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LieAlgebra -- the class of all Lie algebras


This type represents graded Lie algebras. More precisely, the Lie algebras under consideration are graded over ${\mathbb Z}^{+}\times {\mathbb Z}^n\times {\mathbb Z}/2{\mathbb Z}$, where the first component is called the degree, and the last component is called the sign, which is 0 or 1 and have effect on the axioms, see below. The list of components of the grading except the sign is called the weight, and the last component of the weight is called the homological degree. If no differential is defined, then the homological degree will be zero for all elements. Each object of type LieAlgebra is also a type L, and elements in L belong also to the type LieElement, which is the parent of L. Lie multiplication is given by LieElement LieElement.

The axioms for Lie algebras depend on the signs of the generators, which are specified by lieAlgebra(...,Signs=>...). The sign of a homogeneous element can be obtained by the function sign. In the axioms below the sign of an element $a$ is written sign($a$).

Anticommutativity: [$a$, $b$] = -(-1)^{sign($a$) * sign($b$)} [$b$, $a$]

Jacobi identity: [$a$, [$b$, $c$]] = [[$a$, $b$], $c$] + (-1)^{sign($a$) * sign($b$)} [$b$, [$a$, $c$]]

Also, in characteristic 2 and 3, there are in addition the following axioms:

Characteristic 2: [$a$, $a$] = 0

Characteristic 3: [$a$, [$a$, $a$]] = 0

i1 : L = lieAlgebra{a,b}

o1 = L

o1 : LieAlgebra
i2 : a

o2 = a

o2 : L
i3 : class L

o3 = LieAlgebra

o3 : Type
i4 : parent L

o4 = LieElement

o4 : Type
i5 : a b

o5 =  - (b a)

o5 : L
i6 : weight oo

o6 = {2, 0}

o6 : List

See also

Functions and methods returning an object of class LieAlgebra :

Methods that use an object of class LieAlgebra :

For the programmer

The object LieAlgebra is a type, with ancestor classes HashTable < Thing.