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\begin{document}

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\section*{hyperalgebra}

\textbf{Hyperalgebra} of an affine algebraic group $G$ is the [[finite dual]]s of the Hopf algebra of [[representative function]]s of $G$. It can be interpreted (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 algebra]]s have chapters dedicated to this topic e.g.

\begin{itemize}%
\item M. Sweedler, \emph{Hopf algebras, Benjamin, NY 1969}
\item C. Jantzen, \emph{Representations of algebraic groups}, chapter 7
\item $H$. Yanagihara, \emph{Theory of Hopf algebras attached to group schemes}, Springer Lecture Notes in Mathematics 614 (1977)

\end{itemize}
Articles:

\begin{itemize}%
\item J. Sullivan, \emph{Simply connected groups, the hyperalgebra, and Verma's conjecture}, Amer. J. Math. 100 (1978) 1015-1019.
\item M. Takeuchi, \emph{Tangent coalgebras and hyperalgebras I}, Japan. J. Math. 42 (1974) 1-143 \href{https://www.jstage.jst.go.jp/article/jjm1924/42/0/42_0_1/_pdf}{pdf}; \emph{On coverings and hyperalgebras of affine algebraic groups}, Trans. AMS 211 (1975), 249-275; \emph{A hyperalgebraic proof of the isomorphism and isogeny theorems for reductive groups}, J. Algebra 85 (1983), 179-196; \emph{Generators and relations for the hyperalgebras of reductive groups}, 
\item W. J. Haboush, \emph{Central differential operators of split semisimple groups over fields of positive characteristic}, In: S\'e{}minaire d'Alg\`e{}bre Paul Dubreil et Marie-Paule Malliavin, Springer Lecture Notes in Mathematics \textbf{795}, pp 35-85 \href{http;//dx.doi.org/10.1007/BFb0090113}{doi}
\item Edward Cline, Brian Parshall, Leonard Scott, \emph{Cohomology, hyperalgebras, and representations}, Journal of Algebra \textbf{63}:1, 98-123 (1980) 

\end{itemize}
MathOverflow: \href{http://mathoverflow.net/questions/7112/which-is-the-correct-universal-enveloping-algebra-in-positive-characteristic}{which-is-the-correct-universal-enveloping-algebra-in-positive-characteristic}

A quantum version at root of unity is proposed in

\begin{itemize}%
\item Iv\'a{}n Angiono, \emph{A quantum version of the algebra of distributions of $SL(2)$}, \href{http://arxiv.org/abs/1607.04869}{arxiv/1607.04869}

\end{itemize}
and another approach is in

\begin{itemize}%
\item W. Chin, L. Krop, \emph{Quantized hyperalgebra of rank 1}, \href{http://condor.depaul.edu/~wchin/qhyperalgebra.pdf}{pdf}; \emph{Spectra of quantized hyperalgebras}, \href{http://condor.depaul.edu/wchin/Spectra.pdf}{pdf}

\end{itemize}
[[!redirects hyperalgebras]]



\end{document}