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</style><div role="note" class="hatnote navigation-not-searchable">This article is about the algebraic term. For a biological cell having the capacity to develop into only one cell type, see <a href="Cell_potency#Unipotency" title="Cell potency">Cell potency § Unipotency</a>.</div>

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>unipotent element</b><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <i>r</i> of a <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a> <i>R</i> is one such that <i>r</i> − 1 is a <a href="Nilpotent_element" class="mw-redirect" title="Nilpotent element">nilpotent element</a>; in other words, (<i>r</i> − 1)<sup><i>n</i></sup> is zero for some <i>n</i>.

</p><p>In particular, a <a href="Square_matrix" title="Square matrix">square matrix</a> <i>M</i> is a <b>unipotent matrix</b> <a href="If_and_only_if" title="If and only if">if and only if</a> its <a href="Characteristic_polynomial" title="Characteristic polynomial">characteristic polynomial</a> <i>P</i>(<i>t</i>) is a power of <i>t</i> − 1. Thus all the <a href="Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> of a unipotent matrix are 1.

</p><p>The term <b>quasi-unipotent</b> means that some power is unipotent, for example for a <a href="Diagonalizable_matrix" title="Diagonalizable matrix">diagonalizable matrix</a> with eigenvalues that are all <a href="Roots_of_unity" class="mw-redirect" title="Roots of unity">roots of unity</a>.

</p><p>In the theory of <a href="Algebraic_groups" class="mw-redirect" title="Algebraic groups">algebraic groups</a>, a group element is <b>unipotent</b> if it acts unipotently in a certain natural <a href="Group_representation" title="Group representation">group representation</a>. A <b>unipotent affine algebraic group</b> is then a group with all elements unipotent.

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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Definition_with_matrices">Definition with matrices</h3></div>

<p>Consider the <a href="Group_(mathematics)" title="Group (mathematics)">group</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {U} _{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {U} _{n}}</annotation>

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</math></span><img src="./e86c30194acc8d4860819d47d0a9e3845449a850.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.897ex; height:2.509ex;" alt="{\displaystyle \mathbb {U} _{n}}" loading="lazy"></span> of <a href="Upper-triangular_matrix" class="mw-redirect" title="Upper-triangular matrix">upper-triangular matrices</a> with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

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<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

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</math></span><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"></span>'s along the diagonal, so they are the group of <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a><sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {U} _{n}=\left\{{\begin{bmatrix}1&*&\cdots &*&*\\0&1&\cdots &*&*\\\vdots &\vdots &&\vdots &\vdots \\0&0&\cdots &1&*\\0&0&\cdots &0&1\end{bmatrix}}\right\}.}</annotation>

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</math></span><img src="./c01c89d341a4bb14b0bc4cc4438c42f0818a5192.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.838ex; width:31.676ex; height:16.843ex;" alt="{\displaystyle \mathbb {U} _{n}=\left\{{\begin{bmatrix}1&*&\cdots &*&*\\0&1&\cdots &*&*\\\vdots &\vdots &&\vdots &\vdots \\0&0&\cdots &1&*\\0&0&\cdots &0&1\end{bmatrix}}\right\}.}" loading="lazy"></span></dd></dl>

<p>Then, a <b>unipotent group</b> can be defined as a <a href="Subgroup" title="Subgroup">subgroup</a> of some <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {U} _{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {U} _{n}}</annotation>

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</math></span><img src="./e86c30194acc8d4860819d47d0a9e3845449a850.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.897ex; height:2.509ex;" alt="{\displaystyle \mathbb {U} _{n}}" loading="lazy"></span>. Using <a href="Scheme_(mathematics)" title="Scheme (mathematics)">scheme theory</a> the group <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {U} _{n}}">

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<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Spec}}\left({\frac {\mathbb {C} \!\left[x_{11},x_{12},\ldots ,x_{nn},{\frac {1}{\text{det}}}\right]}{(x_{ii}=1,x_{i>j}=0)}}\right)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{Spec}}\left({\frac {\mathbb {C} \!\left[x_{11},x_{12},\ldots ,x_{nn},{\frac {1}{\text{det}}}\right]}{(x_{ii}=1,x_{i>j}=0)}}\right)}</annotation>

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<p>and an affine group scheme is unipotent if it is a closed group scheme of this scheme.

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<div class="mw-heading mw-heading3"><h3 id="Definition_with_ring_theory">Definition with ring theory</h3></div>

<p>An element <i>x</i> of an affine <a href="Algebraic_group" title="Algebraic group">algebraic group</a> is unipotent when its associated right translation operator, <i>r</i><sub><i>x</i></sub>, on the <a href="Affine_coordinate_ring" class="mw-redirect" title="Affine coordinate ring">affine coordinate ring</a> <i>A</i>[<i>G</i>] of <i>G</i> is locally unipotent as an element of the ring of <a href="Linear_map" title="Linear map">linear endomorphism</a> of <i>A</i>[<i>G</i>]. (Locally unipotent means that its restriction to any finite-dimensional stable subspace of <i>A</i>[<i>G</i>] is unipotent in the usual ring-theoretic sense.)

</p><p>An affine algebraic group is called <b>unipotent</b> if all its elements are unipotent. Any unipotent algebraic group is <a href="Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> to a closed subgroup of the group of upper triangular matrices with diagonal entries 1, and <a href="Converse_(logic)" title="Converse (logic)">conversely</a> any such subgroup is unipotent. In particular any unipotent group is a <a href="Nilpotent_group" title="Nilpotent group">nilpotent group</a>, though the converse is not true (counterexample: the <a href="Diagonal_matrices" class="mw-redirect" title="Diagonal matrices">diagonal matrices</a> of GL<sub><i>n</i></sub>(<i>k</i>)).

</p><p>For example, the standard representation of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {U} _{n}}">

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<div class="mw-heading mw-heading3"><h3 id="Definition_with_representation_theory">Definition with representation theory</h3></div>

<p>If a unipotent group acts on an <a href="Affine_variety" title="Affine variety">affine variety</a>, all its orbits are closed, and if it acts linearly on a finite-dimensional <a href="Vector_space" title="Vector space">vector space</a> then it has a non-zero fixed vector. In fact, the latter property characterizes unipotent groups.<sup id="cite_ref-:0_2-1" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> In particular, this implies there are no non-trivial <a href="Semisimple_representation" title="Semisimple representation">semisimple representations</a>.

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<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<p>Of course, the group of matrices <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {U} _{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {U} _{n}=\mathbb {U} _{n}^{(0)}\supset \mathbb {U} _{n}^{(1)}\supset \mathbb {U} _{n}^{(2)}\supset \cdots \supset \mathbb {U} _{n}^{(m)}=e}</annotation>

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</math></span><img src="./0dfcc78589318d6a2339b5cf43ee61b2297f6bce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:41.962ex; height:3.343ex;" alt="{\displaystyle \mathbb {U} _{n}=\mathbb {U} _{n}^{(0)}\supset \mathbb {U} _{n}^{(1)}\supset \mathbb {U} _{n}^{(2)}\supset \cdots \supset \mathbb {U} _{n}^{(m)}=e}" loading="lazy"></span></dd></dl>

<p>where

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</math></span><img src="./d7f416a069728bba921863fa4f0aae18603cd206.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.231ex; height:3.509ex;" alt="{\displaystyle \mathbb {U} _{n}^{(1)}=[\mathbb {U} _{n},\mathbb {U} _{n}]}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {U} _{n}^{(2)}=[\mathbb {U} _{n},\mathbb {U} _{n}^{(1)}]}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {U} _{n}^{(2)}=[\mathbb {U} _{n},\mathbb {U} _{n}^{(1)}]}</annotation>

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<p>there are associated unipotent groups. For example, on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=4}">

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</math></span><img src="./d928ec15aeef83aade867992ee473933adb6139d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=4}" loading="lazy"></span>, the central series are the matrix groups

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {U} _{4}=\left\{{\begin{bmatrix}1&*&*&*\\0&1&*&*\\0&0&1&*\\0&0&0&1\end{bmatrix}}\right\}}</annotation>

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</math></span><img src="./2645d2ea1b3ee40bb54b27bdd7709d606428a071.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:25.432ex; height:12.509ex;" alt="{\displaystyle \mathbb {U} _{4}=\left\{{\begin{bmatrix}1&*&*&*\\0&1&*&*\\0&0&1&*\\0&0&0&1\end{bmatrix}}\right\}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {U} _{4}^{(1)}=\left\{{\begin{bmatrix}1&0&*&*\\0&1&0&*\\0&0&1&0\\0&0&0&1\end{bmatrix}}\right\}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {U} _{4}^{(1)}=\left\{{\begin{bmatrix}1&0&*&*\\0&1&0&*\\0&0&1&0\\0&0&0&1\end{bmatrix}}\right\}}</annotation>

</semantics>

</math></span><img src="./4deea0c44875459cce0ed9e7500511581b0cb17b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:26.712ex; height:12.509ex;" alt="{\displaystyle \mathbb {U} _{4}^{(1)}=\left\{{\begin{bmatrix}1&0&*&*\\0&1&0&*\\0&0&1&0\\0&0&0&1\end{bmatrix}}\right\}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {U} _{4}^{(2)}=\left\{{\begin{bmatrix}1&0&0&*\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}\right\}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {U} _{4}^{(2)}=\left\{{\begin{bmatrix}1&0&0&*\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}\right\}}</annotation>

</semantics>

</math></span><img src="./5e55d073e2d49a30f86097bca803348ad0d3e701.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:26.712ex; height:12.509ex;" alt="{\displaystyle \mathbb {U} _{4}^{(2)}=\left\{{\begin{bmatrix}1&0&0&*\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}\right\}}" loading="lazy"></span>, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {U} _{4}^{(3)}=\left\{{\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}\right\}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {U} _{4}^{(3)}=\left\{{\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}\right\}}</annotation>

</semantics>

</math></span><img src="./19843ac8e3142ebc9240a2923b07321aa9ef3f03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:26.712ex; height:12.509ex;" alt="{\displaystyle \mathbb {U} _{4}^{(3)}=\left\{{\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}\right\}}" loading="lazy"></span></dd></dl>

<p>given some induced examples of unipotent groups.

</p>

<p>The additive group <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {G} _{a}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{a}}</annotation>

</semantics>

</math></span><img src="./0a3a118277795dec18b43d8e6b839e6af5856180.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.91ex; height:2.509ex;" alt="{\displaystyle \mathbb {G} _{a}}" loading="lazy"></span> is a unipotent group through the embedding

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<mrow>

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<annotation encoding="application/x-tex">{\displaystyle a\mapsto {\begin{bmatrix}1&a\\0&1\end{bmatrix}}}</annotation>

</semantics>

</math></span><img src="./8d43c2cbdd6404f5ec572d04dae528ff351c0820.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.765ex; height:6.176ex;" alt="{\displaystyle a\mapsto {\begin{bmatrix}1&a\\0&1\end{bmatrix}}}" loading="lazy"></span></dd></dl>

<p>Notice the matrix multiplication gives

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<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&a\\0&1\end{bmatrix}}\cdot {\begin{bmatrix}1&b\\0&1\end{bmatrix}}={\begin{bmatrix}1&a+b\\0&1\end{bmatrix}}}</annotation>

</semantics>

</math></span><img src="./6f3a879012f724baee5a0a84b9849947b54c199c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.312ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}1&a\\0&1\end{bmatrix}}\cdot {\begin{bmatrix}1&b\\0&1\end{bmatrix}}={\begin{bmatrix}1&a+b\\0&1\end{bmatrix}}}" loading="lazy"></span></dd></dl>

<p>hence this is a group embedding. More generally, there is an embedding <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {G} _{a}^{n}\to \mathbb {U} _{n+1}}">

</mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{a}^{n}\to \mathbb {U} _{n+1}}</annotation>

</semantics>

</math></span><img src="./ea94030e7ca5d536cd159824e7267a06e9579032.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.638ex; height:2.843ex;" alt="{\displaystyle \mathbb {G} _{a}^{n}\to \mathbb {U} _{n+1}}" loading="lazy"></span> from the map

</p>

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<annotation encoding="application/x-tex">{\displaystyle (a_{1},\ldots ,a_{n})\,\mapsto {\begin{bmatrix}1&a_{1}&a_{2}&\cdots &a_{n-1}&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{bmatrix}}}</annotation>

</semantics>

</math></span><img src="./ada2c5b848331e186c855a837231635453197648.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.005ex; width:46.637ex; height:17.176ex;" alt="{\displaystyle (a_{1},\ldots ,a_{n})\,\mapsto {\begin{bmatrix}1&a_{1}&a_{2}&\cdots &a_{n-1}&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{bmatrix}}}" loading="lazy"></span></dd></dl>

<p>Using scheme theory, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {G} _{a}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{a}}</annotation>

</semantics>

</math></span><img src="./0a3a118277795dec18b43d8e6b839e6af5856180.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.91ex; height:2.509ex;" alt="{\displaystyle \mathbb {G} _{a}}" loading="lazy"></span> is given by the <a href="Functor" title="Functor">functor</a>

</p>

</mrow>

<msup>

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</msup>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}:{\textbf {Sch}}^{op}\to {\textbf {Sets}}}</annotation>

</semantics>

</math></span><img src="./9d14c9e9d7e606c423ce16dd7c090dab3dffe19d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.222ex; height:2.343ex;" alt="{\displaystyle {\mathcal {O}}:{\textbf {Sch}}^{op}\to {\textbf {Sets}}}" loading="lazy"></span></dd></dl>

<p>where

</p>

<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle (X,{\mathcal {O}}_{X})\mapsto {\mathcal {O}}_{X}(X)}</annotation>

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</math></span><img src="./87ee97a179b31846ad85bd47564015f16979754e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.191ex; height:2.843ex;" alt="{\displaystyle (X,{\mathcal {O}}_{X})\mapsto {\mathcal {O}}_{X}(X)}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Kernel_of_the_Frobenius">Kernel of the Frobenius</h3></div>

<p>Consider the functor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}}">

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}}</annotation>

</semantics>

</math></span><img src="./d6ae2ed4058fb748a183d9ada8aea50a00d0c89f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.85ex; height:2.176ex;" alt="{\displaystyle {\mathcal {O}}}" loading="lazy"></span> on the <a href="Subcategory" title="Subcategory">subcategory</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {Sch}}/\mathbb {F} _{p}}">

</mrow>

</mrow>

<msub>

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</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\textbf {Sch}}/\mathbb {F} _{p}}</annotation>

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</math></span><img src="./799429666a55d6760bdf6956ee92d6097342a4b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.8ex; height:3.009ex;" alt="{\displaystyle {\textbf {Sch}}/\mathbb {F} _{p}}" loading="lazy"></span>, there is the subfunctor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{p}}">

<msub>

</mrow>

</msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha _{p}}</annotation>

</semantics>

</math></span><img src="./5c48aa9000af59f94d3022f58beadb61cea7d8b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.547ex; height:2.343ex;" alt="{\displaystyle \alpha _{p}}" loading="lazy"></span> where

</p>

<msub>

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</msub>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle \alpha _{p}(X)=\{x\in {\mathcal {O}}(X):x^{p}=0\}}</annotation>

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</math></span><img src="./690844dcdeaa6d248fe00977dff17ecfd7672cb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.156ex; height:3.009ex;" alt="{\displaystyle \alpha _{p}(X)=\{x\in {\mathcal {O}}(X):x^{p}=0\}}" loading="lazy"></span></dd></dl>

<p>so it is given by the kernel of the <a href="Frobenius_endomorphism" title="Frobenius endomorphism">Frobenius endomorphism</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Classification_of_unipotent_groups_over_characteristic_0">Classification of unipotent groups over characteristic 0</h2></div>

<p>Over <a href="Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a> 0 there is a nice classification of unipotent algebraic groups with respect to <a href="Nilpotent_Lie_algebra" title="Nilpotent Lie algebra">nilpotent Lie algebras</a>. Recall that a nilpotent Lie algebra is a subalgebra of some <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {gl}}_{n}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {gl}}_{n}}</annotation>

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</math></span><img src="./7074085f669b257cf21f2f443dd433888e2597b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.042ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {gl}}_{n}}" loading="lazy"></span> such that the iterated adjoint action eventually terminates to the zero-map. In terms of matrices, this means it is a subalgebra <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation>

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</math></span><img src="./40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy"></span> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {n}}_{n}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {n}}_{n}}</annotation>

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</math></span><img src="./38f4b73d08583660882d55f903b2f61d30b5fbb5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.444ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {n}}_{n}}" loading="lazy"></span>, the matrices with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{ij}=0}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle a_{ij}=0}</annotation>

</semantics>

</math></span><img src="./a565e93211d1ad5c06a571ff8952ab3dfcff3638.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.968ex; height:2.843ex;" alt="{\displaystyle a_{ij}=0}" loading="lazy"></span> for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\leq j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i\leq j}</annotation>

</semantics>

</math></span><img src="./894ab6e9c9afcfea7d9370399cebe1557bdf9b2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.859ex; height:2.509ex;" alt="{\displaystyle i\leq j}" loading="lazy"></span>.

</p><p>Then, there is an <a href="Equivalence_of_categories" title="Equivalence of categories">equivalence of categories</a> of finite-dimensional nilpotent Lie algebras and unipotent algebraic groups.<sup id="cite_ref-:0_2-2" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup>page 261</sup> This can be constructed using the <a href="Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula" title="Baker–Campbell–Hausdorff formula">Baker–Campbell–Hausdorff series</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(X,Y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H(X,Y)}</annotation>

</semantics>

</math></span><img src="./e1d87c82092f7817e719251729dc0a55289df0eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.66ex; height:2.843ex;" alt="{\displaystyle H(X,Y)}" loading="lazy"></span>, where given a finite-dimensional nilpotent Lie algebra, the map

</p>

</mrow>

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</mrow>

<mtext> where </mtext>

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<annotation encoding="application/x-tex">{\displaystyle H:{\mathfrak {g}}\times {\mathfrak {g}}\to {\mathfrak {g}}{\text{ where }}(X,Y)\mapsto H(X,Y)}</annotation>

</semantics>

</math></span><img src="./26e96c019768c5ba3e00f366f0f8e4aead47e9bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.95ex; height:2.843ex;" alt="{\displaystyle H:{\mathfrak {g}}\times {\mathfrak {g}}\to {\mathfrak {g}}{\text{ where }}(X,Y)\mapsto H(X,Y)}" loading="lazy"></span></dd></dl>

<p>gives a Unipotent algebraic group structure on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation>

</semantics>

</math></span><img src="./40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy"></span>.

</p><p>In the other direction the <a href="Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">exponential map</a> takes any nilpotent square matrix to a unipotent matrix. Moreover, if <i>U</i> is a commutative unipotent group, the exponential map induces an <a href="Isomorphism" title="Isomorphism">isomorphism</a> from the Lie algebra of <i>U</i> to <i>U</i> itself.

</p>

<div class="mw-heading mw-heading3"><h3 id="Remarks">Remarks</h3></div>

<p>Unipotent groups over an <a href="Algebraically_closed_field" title="Algebraically closed field">algebraically closed field</a> of any given dimension can in principle be classified, but in practice the complexity of the classification increases rapidly with the dimension, so some tend to give up somewhere around dimension 6.

</p>

<div class="mw-heading mw-heading2"><h2 id="Unipotent_radical">Unipotent radical</h2></div>

<p>The <b>unipotent radical</b> of an <a href="Algebraic_group" title="Algebraic group">algebraic group</a> <i>G</i> is the set of unipotent elements in the <a href="Radical_of_an_algebraic_group" title="Radical of an algebraic group">radical</a> of <i>G</i>. It is a connected unipotent normal subgroup of <i>G</i>, and contains all other such subgroups. A group is called reductive if its unipotent radical is trivial. If <i>G</i> is reductive then its radical is a torus.

</p>

<div class="mw-heading mw-heading2"><h2 id="Decomposition_of_algebraic_groups">Decomposition of algebraic groups</h2></div>

<p>Algebraic groups can be decomposed into unipotent groups, multiplicative groups, and <a href="Abelian_varieties" class="mw-redirect" title="Abelian varieties">abelian varieties</a>, but the statement of how they decompose depends upon the characteristic of their base <a href="Field_(mathematics)" title="Field (mathematics)">field</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Characteristic_0">Characteristic 0</h3></div>

<p>Over characteristic 0 there is a nice decomposition theorem of a commutative algebraic group <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math></span><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span> relating its structure to the structure of a <a href="Linear_algebraic_group" title="Linear algebraic group">linear algebraic group</a> and an <a href="Abelian_variety" title="Abelian variety">Abelian variety</a>. There is a <a href="Short_exact_sequence" class="mw-redirect" title="Short exact sequence">short exact sequence</a> of groups<sup id="cite_ref-:1_3-0" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup>page 8</sup>

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\to M\times U\to G\to A\to 0}</annotation>

</semantics>

</math></span><img src="./2e43aa543fabad09fc9158919ff0223d477afbd7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:27.416ex; height:2.176ex;" alt="{\displaystyle 0\to M\times U\to G\to A\to 0}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> is an abelian variety, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math></span><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> is of multiplicative type (meaning, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math></span><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> is, geometrically, a product of tori and algebraic groups of the form <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu _{n}}</annotation>

</semantics>

</math></span><img src="./267d03f9351dcc8d3d3ac7cad59ea3ba4fecbfef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.62ex; height:2.176ex;" alt="{\displaystyle \mu _{n}}" loading="lazy"></span>) and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math></span><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"></span> is a unipotent group.

</p>

<div class="mw-heading mw-heading3"><h3 id="Characteristic_p">Characteristic <i>p</i></h3></div>

<p>When the characteristic of the base field is <i>p</i> there is an analogous statement<sup id="cite_ref-:1_3-1" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> for an algebraic group <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math></span><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span>: there exists a smallest subgroup <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math></span><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"></span> such that

</p>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G/H}</annotation>

</semantics>

</math></span><img src="./21e7e9d6e3072ec8dd48200d755847154ea5d35c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.053ex; height:2.843ex;" alt="{\displaystyle G/H}" loading="lazy"></span> is a unipotent group</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math></span><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"></span> is an extension of an abelian variety <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> by a group <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math></span><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> of multiplicative type.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math></span><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> is unique up to <a href="Commensurability_(group_theory)" title="Commensurability (group theory)">commensurability</a> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math></span><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> is unique up to <a href="Isogeny" title="Isogeny">isogeny</a>.</li></ol>

<div class="mw-heading mw-heading2"><h2 id="Jordan_decomposition">Jordan decomposition</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Jordan%E2%80%93Chevalley_decomposition" title="Jordan–Chevalley decomposition">Jordan–Chevalley decomposition</a></div>

<p>Any element <i>g</i> of a linear algebraic group over a <a href="Perfect_field" title="Perfect field">perfect field</a> can be written uniquely as the product <i>g</i> = <i>g</i><sub><i>u</i></sub> <i>g</i><sub><i>s</i></sub> of commuting unipotent and <a href="Semi-simplicity" title="Semi-simplicity">semisimple</a> elements <i>g</i><sub><i>u</i></sub> and <i>g</i><sub><i>s</i></sub>. In the case of the group GL<sub><i>n</i></sub>(<b>C</b>), this essentially says that any invertible <a href="Complex_number" title="Complex number">complex</a> 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 <a href="Jordan%E2%80%93Chevalley_decomposition" title="Jordan–Chevalley decomposition">Jordan–Chevalley decomposition</a>.

</p><p>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.

</p>

<ul><li><a href="Reductive_group" title="Reductive group">Reductive group</a></li>

<li><a href="Unipotent_representation" title="Unipotent representation">Unipotent representation</a></li>

<li><a href="Deligne%E2%80%93Lusztig_theory" title="Deligne–Lusztig theory">Deligne–Lusztig theory</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Unipotent_element">"Unipotent element - Encyclopedia of Mathematics"</a>. <i>encyclopediaofmath.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-09-23</span></span>.</cite></span>

</li>

<li id="cite_note-:0-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_2-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFMilne" class="citation book cs1">Milne, J. S. <a rel="nofollow" class="external text" href="https://www.jmilne.org/math/CourseNotes/iAG200.pdf"><i>Linear Algebraic Groups</i></a> <span class="cs1-format">(PDF)</span>. pp. <span class="nowrap">252–</span>253, Unipotent algebraic groups.</cite></span>

</li>

<li id="cite_note-:1-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFBrion2016" class="citation arxiv cs1">Brion, Michel (2016-09-27). "Commutative algebraic groups up to isogeny". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1602.00222">1602.00222</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.AG">math.AG</a>].</cite></span>

</li>

</ol></div></div>

<ul><li>A. Borel, <i>Linear algebraic groups</i>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-97370-2</bdi></li>

<li><cite id="CITEREFBorel1956" class="citation cs2"><a href="Armand_Borel" title="Armand Borel">Borel, Armand</a> (1956), "Groupes linéaires algébriques", <i><a href="Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a></i>, Second Series, <b>64</b> (1), Annals of Mathematics: <span class="nowrap">20–</span>82, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1969949">10.2307/1969949</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1969949">1969949</a></cite></li>

<li><cite id="CITEREFPopov2001" class="citation cs2"><a href="Vladimir_L._Popov" class="mw-redirect" title="Vladimir L. Popov">Popov, V.L.</a> (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=unipotent_element">"unipotent element"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite></li>

<li><cite id="CITEREFPopov2001" class="citation cs2"><a href="Vladimir_L._Popov" class="mw-redirect" title="Vladimir L. Popov">Popov, V.L.</a> (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=unipotent_group">"unipotent group"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite></li>

<li><cite id="CITEREFSuprunenko2001" class="citation cs2">Suprunenko, D.A. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=unipotent_matrix">"unipotent matrix"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite></li></ul>

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</style><div id="Matrix_classes596" style="font-size:114%;margin:0 4em"><a href="Matrix_(mathematics)" title="Matrix (mathematics)">Matrix</a> classes</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Explicitly constrained entries</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Alternant_matrix" title="Alternant matrix">Alternant</a></li>

<li><a href="Anti-diagonal_matrix" title="Anti-diagonal matrix">Anti-diagonal</a></li>

<li><a href="Skew-Hermitian_matrix" title="Skew-Hermitian matrix">Anti-Hermitian</a></li>

<li><a href="Skew-symmetric_matrix" title="Skew-symmetric matrix">Anti-symmetric</a></li>

<li><a href="Arrowhead_matrix" title="Arrowhead matrix">Arrowhead</a></li>

<li><a href="Bidiagonal_matrix" title="Bidiagonal matrix">Bidiagonal</a></li>

<li><a href="Bisymmetric_matrix" title="Bisymmetric matrix">Bisymmetric</a></li>

<li><a href="Block-diagonal_matrix" class="mw-redirect" title="Block-diagonal matrix">Block-diagonal</a></li>

<li><a href="Block_matrix" title="Block matrix">Block</a></li>

<li><a href="Block_tridiagonal_matrix" class="mw-redirect" title="Block tridiagonal matrix">Block tridiagonal</a></li>

<li><a href="Boolean_matrix" title="Boolean matrix">Boolean</a></li>

<li><a href="Cauchy_matrix" title="Cauchy matrix">Cauchy</a></li>

<li><a href="Centrosymmetric_matrix" title="Centrosymmetric matrix">Centrosymmetric</a></li>

<li><a href="Conference_matrix" title="Conference matrix">Conference</a></li>

<li><a href="Complex_Hadamard_matrix" title="Complex Hadamard matrix">Complex Hadamard</a></li>

<li><a href="Copositive_matrix" title="Copositive matrix">Copositive</a></li>

<li><a href="Diagonally_dominant_matrix" title="Diagonally dominant matrix">Diagonally dominant</a></li>

<li><a href="Diagonal_matrix" title="Diagonal matrix">Diagonal</a></li>

<li><a href="DFT_matrix" title="DFT matrix">Discrete Fourier Transform</a></li>

<li><a href="Elementary_matrix" title="Elementary matrix">Elementary</a></li>

<li><a href="Equivalent_matrix" class="mw-redirect" title="Equivalent matrix">Equivalent</a></li>

<li><a href="Frobenius_matrix" title="Frobenius matrix">Frobenius</a></li>

<li><a href="Generalized_permutation_matrix" title="Generalized permutation matrix">Generalized permutation</a></li>

<li><a href="Hadamard_matrix" title="Hadamard matrix">Hadamard</a></li>

<li><a href="Hankel_matrix" title="Hankel matrix">Hankel</a></li>

<li><a href="Hermitian_matrix" title="Hermitian matrix">Hermitian</a></li>

<li><a href="Hessenberg_matrix" title="Hessenberg matrix">Hessenberg</a></li>

<li><a href="Hollow_matrix" title="Hollow matrix">Hollow</a></li>

<li><a href="Integer_matrix" title="Integer matrix">Integer</a></li>

<li><a href="Logical_matrix" title="Logical matrix">Logical</a></li>

<li><a href="Matrix_unit" title="Matrix unit">Matrix unit</a></li>

<li><a href="Metzler_matrix" title="Metzler matrix">Metzler</a></li>

<li><a href="Moore_matrix" title="Moore matrix">Moore</a></li>

<li><a href="Nonnegative_matrix" title="Nonnegative matrix">Nonnegative</a></li>

<li><a href="Pentadiagonal_matrix" class="mw-redirect" title="Pentadiagonal matrix">Pentadiagonal</a></li>

<li><a href="Permutation_matrix" title="Permutation matrix">Permutation</a></li>

<li><a href="Persymmetric_matrix" title="Persymmetric matrix">Persymmetric</a></li>

<li><a href="Polynomial_matrix" title="Polynomial matrix">Polynomial</a></li>

<li><a href="Quaternionic_matrix" title="Quaternionic matrix">Quaternionic</a></li>

<li><a href="Signature_matrix" title="Signature matrix">Signature</a></li>

<li><a href="Skew-Hermitian_matrix" title="Skew-Hermitian matrix">Skew-Hermitian</a></li>

<li><a href="Skew-symmetric_matrix" title="Skew-symmetric matrix">Skew-symmetric</a></li>

<li><a href="Skyline_matrix" title="Skyline matrix">Skyline</a></li>

<li><a href="Sparse_matrix" title="Sparse matrix">Sparse</a></li>

<li><a href="Sylvester_matrix" title="Sylvester matrix">Sylvester</a></li>

<li><a href="Symmetric_matrix" title="Symmetric matrix">Symmetric</a></li>

<li><a href="Toeplitz_matrix" title="Toeplitz matrix">Toeplitz</a></li>

<li><a href="Triangular_matrix" title="Triangular matrix">Triangular</a></li>

<li><a href="Tridiagonal_matrix" title="Tridiagonal matrix">Tridiagonal</a></li>

<li><a href="Vandermonde_matrix" title="Vandermonde matrix">Vandermonde</a></li>

<li><a href="Walsh_matrix" title="Walsh matrix">Walsh</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constant</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Exchange_matrix" title="Exchange matrix">Exchange</a></li>

<li><a href="Hilbert_matrix" title="Hilbert matrix">Hilbert</a></li>

<li><a href="Identity_matrix" title="Identity matrix">Identity</a></li>

<li><a href="Lehmer_matrix" title="Lehmer matrix">Lehmer</a></li>

<li><a href="Pascal_matrix" title="Pascal matrix">Pascal</a></li>

<li><a href="Pauli_matrices" title="Pauli matrices">Pauli</a></li>

<li><a href="Redheffer_matrix" title="Redheffer matrix">Redheffer</a></li>

<li><a href="Shift_matrix" title="Shift matrix">Shift</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Conditions on <a href="Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalues or eigenvectors</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Companion_matrix" title="Companion matrix">Companion</a></li>

<li><a href="Convergent_matrix" title="Convergent matrix">Convergent</a></li>

<li><a href="Defective_matrix" title="Defective matrix">Defective</a></li>

<li><a href="Definite_matrix" title="Definite matrix">Definite</a></li>

<li><a href="Diagonalizable_matrix" title="Diagonalizable matrix">Diagonalizable</a></li>

<li><a href="Hurwitz-stable_matrix" title="Hurwitz-stable matrix">Hurwitz-stable</a></li>

<li><a href="Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">Positive-definite</a></li>

<li><a href="Stieltjes_matrix" title="Stieltjes matrix">Stieltjes</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Satisfying conditions on <a href="Matrix_product" class="mw-redirect" title="Matrix product">products</a> or <a href="Inverse_of_a_matrix" class="mw-redirect" title="Inverse of a matrix">inverses</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Matrix_congruence" title="Matrix congruence">Congruent</a></li>

<li><a href="Idempotent_matrix" title="Idempotent matrix">Idempotent</a> or <a href="Projection_(linear_algebra)" title="Projection (linear algebra)">Projection</a></li>

<li><a href="Invertible_matrix" title="Invertible matrix">Invertible</a></li>

<li><a href="Involutory_matrix" title="Involutory matrix">Involutory</a></li>

<li><a href="Nilpotent_matrix" title="Nilpotent matrix">Nilpotent</a></li>

<li><a href="Normal_matrix" title="Normal matrix">Normal</a></li>

<li><a href="Orthogonal_matrix" title="Orthogonal matrix">Orthogonal</a></li>

<li><a href="Unimodular_matrix" title="Unimodular matrix">Unimodular</a></li>

<li><a href="Unitary_matrix" title="Unitary matrix">Unitary</a></li>

<li><a href="Totally_unimodular_matrix" class="mw-redirect" title="Totally unimodular matrix">Totally unimodular</a></li>

<li><a href="Weighing_matrix" title="Weighing matrix">Weighing</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">With specific applications</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Adjugate_matrix" title="Adjugate matrix">Adjugate</a></li>

<li><a href="Alternating_sign_matrix" title="Alternating sign matrix">Alternating sign</a></li>

<li><a href="Augmented_matrix" title="Augmented matrix">Augmented</a></li>

<li><a href="B%C3%A9zout_matrix" title="Bézout matrix">Bézout</a></li>

<li><a href="Carleman_matrix" title="Carleman matrix">Carleman</a></li>

<li><a href="Cartan_matrix" title="Cartan matrix">Cartan</a></li>

<li><a href="Circulant_matrix" title="Circulant matrix">Circulant</a></li>

<li><a href="Cofactor_matrix" class="mw-redirect" title="Cofactor matrix">Cofactor</a></li>

<li><a href="Commutation_matrix" title="Commutation matrix">Commutation</a></li>

<li><a href="Confusion_matrix" title="Confusion matrix">Confusion</a></li>

<li><a href="Coxeter_matrix" class="mw-redirect" title="Coxeter matrix">Coxeter</a></li>

<li><a href="Distance_matrix" title="Distance matrix">Distance</a></li>

<li><a href="Duplication_and_elimination_matrices" title="Duplication and elimination matrices">Duplication and elimination</a></li>

<li><a href="Euclidean_distance_matrix" title="Euclidean distance matrix">Euclidean distance</a></li>

<li><a href="Fundamental_matrix_(linear_differential_equation)" title="Fundamental matrix (linear differential equation)">Fundamental (linear differential equation)</a></li>

<li><a href="Generator_matrix" title="Generator matrix">Generator</a></li>

<li><a href="Hessian_matrix" title="Hessian matrix">Hessian</a></li>

<li><a href="Householder_transformation" title="Householder transformation">Householder</a></li>

<li><a href="Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a></li>

<li><a href="Moment_matrix" title="Moment matrix">Moment</a></li>

<li><a href="Payoff_matrix" class="mw-redirect" title="Payoff matrix">Payoff</a></li>

<li><a href="Random_matrix" title="Random matrix">Random</a></li>

<li><a href="Rotation_matrix" title="Rotation matrix">Rotation</a></li>

<li><a href="Routh%E2%80%93Hurwitz_matrix" title="Routh–Hurwitz matrix">Routh-Hurwitz</a></li>

<li><a href="Seifert_matrix" class="mw-redirect" title="Seifert matrix">Seifert</a></li>

<li><a href="Shear_matrix" class="mw-redirect" title="Shear matrix">Shear</a></li>

<li><a href="Similarity_matrix" class="mw-redirect" title="Similarity matrix">Similarity</a></li>

<li><a href="Symplectic_matrix" title="Symplectic matrix">Symplectic</a></li>

<li><a href="Totally_positive_matrix" title="Totally positive matrix">Totally positive</a></li>

<li><a href="Transformation_matrix" title="Transformation matrix">Transformation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Used in <a href="Statistics" title="Statistics">statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Centering_matrix" title="Centering matrix">Centering</a></li>

<li><a href="Correlation_matrix" class="mw-redirect" title="Correlation matrix">Correlation</a></li>

<li><a href="Covariance_matrix" title="Covariance matrix">Covariance</a></li>

<li><a href="Design_matrix" title="Design matrix">Design</a></li>

<li><a href="Doubly_stochastic_matrix" title="Doubly stochastic matrix">Doubly stochastic</a></li>

<li><a href="Fisher_information_matrix" class="mw-redirect" title="Fisher information matrix">Fisher information</a></li>

<li><a href="Precision_(statistics)" title="Precision (statistics)">Precision</a></li>

<li><a href="Stochastic_matrix" title="Stochastic matrix">Stochastic</a></li>

<li><a href="Stochastic_matrix" title="Stochastic matrix">Transition</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Used in <a href="Graph_theory" title="Graph theory">graph theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Adjacency_matrix" title="Adjacency matrix">Adjacency</a></li>

<li><a href="Biadjacency_matrix" class="mw-redirect" title="Biadjacency matrix">Biadjacency</a></li>

<li><a href="Degree_matrix" title="Degree matrix">Degree</a></li>

<li><a href="Edmonds_matrix" title="Edmonds matrix">Edmonds</a></li>

<li><a href="Incidence_matrix" title="Incidence matrix">Incidence</a></li>

<li><a href="Laplacian_matrix" title="Laplacian matrix">Laplacian</a></li>

<li><a href="Seidel_adjacency_matrix" title="Seidel adjacency matrix">Seidel adjacency</a></li>

<li><a href="Tutte_matrix" title="Tutte matrix">Tutte</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Used in science and engineering</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix" title="Cabibbo–Kobayashi–Maskawa matrix">Cabibbo–Kobayashi–Maskawa</a></li>

<li><a href="Density_matrix" title="Density matrix">Density</a></li>

<li><a href="Fundamental_matrix_(computer_vision)" title="Fundamental matrix (computer vision)">Fundamental (computer vision)</a></li>

<li><a href="Fuzzy_associative_matrix" title="Fuzzy associative matrix">Fuzzy associative</a></li>

<li><a href="Gamma_matrices" title="Gamma matrices">Gamma</a></li>

<li><a href="Hamiltonian_matrix" title="Hamiltonian matrix">Hamiltonian</a></li>

<li><a href="Irregular_matrix" title="Irregular matrix">Irregular</a></li>

<li><a href="Overlap_matrix" class="mw-redirect" title="Overlap matrix">Overlap</a></li>

<li><a href="State-transition_matrix" title="State-transition matrix">State transition</a></li>

<li><a href="Substitution_matrix" title="Substitution matrix">Substitution</a></li>

<li><a href="Z-matrix_(chemistry)" title="Z-matrix (chemistry)">Z (chemistry)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related terms</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Jordan_normal_form" title="Jordan normal form">Jordan normal form</a></li>

<li><a href="Linear_independence" title="Linear independence">Linear independence</a></li>

<li><a href="Matrix_exponential" title="Matrix exponential">Matrix exponential</a></li>

<li><a href="Matrix_representation_of_conic_sections" title="Matrix representation of conic sections">Matrix representation of conic sections</a></li>

<li><a href="Perfect_matrix" title="Perfect matrix">Perfect matrix</a></li>

<li><a href="Pseudoinverse" class="mw-redirect" title="Pseudoinverse">Pseudoinverse</a></li>

<li><a href="Row_echelon_form" title="Row echelon form">Row echelon form</a></li>

<li><a href="Wronskian" title="Wronskian">Wronskian</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><b><span class="nowrap"><span class="skin-invert-image noviewer" typeof="mw:File"></span> </span><a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></b></li>

<li><a href="List_of_matrices" class="mw-redirect" title="List of matrices">List of matrices</a></li>

<li>Category:Matrices (mathematics)</li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

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