|
Adjoint Action
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL(n, \mathbb), the Lie group of real ''n''-by-''n'' invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible ''n''-by-''n'' matrix g to an endomorphism of the vector space of all linear transformations of \mathbb^n defined by: x \mapsto g x g^ . For any Lie group, this natural representation is obtained by linearizing (i.e. taking the differential of) the action of ''G'' on itself by conjugation. The adjoint representation can be defined for linear algebraic groups over arbitrary fields. Definition Let ''G'' be a Lie group, and let :\Psi: G \to \operatorname(G) be the mapping , with Aut(''G'') the automorphism group of ''G'' and given by the inner automorphism (conjugation) :\Psi_g(h)= ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra Automorphism
In abstract algebra, an automorphism of a Lie algebra \mathfrak g is an isomorphism from \mathfrak g to itself, that is, a linear map preserving the Lie bracket. The set of automorphisms of \mathfrak are denoted \text(\mathfrak), the automorphism group of \mathfrak. Inner and outer automorphisms The subgroup of \operatorname(\mathfrak g) generated using the adjoint action e^, x \in \mathfrak g is called the inner automorphism group of \mathfrak g. The group is denoted \operatorname^0(\mathfrak). These form a normal subgroup in the group of automorphisms, and the quotient \operatorname(\mathfrak)/\operatorname^0(\mathfrak) is known as the outer automorphism group. Diagram automorphisms It is known that the outer automorphism group for a simple Lie algebra \mathfrak is isomorphic to the group of diagram automorphisms for the corresponding Dynkin diagram in the classification of Lie algebras. The only algebras with non-trivial outer automorphism group are therefore A_n (n \geq 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
