Graded identities with involution for the algebra of upper triangular matrices.

Let F be a field of characteristic zero and let m ≥ 2 be an integer. In this paper, we prove that if a group grading on U T m (F) admits a graded involution then this grading is a coarsening of a Z ⌊ m 2 ⌋ -grading on U T m (F) and the graded involution is equivalent to the reflection or symplectic involution on U T m (F) , this grading is called the finest grading on U T m (F). Furthermore, if m ≤ 4 the algebra U T m (F) with the finest grading satisfies no non-trivial monomial identities. For the finest grading, a finite basis for the (Z ⌊ m 2 ⌋ , ⁎) -identities is exhibited with the reflection and symplectic involutions and the asymptotic growth of the (Z ⌊ m 2 ⌋ , ⁎) -codimensions is determined. As a consequence, we prove that for any G -grading on U T m (F) and any graded involution the (G , ⁎) -exponent is m. Finally, we study the algebra U T 3 (F). For this algebra, there are, up to equivalence, two non-trivial gradings that admit a graded involution: the canonical Z -grading and the Z 2 -grading induced by (0 , 1 , 0). We determine a basis for the (Z 2 , ⁎) -identities and we compute the codimension sequence for the (Z 2 , ⁎) -graded identities for U T 3 (F). [ABSTRACT FROM AUTHOR]