Let \(F\) be an algebraically closed field of characteristic zero and \(L\) a finite dimensional Lie algebra over \(F\). \(H(L)\) denotes the Hopf algebra of representative functions on \(L\); that is, the subalgebra \(U(L)^0\) of the linear dual \(U(L)^*\) of the \(F\) universal enveloping algebra of \(L\) consisting of functionals vanishing on an ideal of finite codimension. \(H(L)\) is the coordinate ring of a pro-algebraic group, denoted \(G(L)\).
The author is concerned with the question of when two Lie algebras have the same Hopf algebra, or equivalently the same pro-algebraic group. \(H(L)\) contains a canonical finitely generated Hopf subalgebra, called the basic subalgebra; the corresponding affine algebraic group is called the Hochschild basic group and denoted \(B(L)\).
The author proves that \(B(L)\) can be described from \(G(L)\) as the latter modulo the intersection of its radical with the reductive part of its center; and that the Hopf algebra \(H(L)\) is determined by the algebraic group \(B(L)\). He further shows that \(B(L)\) is determined by its Lie algebra \(\text{Lie}(B(L))\). Finally, he provides a construction that produces \(\text{Lie}(B(L))\) from the adjoint representation of \(L\).
These results allow him to give a new, and simpler, characterization of the Hopf algebras of the form \(H(L)\), and to establish that an algebraic Lie algebra \(L\) (Lie algebra of an affine algebraic group) is the Lie algebra of a unique almost simply connected affine group (almost simply connected means that the radical has unipotent center and the quotient by the radical is simply connected), the group in question being \(B(L)\) modulo any direct factor vector subgroup \(Z\) of \(B(L)\) such that \(\text{Lie}(B(L))=L\oplus\text{Lie}(Z)\).