In mathematics, the special linear Lie algebra of order n {\displaystyle n} over a field F {\displaystyle F} , denoted s l n F {\displaystyle {\mathfrak...

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...

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 linear algebra, the trace of a square matrix A, denoted tr(A), is the sum of the elements on its main diagonal, a 11 + a 22 + ⋯ + a n n {\displaystyle...

correspondence between Lie groups and Lie algebras, the notion of a real form can be defined for Lie groups. In the case of linear algebraic groups, the notions...

{\displaystyle R} is a field). These elements are "special" in that they form an algebraic subvariety of the general linear group – they satisfy a polynomial equation...

{sl}}(n+1)=\{x\in {\mathfrak {gl}}(n+1):{\text{tr}}(x)=0\}} , the special linear Lie algebra; B n := o ( 2 n + 1 ) = { x ∈ g l ( 2 n + 1 ) : x + x T = 0 }...

area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group...

standard generators of the special linear Lie algebra sl2. This notion plays an important role in the theory of semisimple Lie algebras, especially in regard...

{\displaystyle M} . Many Lie groups can be viewed as linear algebraic groups over the field of real or complex numbers. (For example, every compact Lie group can be...

abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures...

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...

examples of Lie groups. The projective linear group PGL ⁡ ( n , F ) {\displaystyle \operatorname {PGL} (n,F)} and the projective special linear group PSL...

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...

decomposition Lie groupoid Lie subgroup Complex Lie group Local Lie group Poisson–Lie group Real Lie groups Simple Lie group Solvable Lie algebra Special linear Lie...

linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally in algebraic geometry...

matrix Lie algebra, there is a linear group (matrix Lie group) with this algebra as its Lie algebra. On the other hand, Lie groups with isomorphic Lie algebras...

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...

the Lie algebra of a maximal torus of the compact group. If g {\displaystyle {\mathfrak {g}}} is a linear Lie algebra (a Lie subalgebra of the Lie algebra...

Sheldon Jay (1997), Linear Algebra Done Right (2nd ed.), Springer-Verlag, ISBN 0-387-98259-0. Lay, David C. (2005), Linear Algebra and Its Applications...

mathematics and theoretical physics, a representation of a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation...

F)} , and called the orthogonal Lie algebra or special orthogonal Lie algebra. Over real numbers, these Lie algebras for different n are the compact real...

This is a list of Lie group topics, by Wikipedia page. See Table of Lie groups for a list General linear group, special linear group SL2(R) SL2(C) Unitary...

of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms...

In linear algebra and functional analysis, a projection is a linear transformation P {\displaystyle P} from a vector space to itself (an endomorphism)...

Heisenberg group is a linear space, vectors in the Lie algebra can be canonically identified with vectors in the group. The Lie algebra of the Heisenberg...

classical Chevalley Lie algebra A 6 {\displaystyle A_{6}} through the special linear group and its corresponding special linear Lie algebra. In the third dimension...

variables. Linear algebra is a closely related field that investigates linear equations and combinations of them called systems of linear equations. It...

function), sine lemniscate function Special linear group in mathematics, denoted SLn or SL(n) Special linear Lie algebra, denoted s l n ( F ) {\displaystyle...

Linear algebra is the branch of mathematics concerning linear equations such as a 1 x 1 + ⋯ + a n x n = b , {\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b...