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Let L be a finite-dimensional
Lie algebra over an algebraically closed field F of characteristic 0, let H(L) be the
Hopf algebra of representative functions of L, and let B(L) be the Hochschild basic
group B(L) of L.
By using Hochschild theory of H(L), we show that two such Lie algebras have
the same Hopf algebra if and only if they have the same basic group, or equivalently,
they have the same basic Lie algebra (the Lie algebra of the basic group). This is
shown by first obtaining the following characterization of B(L). If G(L) is the
pro-affine algebraic group associated with H(L), then B(L) is the quotient of
G(L) by the intersection of the radical of G(L) with the reductive part of
the center of G(L). We also show that the basic Lie algebra of L can be
constructed, up to isomorphism, directly from the adjoint representation of
L.
Finally, we apply the theory of basic groups to obtain an intrinsic
characterization of the Hopf algebras (over F) that are isomorphic to H(L) for some
Lie algebra L.
Some applications to algebraic Lie algebras are also considered.
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