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In mathematics, a unipotent element ''r'' of a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
''R'' is one such that ''r'' − 1 is a
nilpotent element In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cla ...
; 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 In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is b ...
its characteristic polynomial ''P''(''t'') is a power of ''t'' − 1. Thus all the
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
s of a unipotent matrix are 1. The term quasi-unipotent means that some power is unipotent, for example for a
diagonalizable matrix In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) ...
with eigenvalues that are all
roots of unity In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in ...
. In the theory of algebraic groups, a group element is unipotent if it acts unipotently in a certain natural
group representation In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to ...
. A unipotent affine algebraic group is then a group with all elements unipotent.


Definition


Definition with matrices

Consider the
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
\mathbb_n of upper-triangular matrices with 1's along the diagonal, so they are the group of
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
:\mathbb_n = \left\. Then, a unipotent group can be defined as a
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of some \mathbb_n. Using
scheme theory In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different sc ...
the group \mathbb_n can be defined as the
group scheme In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have ...
:\text\left( \frac \right) and 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 In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Ma ...
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-theoretic 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 In mathematics, specifically group theory, a nilpotent group ''G'' is a group that has an upper central series that terminates with ''G''. Equivalently, its central series is of finite length or its lower central series terminates with . Intui ...
, though the converse is not true (counterexample: the
diagonal matrices In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal m ...
of GL''n''(''k'')). For example, the standard representation of \mathbb_n on k^n with standard basis e_i has the fixed vector e_1.


Definition with representation theory

If a unipotent group acts on an
affine variety In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime ideal. ...
, all its orbits are closed, and if it acts linearly on a finite-dimensional
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
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


U''n''

Of course, the group of matrices \mathbb_n is unipotent. Using the
lower central series In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, the existence of a centra ...
:\mathbb_n = \mathbb_n^ \supset \mathbb_n^ \supset \mathbb_n^ \supset \cdots \supset \mathbb_n^ = e where :\mathbb_n^ = mathbb_n,\mathbb_n/math> and \mathbb_n^ = mathbb_n, \mathbb_n^/math> there are associated unipotent groups. For example, on n = 4, the central series are the matrix groups :\mathbb_4 = \left\, \mathbb_4^ = \left\, \mathbb_4^ = \left\, and \mathbb_4^ = \left\ given some induced examples of unipotent groups.


Ga''n''

The additive group \mathbb_a is a unipotent group through the embedding :a \mapsto \begin 1 & a\\ 0 & 1 \end Notice the matrix multiplication gives :\begin 1 & a \\ 0 & 1 \end \cdot \begin 1 & b \\ 0 & 1 \end = \begin 1 & a + b \\ 0 & 1 \end hence this is a group embedding. More generally, there is an embedding \mathbb_a^n \to \mathbb_ from the map :(a_1,\ldots, a_n) \,\mapsto \begin 1 & a_1 & a_2 & \cdots & a_ &a_n \\ 0 & 1 & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & & \vdots & \vdots \\ 0 & 0 & 0 & \cdots &1 & 0 \\ 0 & 0 & 0 & \cdots &0 & 1 \end Using scheme theory, \mathbb_a is given by the
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
:\mathcal:\textbf^ \to \textbf where :(X,\mathcal_X) \mapsto \mathcal_X(X)


Kernel of the Frobenius

Consider the functor \mathcal on the subcategory \textbf/\mathbb_p, there is the subfunctor \alpha_p where :\alpha_p(X) = \ so it is given by the kernel of the
Frobenius endomorphism In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphis ...
.


Classification of unipotent groups over characteristic 0

Over characteristic 0 there is a nice classification of unipotent algebraic groups with respect to
nilpotent Lie algebra In mathematics, a Lie algebra \mathfrak is nilpotent if its lower central series terminates in the zero subalgebra. The ''lower central series'' is the sequence of subalgebras : \mathfrak \geq mathfrak,\mathfrak\geq mathfrak,[\mathfrak,\mathfrak ...
s. Recall that a nilpotent Lie algebra is a subalgebra of some \mathfrak_n such that the iterated adjoint action eventually terminates to the zero-map. In terms of matrices, this means it is a subalgebra \mathfrak of \mathfrak_n, the matrices with a_ = 0 for i \leq j. Then, there is an equivalence of categories of finite-dimensional nilpotent Lie algebras and unipotent algebraic groups.page 261 This can be constructed using the Baker–Campbell–Hausdorff formula, Baker–Campbell–Hausdorff series H(X,Y), where given a finite-dimensional nilpotent Lie algebra, the map :H:\mathfrak\times\mathfrak \to \mathfrak \text (X,Y)\mapsto H(X,Y) gives a Unipotent algebraic group structure on \mathfrak. In the other direction the exponential map (Lie theory), 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 In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
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 In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Ma ...
''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 In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group law th ...
, but the statement of how they decompose depends upon the characteristic of their base
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
.


Characteristic 0

Over characteristic 0 there is a nice decomposition theorem of an algebraic group G relating its structure to the structure of a
linear algebraic group In mathematics, a linear algebraic group is a subgroup of the group of invertible n\times n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I_n ...
and an Abelian variety. There is a
short exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context ...
of groupspage 8 :0 \to M\times U \to G \to A \to 0 where A is an abelian variety, M is of multiplicative type, meaning, and U is a unipotent group.


Characteristic ''p''

When the characteristic of the base field is ''p'' there is an analogous statement for an algebraic group G: there exists a smallest subgroup H such that # G/H is a unipotent group # H is an extension of an abelian variety A by a group M of multiplicative type. # M is unique up to commensurability in G and A is unique up to
isogeny In mathematics, in particular, in algebraic geometry, an isogeny is a morphism of algebraic groups (also known as group varieties) that is surjective and has a finite kernel. If the groups are abelian varieties, then any morphism of the underlyi ...
.


Jordan decomposition

Any element ''g'' of a linear algebraic group over a
perfect field In algebra, a field ''k'' is perfect if any one of the following equivalent conditions holds: * Every irreducible polynomial over ''k'' has distinct roots. * Every irreducible polynomial over ''k'' is separable. * Every finite extension of ''k' ...
can be written uniquely as the product ''g'' = ''g''''u''  ''g''''s'' of commuting unipotent and
semisimple In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...
elements ''g''''u'' and ''g''''s''. In the case of the group GL''n''(C), this essentially says that any invertible
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
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 In mathematics, the Jordan–Chevalley decomposition, named after Camille Jordan and Claude Chevalley, expresses a linear operator as the sum of its commuting semisimple part and its nilpotent part. The multiplicative decomposition expresses an inve ...
. 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

*
Reductive group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...
* Unipotent representation *
Deligne–Lusztig theory In mathematics, Deligne–Lusztig theory is a way of constructing linear representations of finite groups of Lie type using ℓ-adic cohomology with compact support, introduced by . used these representations to find all representations of all ...


References

*A. Borel, ''Linear algebraic groups'', * * * * {{Matrix classes Ring theory Matrix theory Algebraic groups