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

Lie superalgebra Orthogonal symmetric Lie algebra Poisson algebra Pre-Lie algebra Quantum groups Moyal algebra Quasi-Frobenius Lie algebra Quasi-Lie algebra...

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...

the theory of holonomy; or algebraically through Lie theory, which allowed Cartan to give a complete classification. Symmetric spaces commonly occur in...

inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact. The orthogonal group in dimension n has two connected...

most familiar Clifford algebras, the orthogonal Clifford algebras, are also referred to as (pseudo-)Riemannian Clifford algebras, as distinct from symplectic...

decomposition of the orthogonal symmetric Lie algebra yields a direct product decomposition of the corresponding compact symmetric space H / K when H is...

nondegenerate, symmetric bilinear form of signature (p, q), where n = p + q. It is also called the pseudo-orthogonal group or generalized orthogonal group. The...

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...

semisimple Lie algebras, Cartan's theory of symmetric spaces, and Hermann Weyl's description of representations of compact and semisimple Lie groups using...

In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express...

the field of real and complex numbers. A real Lie algebra g0 is called a real form of a complex Lie algebra g if g is the complexification of g0: g ≃ g...

In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its...

{\displaystyle \mathbb {H} } together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms...

Monster Lie algebra Nilpotent Lie algebra Nilradical of a Lie algebra Orthogonal symmetric Lie algebra Parabolic Lie algebra Pre-Lie algebra Quadratic Lie algebra...

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...

pair ( g {\displaystyle {\mathfrak {g}}} , σ) is called an orthogonal symmetric Lie algebra of compact type. Any inner product on g {\displaystyle {\mathfrak...

In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group...

field of Lie theory, there are two definitions of a compact Lie algebra. Extrinsically and topologically, a compact Lie algebra is the Lie algebra of a compact...

example, orthogonal groups, general linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups...

represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group Sp(2n, C) is denoted Cn, and Sp(n) is the...

after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion...

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...

and Fourier analysis (orthogonal basis). Nearly all scientific computations involve linear algebra. Consequently, linear algebra algorithms have been highly...

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

matrices with positive determinant. This is also a Lie group of dimension n2; it has the same Lie algebra as GL(n, R). The polar decomposition, which is unique...

In the theory of Lie groups, the exponential map is a map from the Lie algebra g {\displaystyle {\mathfrak {g}}} of a Lie group G {\displaystyle G} to...

Group ring Localization of a ring Tensor algebra Symmetric algebra, Exterior algebra, Clifford algebra Free algebra Completion (ring theory) Types Field (mathematics)...

Multilinear algebra Symmetric algebra, the symmetric analog Tensor algebra Weyl algebra, a quantum deformation of the symmetric algebra by a symplectic...

equipped with a nonsingular real symmetric bilinear form ( ⋅ , ⋅ ) {\displaystyle (\cdot ,\cdot )} instead, the unital *-algebra generated by the elements of...