nilpotent

English

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.

Pronunciation

• IPA^(key): /nɪlˈpəʊtənt/

Adjective

nilpotent (not comparable)

1. (mathematics, algebra, ring theory, of an element x of a ring) Such that, for some positive integer n, xⁿ = 0.

   If a square matrix is upper triangular, then it is nilpotent (under the usual matrix multiplication).

   • 2012, Martin W. Liebeck, Gary M. Seitz, Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras, American Mathematical Society, page 129:

     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.

2. (mathematics, algebra) 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.
   1. (Lie theory, of an element x of a Lie algebra L) Belonging to the derived algebra of L and such that the adjoint action of x is nilpotent (as a linear transformation on L).
   2. (Lie theory, of a Lie algebra) Such that the lower central series terminates.
   3. (group theory, of a group) Admitting a central series of finite length.
   4. (ring theory, of an ideal I) Such that there exists a natural number k with I^k = 0.
   5. (semigroup theory, of a semigroup with zero) Containing only nilpotent elements.
   6. (of an algebra over a commutative ring) 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.

Coordinate terms

• idempotent

Derived terms

• nilpotent algebra
• nilpotent ideal
• nilpotently
• nilpotent orbit
• nilpotent semigroup

Related terms

• nilpotence
• nilpotency
• idempotent
• nullipotent
• unipotent

Translations

(algebra)

• Czech: nilpotentní
• Danish: nilpotent
• Esperanto: nilpotenta, nulpotenca
• Finnish: nilpotentti
• French: nilpotent (fr)
• Russian: нильпотент (ru) (nilʹpotent)
• Spanish: nilpotente

Noun

nilpotent (plural nilpotents)

1. (algebra) A nilpotent element.
   • 2015, Garret Sobczyk, “Part I: Vector Analysis of Spinors”, in arXiv‎^[1]:

     The so-called spinor algebra of C(2), the language of the quantum mechanics, is formulated in terms of the idempotents and nilpotents of the geometric algebra of space, including its beautiful representation on the Riemann sphere, and a new proof of the Heisenberg uncertainty principle.