Theorem
[arXiv:1506.01552].
If a division grading by an abelian group on a
finite-dimensional
real associative algebra is fine,
then either all the homogeneous components have dimension 1,
or it is a grading as a complex algebra.
Example
[arXiv:1707.05526].
The following decomposition defines
a division grading on the algebra
\( M_2(\mathbb{R}) \times \mathbb{H} \)
by the group
\( \mathbb{Z}_2 \times \mathbb{Z}_4 \):
Equation
[arXiv:1801.07002].
The isomorphism class of the real Clifford algebra
\( \operatorname{Cl}_{p,q}(\mathbb{R}) \)
is determined by the following formula:
Theorem
[arXiv:2007.13426].
Any inner automorphism in the stabilizer of a
graded-simple
unital associative algebra
whose grading group is abelian
is the conjugation by a homogeneous element.
