Unipotent

In mathematics, a unipotent element "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. Thus all the eigenvalues of a unipotent matrix 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.
In a unipotent affine algebraic group, all elements are unipotent (see below for the definition of an element being unipotent in such a group).
Definition.
Definition with matrices.
Consider the group formula_1 of upper-triangular matrices with formula_2's along the diagonal, so they are the group of matricesformula_3then, a unipotent group can be defined as a subgroup of some formula_1. Using scheme theory the group formula_1 can be defined as the group schemeformula_6and an affine group scheme is unipotent if it is a closed group scheme of this scheme.
Definition with ring theory.
An element, "x", of an affine algebraic group is unipotent when its associated right translation operator, "r""x", 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, though the converse is not true (counterexample: the diagonal matrices of GL"n"("k")).
For example, the standard representation of formula_1 on formula_8 with standard basis formula_9 has the fixed vector formula_10.
Definition with representation theory.
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. In particular, this implies there are no non-trivial semisimple representations.
Examples.
Un.
Of course, the group of matrices formula_1 is unipotent. Using the Lower Central Seriesformula_12whereformula_13 and formula_14there are associated unipotent groups. For example, on formula_15, the central series are the matrix groupsformula_16, formula_17, formula_18, and formula_19given some induced examples of unipotent groups.
Gan.
The additive group formula_20 is a unipotent group through the embeddingformula_21Notice the matrix multiplication givesformula_22hence this is a group embedding. More generally, there is an embedding formula_23 from the mapformula_24
Using scheme theory, formula_20 is given by the functorformula_26whereformula_27
Kernel of the Frobenius.
Consider the functor formula_28 on the subcategory formula_29, there is the subfunctor formula_30 whereformula_31so it is given by the kernel of the Frobenius endomorphism.
Classification of unipotent groups over characteristic 0.
Over characteristic formula_32 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 formula_33 such that the iterated adjoint action eventually terminates to the zero-map. In terms of matrices, this means it is a subalgebra formula_34 of formula_35, the matrices with formula_36 for formula_37.
Then, there is an equivalence of categories of finite-dimensional nilpotent Lie algebras and unipotent algebraic groupspage 261. This can be constructed using the Baker–Campbell–Hausdorff series formula_38, where given a finite-dimensional nilpotent Lie algebra, the mapformula_39gives a Unipotent algebraic group structure on formula_34.
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.
Remarks.
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.
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.
Decomposition of algebraic groups.
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.
Characteristic 0.
Over characteristic formula_32 there is a nice decomposition theorem of an algebraic group formula_42 relating its structure to the structure of a linear algebraic group and an Abelian variety. There is a short exact sequence of groupspage 8formula_43where formula_44 is an abelian variety, formula_45 is of multiplicative type, meaning, and formula_46 is a unipotent group.
Characteristic p.
When the characteristic of the base field is formula_47 there is an analogous statement for an algebraic group formula_42: there exists a smallest subgroup formula_49 such that
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–Chevalley decomposition.
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.