It is known that the category of Lie algebras over a ring admits algebraic exponents. The aim of this paper is to show that the same is true for the category of internal Lie algebras in an additive, cocomplete, symmetric, closed, monoidal category. In this way, we add some new examples to the brief list of known locally algebraically cartesian closed categories, including the categories of Lie superalgebras and differentially graded Lie algebras amongst others. Note that we are mainly interested in the case where the underlying category is abelian, as is the case in all our examples, but do not impose this condition since not requiring it adds no complexity to our arguments.
Keywords: locally algebraically cartesian closed, semi-abelian category, algebraic exponentiation, Lie algebra
2020 MSC: 18E13,16W25,17A32, 18M05
Theory and Applications of Categories, Vol. 36, 2021, No. 11, pp 288-305.