See nilpotent in All languages combined, or Wiktionary
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Liebeck, Gary M. Seitz, Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras, American Mathematical Society, page 129:", "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.", "type": "quote" } ], "glosses": [ "Such that, for some positive integer n, xⁿ = 0." ], "links": [ [ "mathematics", "mathematics" ], [ "algebra", "algebra" ], [ "element", "element" ], [ "ring", "ring" ] ], "qualifier": "ring theory", "raw_glosses": [ "(mathematics, algebra, ring theory, of an element x of a ring) Such that, for some positive integer n, xⁿ = 0." ], "raw_tags": [ "of an element x of a ring" ], "tags": [ "not-comparable" ], "topics": [ "algebra", "mathematics", "sciences" ] }, { "categories": [ "en:Algebra", "en:Mathematics" ], "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)." ], "links": [ [ "mathematics", "mathematics" ], [ "algebra", "algebra" ], [ "algebraic structure", "algebraic structure" ], [ "Lie theory", "Lie theory" ], [ "Lie algebra", "Lie algebra" ], [ "derived algebra", "derived algebra" ], [ "adjoint", "adjoint" ], [ "action", "action" ], [ "linear transformation", "linear transformation" ] ], "qualifier": "Lie theory", "raw_glosses": [ "(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.", "(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)." ], "raw_tags": [ "of an element x of a Lie algebra L" ], "tags": [ "not-comparable" ], "topics": [ "algebra", "mathematics", "sciences" ] }, { "categories": [ "en:Algebra", "en:Mathematics" ], "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." ], "links": [ [ "mathematics", "mathematics" ], [ "algebra", "algebra" ], [ "algebraic structure", "algebraic structure" ], [ "Lie theory", "Lie theory" ], [ "lower central series", "lower central series" ] ], "qualifier": "Lie theory", "raw_glosses": [ "(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.", "(Lie theory, of a Lie algebra) Such that the lower central series terminates." ], "raw_tags": [ "of a Lie algebra" ], "tags": [ "not-comparable" ], "topics": [ "algebra", "mathematics", "sciences" ] }, { "categories": [ "en:Algebra", "en:Group theory", "en:Mathematics" ], "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." ], "links": [ [ "mathematics", "mathematics" ], [ "algebra", "algebra" ], [ "algebraic structure", "algebraic structure" ], [ "group theory", "group theory" ], [ "group", "group" ], [ "central series", "central series" ] ], "raw_glosses": [ "(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.", "(group theory, of a group) Admitting a central series of finite length." ], "raw_tags": [ "of a group" ], "tags": [ "not-comparable" ], "topics": [ "algebra", "group-theory", "mathematics", "sciences" ] }, { "categories": [ "en:Algebra", "en:Mathematics" ], "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." ], "links": [ [ "mathematics", "mathematics" ], [ "algebra", "algebra" ], [ "algebraic structure", "algebraic structure" ], [ "ring theory", "ring theory" ], [ "ideal", "ideal" ], [ "natural number", "natural number" ] ], "qualifier": "ring theory", "raw_glosses": [ "(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.", "(ring theory, of an ideal I) Such that there exists a natural number k with Iᵏ = 0." ], "raw_tags": [ "of an ideal I" ], "tags": [ "not-comparable" ], "topics": [ "algebra", "mathematics", "sciences" ] }, { "categories": [ "en:Algebra", "en:Mathematics" ], "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." ], "links": [ [ "mathematics", "mathematics" ], [ "algebra", "algebra" ], [ "algebraic structure", "algebraic structure" ], [ "semigroup", "semigroup" ], [ "zero", "zero" ] ], "qualifier": "semigroup theory", "raw_glosses": [ "(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.", "(semigroup theory, of a semigroup with zero) Containing only nilpotent elements." ], "raw_tags": [ "of a semigroup with zero" ], "tags": [ "not-comparable" ], "topics": [ "algebra", "mathematics", "sciences" ] }, { "categories": [ "en:Algebra", "en:Mathematics" ], "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." ], "links": [ [ "mathematics", "mathematics" ], [ "algebra", "algebra" ], [ "algebraic structure", "algebraic structure" ], [ "commutative", "commutative" ], [ "ring", "ring" ], [ "natural number", "natural number" ], [ "index", "index" ] ], "raw_glosses": [ "(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.", "(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." ], "raw_tags": [ "of an algebra over a commutative ring" ], "tags": [ "not-comparable" ], "topics": [ "algebra", "mathematics", "sciences" ] } ], "sounds": [ { "ipa": "/nɪlˈpəʊtənt/" }, { "audio": "LL-Q1860 (eng)-Flame, not lame-nilpotent.wav", "mp3_url": "https://upload.wikimedia.org/wikipedia/commons/transcoded/d/d0/LL-Q1860_%28eng%29-Flame%2C_not_lame-nilpotent.wav/LL-Q1860_%28eng%29-Flame%2C_not_lame-nilpotent.wav.mp3", "ogg_url": "https://upload.wikimedia.org/wikipedia/commons/transcoded/d/d0/LL-Q1860_%28eng%29-Flame%2C_not_lame-nilpotent.wav/LL-Q1860_%28eng%29-Flame%2C_not_lame-nilpotent.wav.ogg" } ], "translations": [ { "code": "cs", "lang": "Czech", "sense": "(algebra)", "word": "nilpotentní" }, { "code": "da", "lang": "Danish", "sense": "(algebra)", "word": "nilpotent" }, { "code": "eo", "lang": "Esperanto", "sense": "(algebra)", "word": "nilpotenta" }, { "code": "eo", "lang": "Esperanto", "sense": "(algebra)", "word": "nulpotenca" }, { "code": "fi", "lang": "Finnish", "sense": "(algebra)", "word": "nilpotentti" }, { "code": "fr", "lang": "French", "sense": "(algebra)", "word": "nilpotent" }, { "code": "ru", "lang": "Russian", "roman": "nilʹpotent", "sense": "(algebra)", "word": "нильпотент" }, { "code": "es", "lang": "Spanish", "sense": "(algebra)", "word": "nilpotente" } ], "wikipedia": [ "Benjamin Peirce", "nilpotent" ], "word": "nilpotent" } { "categories": [ "English adjectives", "English compound terms", "English countable nouns", "English entries with incorrect language header", "English lemmas", "English nouns", "English terms derived from Latin", "English uncomparable adjectives", "English undefined derivations", "Entries with translation boxes", "Pages with 1 entry", "Pages with entries", "Terms with Czech translations", "Terms with Danish translations", "Terms with Esperanto translations", "Terms with Finnish translations", "Terms with French translations", "Terms with Russian translations", "Terms with Spanish translations" ], "etymology_templates": [ { "args": { "1": "en", "2": "nil", "3": "potent", "t1": "not any", "t2": "having power" }, "expansion": "nil (“not any”) + potent (“having power”)", "name": "compound" }, { "args": { "1": "en", "2": "la", "3": "-" }, "expansion": "Latin", "name": "uder" } ], "etymology_text": "From nil (“not any”) + potent (“having power”) with literal meaning “having zero power” - bearing Latin roots nil and potens.\nCoined in 1870, along with idempotent, by American mathematician Benjamin Peirce to describe elements of associative algebras.", "forms": [ { "form": "nilpotents", "tags": [ "plural" ] } ], "head_templates": [ { "args": {}, "expansion": "nilpotent (plural nilpotents)", "name": "en-noun" } ], "lang": "English", "lang_code": "en", "pos": "noun", "senses": [ { "categories": [ "English terms with quotations", "en:Algebra" ], "examples": [ { "ref": "2015, Garret Sobczyk, “Part I: Vector Analysis of Spinors”, in arXiv:", "text": "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.", "type": "quote" } ], "glosses": [ "A nilpotent element." ], "links": [ [ "algebra", "algebra" ] ], "raw_glosses": [ "(algebra) A nilpotent element." ], "topics": [ "algebra", "mathematics", "sciences" ] } ], "sounds": [ { "ipa": "/nɪlˈpəʊtənt/" }, { "audio": "LL-Q1860 (eng)-Flame, not lame-nilpotent.wav", "mp3_url": "https://upload.wikimedia.org/wikipedia/commons/transcoded/d/d0/LL-Q1860_%28eng%29-Flame%2C_not_lame-nilpotent.wav/LL-Q1860_%28eng%29-Flame%2C_not_lame-nilpotent.wav.mp3", "ogg_url": "https://upload.wikimedia.org/wikipedia/commons/transcoded/d/d0/LL-Q1860_%28eng%29-Flame%2C_not_lame-nilpotent.wav/LL-Q1860_%28eng%29-Flame%2C_not_lame-nilpotent.wav.ogg" } ], "wikipedia": [ "Benjamin Peirce", "nilpotent" ], "word": "nilpotent" }
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