R or C. Our approach also leads to non-associative graded commutative algebras extending the Clifford algebras.