148 research outputs found

    On the Graded Identities for Elementary Gradings in Matrix Algebras over Infinite Fields

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    We find a basis for the GG-graded identities of the nΓ—nn\times n matrix algebra Mn(K)M_n(K) over an infinite field KK of characteristic p>0p>0 with an elementary grading such that the neutral component corresponds to the diagonal of Mn(K)M_n(K)

    Graded identities with involution for the algebra of upper triangular matrices

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    Let FF be a field of characteristic zero. We prove that if a group grading on UTm(F)UT_m(F) admits a graded involution then this grading is a coarsening of a Z⌊m2βŒ‹\mathbb{Z}^{\lfloor\frac{m}{2}\rfloor}-grading on UTm(F)UT_m(F) and the graded involution is equivalent to the reflection or symplectic involution on UTm(F)UT_m(F). A finite basis for the (Z⌊m2βŒ‹,βˆ—)(\mathbb{Z}^{\lfloor\frac{m}{2}\rfloor},\ast)-identities is exhibited for the reflection and symplectic involutions and the asymptotic growth of the (Z⌊m2βŒ‹,βˆ—)(\mathbb{Z}^{\lfloor\frac{m}{2}\rfloor},\ast)-codimensions is determined. As a consequence we prove that for any GG-grading on UTm(F)UT_m(F) and any graded involution the (G,βˆ—)(G,\ast)-exponent is mm if mm is even and either mm or m+1m+1 if mm is odd. For the algebra UT3(F)UT_3(F) there are, up to equivalence, two non-trivial gradings that admit a graded involution: the canonical Z\mathbb{Z}-grading and the Z2\mathbb{Z}_2-grading induced by (0,1,0)(0,1,0). We determine a basis for the (Z2,βˆ—)(\mathbb{Z}_2,\ast)-identities and prove that the exponent is 33. Hence we conclude that the ordinary βˆ—\ast-exponent for UT3(F)UT_3(F) is 33
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