Semisimple element information

context: A semisimple element in the endomorphism ring of a vector space is a semisimple operator. In a semisimple Lie algebra, an element is semisimple if its...

regular element of a Lie algebra or Lie group is an element whose centralizer has dimension as small as possible. For example, in a complex semisimple Lie...

In 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...

mathematics, a semisimple algebra is an associative artinian algebra over a field which has trivial Jacobson radical (only the zero element of the algebra...

the order of Δ, the semisimple rank of G. Every parabolic subgroup of G is conjugate to a subgroup containing B by some element of G(k). As a result...

that the coefficients of the characteristic equation of a regular semisimple element of a Lie algebra are invariant under the adjoint group, from which...

semisimplicity. Hence, g {\displaystyle {\mathfrak {g}}} must have a nonzero semisimple element, say x; the linear span of x is then a toral subalgebra. Maximal torus...

Lie algebra is a direct sum of a commutative algebra and a semisimple algebra. An element v of V is said to be g {\displaystyle {\mathfrak {g}}} -invariant...

In mathematics, the representation theory of semisimple Lie algebras is one of the crowning achievements of the theory of Lie groups and Lie algebras....

Killing form, the corresponding Casimir element is uniquely defined up to a constant. For a general semisimple Lie algebra, the space of invariant bilinear...

groups over F1: Given a Dynkin diagram for a semisimple algebraic group, its Weyl group is the semisimple algebraic group over F1. The affine scheme Spec...

Any element g of a linear algebraic group over a perfect field can be written uniquely as the product g = gu  gs of commuting unipotent and semisimple elements...

matrices implies that every element g of GL(n,k) can be written uniquely as a product g = gssgu such that gss is semisimple, gu is unipotent, and gss and...

in module theory, the largest semisimple quotient of a module Top, written ⊤ or 1, in lattice theory, the greatest element in a partially ordered set Top...

characteristic zero, every finite-dimensional representation of a semisimple Lie algebra is semisimple (that is, a direct sum of irreducible representations). A...

(left)-semisimple if it is semisimple as a left module over itself. Surprisingly, a left-semisimple ring is also right-semisimple and vice versa. The left/right...

half the number of roots; and 2m+n is the dimension of the corresponding semisimple Lie algebra. If the highest root is ∑ m i α i {\displaystyle \sum m_{i}\alpha...

simple and also semisimple algebras over fields. Simple algebras are building blocks of semisimple algebras: any finite-dimensional semisimple algebra is a...

highest weight classifies the irreducible representations of a complex semisimple Lie algebra g {\displaystyle {\mathfrak {g}}} . There is a closely related...

whose L-group is the connected component of the centralizer of a semisimple element of the L-group of G. In the stable trace formula, unstable orbital...

semisimple Lie algebra, an element is semisimple if its image under the adjoint representation is semisimple; see Semisimple Lie algebra#Jordan decomposition...

representation theory of real and complex semisimple Lie groups and semisimple Lie algebras. An element X of a semisimple Lie algebra g is called nilpotent if...

integral element. This integrality condition is related to, but not identical to, the notion of integral element in the setting of semisimple Lie algebras...

important examples of these. The Weyl group of a semisimple Lie group, a semisimple Lie algebra, a semisimple linear algebraic group, etc. is the Weyl group...