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In mathematics, a Lie algebra is nilpotent if its lower central series terminates in the zero subalgebra. The lower central series is the sequence of subalgebras. We write , and for all . If the lower central series eventually arrives at the zero subalgebra, then the Lie algebra is called nilpotent. The lower central series for Lie algebras is ...
Nilpotent algebra. In mathematics, specifically in ring theory, a nilpotent algebra over a commutative ring is an algebra over a commutative ring, in which for some positive integer n every product containing at least n elements of the algebra is zero. The concept of a nilpotent Lie algebra has a different definition, which depends upon the Lie ...
t. e. In mathematics, a Lie algebra (pronounced / liː / LEE) is a vector space together with an operation called the Lie bracket, an alternating bilinear map , that satisfies the Jacobi identity. In other words, a Lie algebra is an algebra over a field for which the multiplication operation (called the Lie bracket) is alternating and satisfies ...
As linear operators form an associative algebra and thus a ring, this is a special case of the initial definition. [ 4 ] [ 5 ] More generally, in view of the above definitions, an operator Q {\displaystyle Q} is nilpotent if there is n ∈ N {\displaystyle n\in \mathbb {N} } such that Q n = 0 {\displaystyle Q^{n}=0} (the zero function ).
Definition. An element X of a semisimple Lie algebra g is called nilpotent if its adjoint endomorphism. ad X: g → g, ad X (Y) = [X, Y] is nilpotent, that is, (ad X) n = 0 for large enough n. Equivalently, X is nilpotent if its characteristic polynomial pad X (t) is equal to tdim g. A semisimple Lie group or algebraic group G acts on its Lie ...
In mathematics, the Baker–Campbell–Hausdorff formula gives the value of that solves the equation for possibly noncommutative X and Y in the Lie algebra of a Lie group. There are various ways of writing the formula, but all ultimately yield an expression for in Lie algebraic terms, that is, as a formal series (not necessarily convergent) in ...
t. e. In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra of a Lie algebra that is self-normalising (if for all , then ). They were introduced by Élie Cartan in his doctoral thesis. It controls the representation theory of a semi-simple Lie algebra over a field of characteristic .
In algebra, the nilradical of a Lie algebra is a nilpotent ideal, which is as large as possible. The nilradical of a finite-dimensional Lie algebra is its maximal nilpotent ideal, which exists because the sum of any two nilpotent ideals is nilpotent. It is an ideal in the radical of the Lie algebra . The quotient of a Lie algebra by its ...