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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 ...
The definition of a Lie algebra can be reformulated more abstractly in the language of category theory. Namely, one can define a Lie algebra in terms of linear maps—that is, morphisms in the category of vector spaces—without considering individual elements. (In this section, the field over which the algebra is defined is assumed to be of ...
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 ...
The two-dimensional dual numbers contain a nilpotent space. Other algebras and numbers that contain nilpotent spaces include split-quaternions (coquaternions), split-octonions, biquaternions, and complex octonions. If a nilpotent infinitesimal is a variable tending to zero, it can be shown that any sum of terms for which it is the subject is an ...
Central series. In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, the existence of a central series means it is a nilpotent group; for matrix rings (considered as Lie algebras ...
A Lie group can be defined as a (Hausdorff) topological group that, near the identity element, looks like a transformation group, with no reference to differentiable manifolds. [14] First, we define an immersely linear Lie group to be a subgroup G of the general linear group such that.
The double cover H of SL(2,R), known as the metaplectic group, is a Lie group that cannot be viewed as a linear algebraic group over R. More strongly, H has no faithful finite-dimensional representation. Anatoly Maltsev showed that every simply connected nilpotent Lie group can be viewed as a unipotent algebraic group G over R in a unique way. [27]
2. A simple Lie algebra is a Lie algebra that is non abelian and has only two ideals, itself and. { 0 } {\displaystyle \ {0\}} . 3. simply laced group (a simple Lie group is simply laced when its Dynkin diagram is without multiple edges). 4. simple root. A subset. Δ {\displaystyle \Delta } of a root system.