View source for Unipotent element

An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u0954001.png" /> of a [[Linear algebraic group|linear algebraic group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u0954002.png" /> which coincides with the unipotent part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u0954003.png" /> of its [[Jordan decomposition|Jordan decomposition]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u0954004.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u0954005.png" /> is realized as a closed subgroup of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u0954006.png" /> of automorphisms of a finite-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u0954007.png" /> over an algebraically closed ground field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u0954008.png" />, then a unipotent element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u0954009.png" /> is precisely an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540010.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540012.png" />, or, equivalently, is such that its matrix relative to a certain basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540013.png" /> is upper triangular, with 1's on the main diagonal. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540014.png" /> of all unipotent elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540015.png" /> is closed in the Zariski topology. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540016.png" />, then every unipotent element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540017.png" /> has infinite order. In that case the smallest algebraic subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540018.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540019.png" /> is a one-dimensional [[Unipotent group|unipotent group]]. If, however, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540020.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540021.png" /> will be unipotent precisely when it has finite order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540022.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540023.png" />. A connected group contains no unipotent element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540024.png" /> if and only if it is an [[Algebraic torus|algebraic torus]].

A criterion for anisotropy may be given in terms of unipotent elements (cf. [[Anisotropic group|Anisotropic group]]).

Unipotent elements play an important role in the theory of discrete subgroups (cf. [[Discrete subgroup|Discrete subgroup]]) of algebraic groups and Lie groups. The presence of unipotent elements in a discrete group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540025.png" /> of motions of a symmetric space, having a non-compact fundamental domain of finite volume, is an important tool for studying the structure of such groups and their fundamental domains, cf. [[#References|[5]]]; the existence of unipotent elements in such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540026.png" /> was proved in [[#References|[4]]].

The variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540027.png" /> is invariant under inner automorphisms of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540028.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540029.png" /> be connected and semi-simple. Then the number of conjugacy classes of unipotent elements is finite and for every simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540030.png" /> there is a complete description of them (as well as a description of the centralizer of a unipotent element), cf. [[#References|[7]]]. In the classical groups such a description is obtained by means of the Jordan form of matrices, [[#References|[2]]]. E.g. for the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540031.png" /> there exists a bijection between the conjugacy classes of unipotent elements and the partitions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540032.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540033.png" /> into positive integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540035.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540037.png" /> are two partitions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540038.png" />, then the class corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540039.png" /> contains in its closure the class corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540040.png" /> precisely if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540041.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540042.png" />. The dimension of the class corresponding to the partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540043.png" /> (as an algebraic variety) is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540044.png" />.

The set of all simple points of the algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540045.png" /> forms one conjugacy class of unipotent elements, the regular unipotent elements. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540046.png" /> is simple, then the variety of singular points in the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540047.png" /> also contains a Zariski-open conjugacy class of unipotent elements — the subregular unipotent elements. For a study of the singular points of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095400/u09540048.png" /> see also [[#References|[6]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , Springer (1991) {{MR|1102012}} {{ZBL|0726.20030}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Borel (ed.) R. Carter (ed.) C.W. Curtis (ed.) N. Iwahori (ed.) T.A. Springer (ed.) R. Steinberg (ed.) , ''Seminar on algebraic groups and related finite groups'' , ''Lect. notes in math.'' , '''131''' , Springer (1970)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J.E. Humphreys, "Linear algebraic groups" , Springer (1981) {{MR|0610979}} {{MR|0396773}} {{ZBL|0471.20029}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> D.A. Kazhdan, G.A. Margulis, "A proof of Selberg's hypothesis" ''Math. USSR Sb.'' , '''4''' : 1 (1969) pp. 147–152 ''Mat. Sb.'' , '''75''' : 1 (1968) pp. 163–168</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> A. Selberg, "Recent developments in the theory of discontinuous groups of motions of symmetric spaces" , ''Proc. 15 Scand. Congress (Oslo, 1968)'' , ''Lect. notes in math.'' , '''118''' , Springer (1970) pp. 99–120 {{MR|0263996}} {{ZBL|0197.18002}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> P.J. Slodowy, "Simple singularities and simple algebraic groups" , ''Lect. notes in math.'' , '''815''' , Springer (1980) {{MR|0584445}} {{ZBL|0441.14002}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> N. Spaltenstein, "Classes unipotentes et sousgroupes de Borel" , ''Lect. notes in math.'' , '''946''' , Springer (1982) {{MR|672610}} {{ZBL|}} </TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> T.A. Springer, "Linear algebraic groups" W. Jäger (ed.) J. Moser (ed.) R. Remmert (ed.) , ''Perspectives in Mathematics'' , Birkhäuser (1984) pp. 455–496 {{MR|0779686}} {{ZBL|0564.20024}} </TD></TR></table>