nilpotent
Language: English
Part of speech: adj
Pronunciation: ipa: /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.
Senses: examples: text: If a square matrix is upper triangular and has zeros on the diagonal, then it is nilpotent (under the usual matrix multiplication).
text: 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.
ref: 2012, Martin W. Liebeck, Gary M. Seitz, Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras, American Mathematical Society, page 129:
topics: algebra
mathematics
sciences
qualifier: ring theory
glosses: Such that, for some positive integer n, xⁿ = 0.
raw_tags: of an element x of a ring
topics: algebra
mathematics
sciences
glosses: 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).
qualifier: Lie theory
raw_tags: of an element x of a Lie algebra L
topics: algebra
mathematics
sciences
glosses: 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.
qualifier: Lie theory
raw_tags: of a Lie algebra
topics: algebra
group-theory
mathematics
sciences
glosses: 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.
raw_tags: of a group
topics: algebra
mathematics
sciences
glosses: 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.
qualifier: ring theory
raw_tags: of an ideal I
topics: algebra
mathematics
sciences
glosses: 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.
qualifier: semigroup theory
raw_tags: of a semigroup with zero
topics: algebra
mathematics
sciences
glosses: 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.
raw_tags: of an algebra over a commutative ring
Coordinate terms: idempotent
Derived terms: nilpotent algebra
nilpotent ideal
nilpotently
nilpotent orbit
nilpotent semigroup
quasinilpotent
Related terms: nilpotence
nilpotency
idempotent
nullipotent
unipotent