nilpotent en · ADJ
Pronunciation
- (US) audio
- /nɪlˈpəʊtənt/
Etymology
From nil (“not any”) + potent (“having power”) with literal meaning “having zero power” - bearing Latin roots nil and potens. Coined in 1870, along with idempotent, by American mathematician Benjamin Peirce to describe elements of associative algebras.
Meanings
-
(not-comparable, of an element x of a ring, qualifier:ring theory) Such that, for some positive integer n, xⁿ = 0.
The rest of this book is devoted to determining the conjugacy classes and centralizers of nilpotent elements in L(G) and unipotent elements in G, where G is an exceptional algebraic group of type E₈,E₇, E₆, F₄ or G₂ over an algebraically closed field K of characteristic p. This chapter contains statements of the main results for nilpotent elements.
2012, Martin W. Liebeck, Gary M. Seitz, Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras, American Mathematical Society, page 129:If a square matrix is upper triangular and has zeros on the diagonal, then it is nilpotent (under the usual matrix multiplication).
- (not-comparable, of an element x of a Lie algebra L, qualifier:Lie theory) In any of several technical senses: behaving analogously to nilpotent ring elements as an element of some other algebraic structure; composed of elements displaying such behavior. ▸ Belonging to the derived algebra of L and such that the adjoint action of x is nilpotent (as a linear transformation on L).
- (not-comparable, of a Lie algebra, qualifier:Lie theory) In any of several technical senses: behaving analogously to nilpotent ring elements as an element of some other algebraic structure; composed of elements displaying such behavior. ▸ Such that the lower central series terminates.
- (not-comparable, of a group) In any of several technical senses: behaving analogously to nilpotent ring elements as an element of some other algebraic structure; composed of elements displaying such behavior. ▸ Admitting a central series of finite length.
- (not-comparable, of an ideal I, qualifier:ring theory) In any of several technical senses: behaving analogously to nilpotent ring elements as an element of some other algebraic structure; composed of elements displaying such behavior. ▸ Such that there exists a natural number k with Iᵏ = 0.
- (not-comparable, of a semigroup with zero, qualifier:semigroup theory) In any of several technical senses: behaving analogously to nilpotent ring elements as an element of some other algebraic structure; composed of elements displaying such behavior. ▸ Containing only nilpotent elements.
- (not-comparable, of an algebra over a commutative ring) In any of several technical senses: behaving analogously to nilpotent ring elements as an element of some other algebraic structure; composed of elements displaying such behavior. ▸ Such that there exists some natural number n (called the index of the algebra) such that all products (of elements in the given algebra) of length n are zero.
Coordinates
Deriveds
quasinilpotent · nilpotent ideal · nilpotently · nilpotent algebra · nilpotent semigroup · nilpotent orbit
Relateds
idempotent · unipotent · nullipotent · nilpotency · nilpotence
Translations (8)
fr nilpotent ((algebra)) · ru нильпотент ((algebra)) · eo nulpotenca ((algebra)) · da nilpotent ((algebra)) · cs nilpotentní ((algebra)) · es nilpotente ((algebra)) · eo nilpotenta ((algebra)) · fi nilpotentti ((algebra))