Universal coacting Hopf algebra of a finite-dimensional algebra over an operad

A.L. Agore and G. Militaru constructed a new invariant (a “universal coacting Hopf algebra”) for some finite-dimensional binary quadratic algebras such as Lie/Leibniz algebras, associative algebras, and Poisson algebras with prominent applications. In our recent work, we extended their construction from the binary case to Lie-Yamaguti algebras (an algebra with a binary and a ternary bracket). In this paper, we give a construction of universal coacting bi/Hopf algebra for any finite-dimensional algebra over a symmetric operad $P$. Precisely, we construct a universal algebra $C\left(a\right)$ for a finite-dimensional $P$-algebra $a$. Furthermore, we show that the category of finite dimensional $P$-algebras is enriched over the dual category of commutative algebras. This enrichment gives a unique bialgebra structure on the universal algebra $C\left(a\right)$, making it a universal coacting bialgebra of the $P$-algebra $a$. Subsequently, we obtain a universal coacting Hopf algebra of the $P$-algebra $a$. We also show that universal coacting Hopf algebra constructed here coincides with the existing cases of Lie/Leibniz, Poisson, and associative algebras. Furthermore, our operadic approach helps us construct a universal coacting algebra for algebras over a graded symmetric operad (graded algebras with finite-dimensional homogeneous components). This allows us to discuss the universal constructions for k-ary quadratic algebras and graded algebras like graded Leibniz, graded Poisson algebras, Gerstenhaber algebras, BV algebras, etc. In the end, we characterize $P$-algebra automorphisms in terms of the invertible group-like elements of the finite dual bialgebra $C\left(a\right)$. We also give a characterization of the abelian group gradings of finite dimensional $P$-algebras.