nilpotent

1. not comparablering theoryalgebraSuch that, for some positive integer n, xⁿ = 0.

   If a square matrix is upper triangular and has zeros on the diagonal, then it is nilpotent (under the usual matrix multiplication).

   1 more example

   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

2. 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. not comparableLie theoryalgebraBelonging to the derived algebra of L and such that the adjoint action of x is nilpotent (as a linear transformation on L).
   2. not comparableLie theoryalgebraSuch that the lower central series terminates.
   3. not comparablealgebraAdmitting a central series of finite length.
   4. not comparablering theoryalgebraSuch that there exists a natural number k with Iᵏ = 0.
   5. not comparablesemigroup theoryalgebraContaining only nilpotent elements.
   6. not comparablealgebraSuch 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.

plural nilpotents

1. algebraA nilpotent element.

   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.

   2015, Garret Sobczyk, “Part I: Vector Analysis of Spinors”, in arXiv