Abstract
Let A be an n-dimensional algebra over a field k and a(A) its quantum symmetry semigroup. We prove that the automorphisms group \(\mathrm{Aut}_{\mathrm{Alg}} (A)\) of A is isomorphic to the group \(U \bigl ( G(a (A)^{\mathrm{o}} ) \bigl )\) of all invertible group-like elements of the finite dual \(a (A)^{\mathrm{o}}\). For a group G, all G-gradings on A are explicitly described and classified: the set of isomorphisms classes of all G-gradings on A is in bijection with the quotient set \( \mathrm{Hom}_{\mathrm{BiAlg}} \, \bigl ( a (A) , \, k[G] \bigl )/\approx \) of all bialgebra maps \(a (A) \, \rightarrow k[G]\), via the equivalence relation implemented by the conjugation with an invertible group-like element of \(a (A)^\mathrm{o}\).
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This work was supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS/CCCDI–UEFISCDI, Project Number PN-III-P4-ID-PCE-2020-0458, within PNCDI III.
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Militaru, G. The Automorphisms Group and the Classification of Gradings of Finite Dimensional Associative Algebras. Results Math 77, 13 (2022). https://doi.org/10.1007/s00025-021-01509-z
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DOI: https://doi.org/10.1007/s00025-021-01509-z