In mathematics, a unipotent element[1] r of a ring R is one such that r − 1 is a nilpotent element; in other words, (r − 1)n is zero for some n. 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.
The term quasi-unipotent means that some power is unipotent, for example for a diagonalizable matrix with eigenvalues that are all roots of unity.
An element x of an affine algebraic group is unipotent when its associated right translation operator, rx, on the affine coordinate ring A[G] of G is locally unipotent as an element of the ring of linear endomorphism of A[G].
(Locally unipotent means that its restriction to any finite-dimensional stable subspace of A[G] is unipotent in the usual ring-theoretic sense.)
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 GLn(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.
[2] In particular, this implies there are no non-trivial semisimple representations.
Using the lower central series where there are associated unipotent groups.
is a unipotent group through the embedding Notice the matrix multiplication gives hence this is a group embedding.
Over characteristic 0 there is a nice classification of unipotent algebraic groups with respect to nilpotent Lie algebras.
Recall that a nilpotent Lie algebra is a subalgebra of some
such that the iterated adjoint action eventually terminates to the zero-map.
In terms of matrices, this means it is a subalgebra
Then, there is an equivalence of categories of finite-dimensional nilpotent Lie algebras and unipotent algebraic groups.
, where given a finite-dimensional nilpotent Lie algebra, the map gives a Unipotent algebraic group structure on
In the other direction 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.
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[who?]
A group is called reductive if its unipotent radical is trivial.
Algebraic groups can be decomposed into unipotent groups, multiplicative groups, and abelian varieties, but the statement of how they decompose depends upon the characteristic of their base field.
Over characteristic 0 there is a nice decomposition theorem of a commutative algebraic group
There is a short exact sequence of groups[3]page 8 where
is, geometrically, a product of tori and algebraic groups of the form
When the characteristic of the base field is p there is an analogous statement[3] for an algebraic group
such that Any element g of a linear algebraic group over a perfect field can be written uniquely as the product g = gu gs of commuting unipotent and semisimple elements gu and gs.
In the case of the group GLn(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–Chevalley decomposition.