mathematical field of Lie theory, the radical of a Lie algebra g {\displaystyle {\mathfrak {g}}} is the largest solvable ideal of g . {\displaystyle {\mathfrak...

mathematics, a Lie algebra (pronounced /liː/ LEE) is a vector space g {\displaystyle {\mathfrak {g}}} together with an operation called the Lie bracket, an...

mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero...

algebra, a concept in Lie theory Nilradical of a Lie algebra, a nilpotent ideal which is as large as possible Left (or right) radical of a bilinear form...

mathematics, a Lie algebra g {\displaystyle {\mathfrak {g}}} is solvable if its derived series terminates in the zero subalgebra. The derived Lie algebra of the...

a direct sum of a semisimple Lie algebra and an abelian Lie algebra: g = s ⊕ a ; {\displaystyle {\mathfrak {g}}={\mathfrak {s}}\oplus {\mathfrak {a}};}...

associative algebra A over a commutative ring (often a field) K is a ring A together with a ring homomorphism from K into the center of A. This is thus...

radical r a d ( g ) {\displaystyle {\mathfrak {rad}}({\mathfrak {g}})} of the Lie algebra g {\displaystyle {\mathfrak {g}}} . The quotient of a Lie algebra...

algebras. Analogues of the Jordan–Chevalley decomposition also exist for elements of Linear algebraic groups and Lie groups via a multiplicative reformulation...

finite-dimensional Lie algebra g over a field of characteristic zero is the semidirect product of a solvable ideal and a semisimple subalgebra. One is its radical, a maximal...

defect of surjectivity is measured by Galois cohomology). Similarly to the Lie group–Lie algebra correspondence, to an algebraic group over a field k...

In ring theory, a branch of mathematics, the radical of an ideal I {\displaystyle I} of a commutative ring is another ideal defined by the property that...

ring theory, a branch of mathematics, a semisimple algebra is an associative Artinian algebra over a field which has trivial Jacobson radical (only the zero...

is a glossary for the terminology applied in the mathematical theories of Lie groups and Lie algebras. For the topics in the representation theory of Lie...

used for the study of algebraic varieties, which are solution sets of systems of polynomial equations. Weyl algebras and Lie algebras may be considered...

ring Baer ring, Rickart ring Lie ring, Lie algebra Ideal (Lie algebra) Jordan algebra Differential algebra Banach algebra Examples Rational number, Real...

M^{T}} is the transpose of M {\displaystyle M} . Many Lie groups can be viewed as linear algebraic groups over the field of real or complex numbers....

In algebra, Weyl's theorem on complete reducibility is a fundamental result in the theory of Lie algebra representations (specifically in the representation...

arise as Hopf algebras depending on an auxiliary parameter q or h, which become universal enveloping algebras of a certain Lie algebra, frequently semisimple...

group Algebraic torus Reductive group Borel subgroup Radical of an algebraic group Unipotent radical Lie-Kolchin theorem Haboush's theorem (also known as...

annihilated by the Jacobson radical of R. In the context of Lie algebras, a socle of a symmetric Lie algebra is the eigenspace of its structural automorphism...

algebraic groups with respect to nilpotent Lie algebras. Recall that a nilpotent Lie algebra is a subalgebra of some g l n {\displaystyle {\mathfrak {gl}}_{n}}...

the simple algebraic groups are classified by Dynkin diagrams, as in the theory of compact Lie groups or complex semisimple Lie algebras. Reductive groups...

the article Radical of a ring for more on this. The nilradical of a Lie algebra is similarly defined for Lie algebras. The nilradical of a commutative...

nested radicals Solvable Lie algebra, a Lie algebra whose derived series reaches the zero algebra in finitely many steps Solvable problem, a computational...

can be applied to any conformal vertex algebra of central charge 26, and the output naturally has a Lie algebra structure. The Goddard–Thorn theorem can...

Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both...

that nilradical is the intersection of all prime ideals. Let g {\displaystyle {\mathfrak {g}}} be a Lie algebra. Then an element x ∈ g {\displaystyle...

mathematics, specifically the theory of Lie algebras, Lie's theorem states that, over an algebraically closed field of characteristic zero, if π : g → g...

the Jacobson radical lies in a few different cases: namely its applications and the resulting geometric interpretations, and its algebraic interpretations...