Hyperalgebra of an affine algebraic group$G$ is the finite dual of the Hopf algebra of representative functions of (irreducible component containing the unit element in) $G$. It can be interpreted as (and is sometimes called) the algebra of distributions supported at unit. This algebra comes with a natural filtration. In characteristic $0$ it coincides (by L. Schwarz’s theorem) with the universal enveloping algebra of the Lie algebra of $G$, but it is much bigger in positive characteristic. It can also be obtained by base change from the Kostant’s integral form of the universal enveloping algebra of the complex Lie algebra associated to $G$.

Some books on algebraic groups and on Hopf algebras have chapters dedicated to this topic e.g.

M. Sweedler, Hopf algebras, Benjamin, NY 1969

C. Jantzen, Representations of algebraic groups, chapter 7

Η. Yanagihara, Theory of Hopf algebras attached to group schemes, Springer Lecture Notes in Mathematics 614 (1977)

Articles:

J. Sullivan, Simply connected groups, the hyperalgebra, and Verma’s conjecture, Amer. J. Math. 100 (1978) 1015-1019.

Mitsuhiro Takeuchi, Tangent coalgebras and hyperalgebras I, Japan. J. Math. 42 (1974) 1–143 pdf; On coverings and hyperalgebras of affine algebraic groups, Trans. AMS

211 (1975), 249–275; A hyperalgebraic proof of the isomorphism and isogeny theorems for reductive groups, J. Algebra 85 (1983) 179–196; Generators and relations for the hyperalgebras of reductive groups, doi

W. J. Haboush, Central differential operators of split semisimple groups over fields of positive characteristic, In: Séminaire d’Algèbre Paul Dubreil et Marie-Paule Malliavin, Springer Lecture Notes in Mathematics 795, pp 35–85 doi

Edward Cline, Brian Parshall, Leonard Scott, Cohomology, hyperalgebras, and representations, Journal of Algebra 63:1, 98-123 (1980) doi