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On the Lie transformation algebra of monoids in symmetric monoidal categories. (English) Zbl 1336.18003

Let \(K\) be a field. The main goal of this paper is to extend the theory of inner derivations for nonassociative algebras developed by R. D. Schafer [Bull. Am. Math. Soc. 55, 769–776 (1949; Zbl 0033.34803)] to monoids over a \(K\)-linear symmetric monoidal category. Moreover, if \(A\) is an associative monoid, the author describes the Lie transformation algebra (Proposition 2.2) and inner derivations of \(A\) (Proposition 2.3). Finally, he shows (Proposition 2.5) that derivations preserve the nucleus of the monoid \(A\).

MSC:

18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
17A36 Automorphisms, derivations, other operators (nonassociative rings and algebras)

Citations:

Zbl 0033.34803
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References:

[1] H.-J. BAUES, M. JIBLADZE, A. TONKS, Cohomology of monoids in monoidal categories. Operads: Proceedings of Renaissance Conferences (Hartford, CT/ Luminy, 1995), 137-165, Contemp. Math., 202, Amer. Math. Soc., Providence, RI, 1997. · Zbl 0860.18006
[2] N. JACOBSON, Derivation algebras and multiplication algebras of semi-simple Jordan algebras. Ann. of Math. (2) 50 (1949), 866-874. · Zbl 0039.02802
[3] O. LOOS, P. H. PETERSSON, M. L. RACINE, Inner derivations of alternative algebras over commutative rings. Algebra Number Theory 2 (2008), no. 8, 927-968. · Zbl 1191.17011
[4] R. D. SCHAFER, Inner derivations of non-associative algebras. Bull. Amer. Math. Soc. 55 (1949), 769-776. · Zbl 0033.34803
[5] R. D. SCHAFER, An introduction to nonassociative algebras. Pure and Applied Mathematics, 22, Academic Press, New York-London 1966. · Zbl 0145.25601
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