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In mathematics, a Lie algebra (pronounced / liː / LEE) is a vector space together with an operation called the Lie bracket, an alternating bilinear map , that satisfies the Jacobi identity.
Examples 1.6. (a)Any vector space V is a Lie algebra for the zero bracket. (b)For any associative unital algebra Aover R, the space of matrices with entries in A, gl(n;A) = Mat n(A), is a Lie algebra, with bracket the commutator. In particular, we have Lie algebras gl(n;R); gl(n;C); gl(n;H): (c)If Ais commutative, then the subspace sl(n;A) gl(n ...
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- Introduction.
- Vocabulary.
- Classical (Simple) Lie Algebras.
- Special Linear Lie Algebra.
- Orthogonal Lie Algebra on Odd Dimensional Spaces.
- Symplectic Lie Algebra.
- Exceptional Lie Algebras.
This article is meant to provide a quick reference guide to Lie algebras: the terminology, important theorems, and a brief overview of the subject. Physicists usually call the elements of Lie algebras generators, as for them they are merely differentials of trajectories, tangent vector fields generated by some operators. Thus the distinction betwee...
A connected linear algebraic group over an algebraically closed field is called semisimple if every smooth connected solvable normal subgroup of is trivial. A connected linear algebraic group over an algebraically closed field is called reductive if every smooth connected unipotent ( upper triangular matrices with s on the diagonal), normal subgrou...
The following subalgebras of of linear transformations on a finite, dimensional vector space are called the classical Lie algebras . They are all simple, and plus five exceptional Lie algebras () all simple ones there are. Let be a Cartan subalgebra of dimension . Let us further define as the matrix whose entry in the i-th row and j-th column is an...
Type: are all linear transformations on , i.e. matrices with vanishing trace. It is thus of dimension Basis: The ‘special unitary’ Lie algebras of skew-Hermtian complex matrices with trace are of this type. There is a complex basis transformation of the real vector spaces This means they are the same real Lie algebra. The basis transformation, howe...
Type: Let be the nondegenerate, symmetric bilinear form on whose matrix is Then the orthogonal algebra is Basis: (according to the choice of )
Type: Let be the nondegenerate, skew-symmetric bilinear form on whose matrix is Then the symplectic algebra is Basis: (according to the choice of )
The actual construction of the exceptional Lie algebras uses concepts like Jordan algebras, octonions and their derivation algebras which will lead too far, so let us summarize them as a list: Many of these simple Lie algebras contain other simple Lie algebras as subalgebras, e.g. or see the info graphic on Wikipedia for Whenever we speak of semisi...
Apr 23, 2016 · A Lie algebra is a unitary k -module L over a commutative ring k with a unit that is endowed with a bilinear mapping (x, y) ↦ [x, y] of L × L into L having the following two properties: 1) [x, x] = 0 (hence the anti-commutative law [x, y] = − [y, x] ); 2) [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 (the Jacobi identity).
Lie Algebras 6 1.7 Definition(Lie Ideal). I⊂g is a Lie ideal if for all x∈g and a∈I, [x,a] ∈I. Claiming instead that [a,x] ∈Iyields the same definition. 1.8 Definition(Center). The center of a Lie algebra g is {x∈g : ∀a, [x,a] = 0}. 1.9 Definition(Adjoint Homomorphism). For g a lie algebra, we define the adjoint
Mar 15, 2024 · Lie Algebra. A nonassociative algebra obeyed by objects such as the Lie bracket and Poisson bracket. Elements , , and of a Lie algebra satisfy. (1) (2) and. (3) (the Jacobi identity ). The relation implies. (4) For characteristic not equal to two, these two relations are equivalent. The binary operation of a Lie algebra is the bracket. (5)
tive multiplication X·Y). We make Ainto a Lie algebra L(also called Aas Lie algebra) by defining [XY] = X·Y−Y·X. The Jacobi identity holds; just “multiply out”. As a simple case, F L is the trivial Lie algebra, of dimension 1 and Abelian. For another “concrete” case see Example 12. Example 2: A special case of Example 1: Take for ...