## 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)

### Keywords:

inner derivations; Lie transformation algebra

Zbl 0033.34803
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### References:

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