152 research outputs found
On the Graded Identities for Elementary Gradings in Matrix Algebras over Infinite Fields
We find a basis for the -graded identities of the matrix
algebra over an infinite field of characteristic with an
elementary grading such that the neutral component corresponds to the diagonal
of
Graded identities with involution for the algebra of upper triangular matrices
Let be a field of characteristic zero. We prove that if a group grading
on admits a graded involution then this grading is a coarsening of a
-grading on and the graded
involution is equivalent to the reflection or symplectic involution on
. A finite basis for the
-identities is exhibited for the
reflection and symplectic involutions and the asymptotic growth of the
-codimensions is determined. As
a consequence we prove that for any -grading on and any graded
involution the -exponent is if is even and either or
if is odd. For the algebra there are, up to equivalence,
two non-trivial gradings that admit a graded involution: the canonical
-grading and the -grading induced by . We
determine a basis for the -identities and prove that the
exponent is . Hence we conclude that the ordinary -exponent for
is
- β¦