UNIPOTENT

In mathematics, a 'unipotent element' ''r'' of a ring ''R'' is one such that
''r'' − 1 is a nilpotent element, in other words such that some power (''r'' − 1)^''n'' is zero.
In particular a square matrix ''M'' is a 'unipotent matrix' if and only if its characteristic polynomial ''P''(''t'') is a power of ''t'' − 1. Equivalently ''M'' is unipotent if all its eigenvalues are 1.
The term 'quasi-unipotent' means that some power is unipotent, for example for a diagonalizable matrix with eigenvalues that are all roots of unity.
A 'unipotent algebraic group' is one all of whose elements are unipotent.

Contents

Unipotent algebraic groups

The unipotent radical

The Jordan decomposition

See also

References

Unipotent algebraic groups

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.

The unipotent radical

The 'unipotent radical' of an algebraic group ''G'' is the set of unipotent elements in the radical of ''G''. It is a connected unipotent normal subgroup of ''G'', and contains all other such subgroups. A group is called 'reductive' if its unipotent radical is trivial. If ''G'' is reductive then its radical is a torus.

The Jordan decomposition

Any element ''g'' of a linear algebraic group over a perfect field can be written uniquely as the product ''g'' = ''g''_''u''''g''_''s'' of commuting unipotent and semisimple elements ''g''_''u'' and ''g''_''s''. In the case of the group GL_''n''('C'), this essentially says that any invertible complex matrix is conjugate to the product of a diagonal matrix and an upper triangular one, which is (more or less) the multiplicative version of the Jordan decomposition.
Compare with the Iwasawa decomposition and Cartan decomposition (of a group and algebra).
There is also a version of the Jordan decomposition for groups:
any commutative linear algebraic group over a perfect field is the product of a unipotent group and a semisimple group.

See also

★ Deligne-Lusztig theory (for unipotent characters)

References

A. Borel, ''Linear algebraic groups'', ISBN 0-387-97370-2