Unipotent

Unipotent

In mathematics, a unipotent element "r" of a ring "R" is one such that"r" − 1 is a nilpotent element, in other words such that some power ("r" − 1)"n" is zero.

In particular a square matrix "M" is a unipotent matrix if and only if its characteristic polynomial "P"("t") is a power of "t" − 1. Equivalently, "M" is unipotent if all its eigenvalues are 1.

The term quasi-unipotent means that some power is unipotent, for example for a diagonalizable matrix with eigenvalues that are all roots of unity.

A unipotent algebraic group is one all of whose elements are unipotent.

Unipotent algebraic groups

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 GLn("k")).

If a unipotent group acts on an affine variety, all its orbits are closed, and if it acts linearly on a finite dimensional vector space then it has a non-zero fixed vector. In fact, the latter property characterizes unipotent groups.

Unipotent groups over an algebraically closed field of any given dimension can in principle be classified, but in practice the complexity of the classification increases very rapidly with the dimension, so people tend to give up somewhere around dimension 6.

Over the real numbers (or more generally any field of characteristic 0) the exponential map takes any nilpotent square matrix to a unipotent matrix. Moreover, if "U" is a commutative unipotent group, the exponential map induces an isomorphism from the Lie algebra of "U" to "U" itself.

Unipotent radical

The unipotent radical of an algebraic group "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.

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.

ee also

*Deligne-Lusztig theory (for unipotent characters)

References

* A. Borel, "Linear algebraic groups", ISBN 0-387-97370-2
*
*springer|id=U/u095400|first=V.L. |last=Popov|title=unipotent element
*springer|id=U/u095410|first=V.L. |last=Popov|title=unipotent group
*springer|id=U/u095420|first=D.A.|last= Suprunenko|title=unipotent matrix


Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

  • unipotent — [yo͞o nip′ə tənt] adj. [ UNI + POTENT] Biol. capable of developing into only a single type of cell or tissue: said of certain, esp. embryonic, cells …   English World dictionary

  • unipotent — Referring to those cells that produce a single type of daughter cell; e.g., a u. stem cell. Cf.:pluripotent cells, under cell. * * * unip·o·tent yü nip ət ənt adj having power in one way only esp capable of developing only in one direction or to… …   Medical dictionary

  • unipotent — yüˈnipəd.ənt adjective Etymology: uni + potent : having power in one way only; especially : capable of developing only in one direction or to one end product unipotent cells …   Useful english dictionary

  • Unipotent representation — In mathematics, a unipotent representation of a reductive group is a representation that has some similarities with unipotent conjugacy classes of groups. Informally, Langlands philosophy suggests that there should be a correspondence between… …   Wikipedia

  • unipotent — /yooh nip euh teuhnt/, adj. Biol. (of cells) capable of developing into only one type of cell or tissue. [UNI + POTENT] * * * …   Universalium

  • unipotent — adjective a) Having the capacity to develop into only one type of cell or tissue b) Having a single idempotent element See Also: idempotent, nilpotent, nullipotent …   Wiktionary

  • unipotent — adj. developing into one type of cell or tissue (Biology) …   English contemporary dictionary

  • unipotent — unip·o·tent …   English syllables

  • 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 Deligne Lusztig (1976). Lusztig (1984) used these representations to… …   Wikipedia

  • Ratner's theorems — In mathematics, Ratner s theorems is a group of major theorems in ergodic theory concerning unipotent flows on homogeneous spaces proved by Marina Ratner around 1990. The study of the dynamics of unipotent flows played decisive role in the proof… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”