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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 ...
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 ).
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 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 .
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 ...