{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,26]],"date-time":"2025-06-26T04:10:36Z","timestamp":1750911036697,"version":"3.41.0"},"reference-count":20,"publisher":"World Scientific Pub Co Pte Ltd","issue":"08","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2017,12]]},"abstract":"<jats:p>This research is a continuation of [Tsurkov, Automorphic equivalence of linear algebras, J. Algebra Appl. 13(7) (2014), doi:10.1142\/S0219498814500261]. In this paper, we consider some classical varieties of linear algebras over the field [Formula: see text] such that [Formula: see text]. We study the relation between the geometric equivalence and automorphic equivalence of the algebras of these varieties.<\/jats:p><jats:p>If we denote by [Formula: see text] one of these varieties, then [Formula: see text] is a category of the finite generated free algebras of the variety [Formula: see text]. In this paper, we calculate for the considered varieties the quotient group [Formula: see text], where [Formula: see text] is a group of all the automorphisms of the category [Formula: see text] and [Formula: see text] is a subgroup of all inner automorphisms of this category. The quotient group [Formula: see text] measures the possible difference between the geometric equivalence and automorphic equivalence of algebras from the variety [Formula: see text]. The results of this paper and [Tsurkov, Automorphic equivalence of linear algebras, J. Algebra Appl. 13(7) (2014), doi: 10.1142\/S0219498814500261] are summarized in the table at the end of Sec. 5.<\/jats:p><jats:p>We can see from this table that in all considered varieties of the linear algebras the group [Formula: see text] is generated by cosets which are presented by no more than two types of the strongly stable automorphisms of the category [Formula: see text]. The first type of automorphisms is connected with the changing of the multiplication by scalar and a the second type is connected with the changing of the multiplication of the elements of the algebras.<\/jats:p><jats:p>In Sec. 6, we present some examples of the pairs of linear algebras such that the considered strongly stable automorphisms provide the automorphic equivalence of these algebras, but these algebras are not geometrically equivalent.<\/jats:p>","DOI":"10.1142\/s021819671750045x","type":"journal-article","created":{"date-parts":[[2017,9,22]],"date-time":"2017-09-22T02:13:25Z","timestamp":1506046405000},"page":"973-999","source":"Crossref","is-referenced-by-count":6,"title":["Automorphic equivalence in the classical varieties of linear algebras"],"prefix":"10.1142","volume":"27","author":[{"given":"A.","family":"Tsurkov","sequence":"first","affiliation":[{"name":"Institute of Mathematics and Statistics, University S\u00e3o Paulo, Rua do Mat\u00e3o, 1010, Cidade Universit\u00e1ria, CEP 05508-090, S\u00e3o Paulo, SP, Brazil"},{"name":"CCET, Mathematics Department, UFRN, Av. 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