Unipotent - Unipotent Algebraic Groups

Unipotent Algebraic Groups

An element x of an affine algebraic group is unipotent when its associated right translation operator r_x on the affine coordinate ring A of G is locally unipotent as an element of the ring of linear endomorphism of A (Locally unipotent means that its restriction to any finite dimensional stable subspace of A is unipotent in the usual ring sense).

An affine algebraic group is called unipotent if all its elements are unipotent. Any unipotent algebraic group is isomorphic to a closed subgroup of the group of upper triangular matrices with diagonal entries 1, and conversely any such subgroup is unipotent. In particular any unipotent group is a nilpotent group, though the converse is not true (counterexample: the diagonal matrices of GL_n(k)).

If a unipotent group acts on an affine variety, all its orbits are closed, and if it acts linearly on a finite dimensional vector space then it has a non-zero fixed vector. In fact, the latter property characterizes unipotent groups.

Unipotent groups over an algebraically closed field of any given dimension can in principle be classified, but in practice the complexity of the classification increases very rapidly with the dimension, so people tend to give up somewhere around dimension 6.

Over the real numbers (or more generally any field of characteristic 0) the exponential map takes any nilpotent square matrix to a unipotent matrix. Moreover, if U is a commutative unipotent group, the exponential map induces an isomorphism from the Lie algebra of U to U itself.