Unipotent
Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a unipotent element r of a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

 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
Characteristic polynomial
In linear algebra, one associates a polynomial to every square matrix: its characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace....

 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
Diagonalizable matrix
In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1AP is a diagonal matrix...

 with eigenvalues that are all roots of unity.

A unipotent affine algebraic group is one all of whose elements are unipotent (see below for the definition of an element being unipotent in such a group).

Unipotent algebraic groups

An element x of an affine algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...

 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 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
Nilpotent group
In mathematics, more specifically in the field of group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute...

, 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.

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
Exponential map
In differential geometry, the exponential map is a generalization of the ordinary exponential function of mathematical analysis to all differentiable manifolds with an affine connection....

 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 radical

The unipotent radical of an algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...

 G is the set of unipotent elements in the radical
Radical of an algebraic group
The radical of an algebraic group is the identity component of its maximal normal solvable subgroup.- External links :*, Encyclopaedia of Mathematics...

 of G. It is a connected unipotent normal subgroup of G, and contains all other such subgroups. A group is called reductive
Reductive group
In mathematics, a reductive group is an algebraic group G over an algebraically closed field such that the unipotent radical of G is trivial . Any semisimple algebraic group is reductive, as is any algebraic torus and any general linear group...

if its unipotent radical is trivial. If G is reductive then its radical is a torus.

Jordan decomposition

Any element g of a linear algebraic group over a perfect field
Perfect field
In algebra, a field k is said to be perfect if any one of the following equivalent conditions holds:* Every irreducible polynomial over k has distinct roots.* Every polynomial over k is separable.* Every finite extension of k is separable...

 can be written uniquely as the product g = gugs 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.

There is also a version of the Jordan decomposition for groups:
any commutative linear algebraic group over a perfect field
Perfect field
In algebra, a field k is said to be perfect if any one of the following equivalent conditions holds:* Every irreducible polynomial over k has distinct roots.* Every polynomial over k is separable.* Every finite extension of k is separable...

is the product of a unipotent group and a semisimple group.
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