In abstract algebra, an automorphism of a Lie algebra g {\displaystyle {\mathfrak {g}}} is an isomorphism from g {\displaystyle {\mathfrak {g}}} to itself...

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

In abstract algebra, an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called...

trivial automorphism. The automorphisms of an object X form a group under composition of morphisms, which is called the automorphism group of X. This...

affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given...

mathematics, Lie group–Lie algebra correspondence allows one to correspond a Lie group to a Lie algebra or vice versa, and study the conditions for such a relationship...

used to read off the list of simple Lie algebras and Riemannian symmetric spaces. Together with the commutative Lie group of the real numbers, R {\displaystyle...

mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure...

self-isomorphisms or "automorphisms". Diagram automorphisms correspond to outer automorphisms of the Lie algebra, meaning that the outer automorphism group Out =...

F)} is a Lie subgroup of GL ⁡ ( n , F ) {\displaystyle \operatorname {GL} (n,F)} of dimension n 2 − 1 {\displaystyle n^{2}-1} . The Lie algebra of SL ⁡...

phrase group of Lie type usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with...

The underlying real Lie algebra of the complex Lie algebra G2 has dimension 28. It has complex conjugation as an outer automorphism and is simply connected...

identity element of the group G). Since Ψ g {\displaystyle \Psi _{g}} is a Lie group automorphism, Adg is a Lie algebra automorphism; i.e., an invertible...

In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation...

automorphism (corresponding to an automorphism of the Dynkin diagram), and a field automorphism (coming from an automorphism of the field k). For a k-simple...

mathematics, E7 is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e7, all of which have dimension 133; the...

outer automorphism, that is, an automorphism that is not in the image of Ad. The derived subalgebra of a finite dimensional solvable Lie algebra over a field...

the theory of real and complex Lie algebras. This difference can be traced to the properties of Frobenius automorphism and to the failure of the exponential...

an automorphism of a conjugacy class is always a conjugacy class (the same or another). The composition of two automorphisms is again an automorphism, and...

This (real) Lie algebra has dimension n2 − 1. More information about the structure of this Lie algebra can be found below in § Lie algebra structure. In...

In mathematics, a restricted Lie algebra (or p-Lie algebra) is a Lie algebra over a field of characteristic p>0 together with an additional "pth power"...

mathematics, the outer automorphism group of a group, G, is the quotient, Aut(G) / Inn(G), where Aut(G) is the automorphism group of G and Inn(G) is the...

Its Lie algebra is a quotient of the complexification of the Lie algebra of the original group. They are isomorphic if the original group has a quotient...

complex automorphism. As well as the complex Lie group of type E6, there are five real forms of the Lie algebra, and correspondingly five real forms of the...

the group of isotopies described below. See also: PSL(2,7) – the automorphism group of the Fano plane. An isotopy of an algebra is a triple of bijective...

an orthogonal symmetric Lie algebra is a pair ( g , s ) {\displaystyle ({\mathfrak {g}},s)} consisting of a real Lie algebra g {\displaystyle {\mathfrak...

Unlike ordinary rings, vertex algebras admit a notion of twisted module attached to an automorphism. For an automorphism σ of order N, the action has the...

In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations...

In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional...

mathematical study of Lie algebras and Lie groups, Satake diagrams are a generalization of Dynkin diagrams that classify involutions of root systems that...