LieAlgebras[QuotientAlgebra] - find the structure equations for a quotient algebra of a Lie algebra by an ideal
QuotientAlgebra(h, m, Algname, keyword)
h - a list of independent vectors defining an ideal 𝔥 in a Lie algebra 𝔤
m - a list of independent vectors defining a vector space complement to 𝔥 in 𝔤
Algname - a name or a string, the name assigned to the quotient algebra 𝔤/𝔥
keyword - (optional) the keyword "Matrix"
Let 𝔤 be a Lie algebra and 𝔥 an ideal in 𝔤. Then elements of the quotient algebra 𝔤/𝔥are the cosets x +𝔥 , where x ∈ 𝔤. The Lie bracket on 𝔤/𝔥 is defined by x +𝔥, y + 𝔥 = x, y + 𝔥. If vectors y1, y2, ... ,yr form a basis for a complement to 𝔥, then the cosets y1+ 𝔥, y2 + 𝔥, ... ,yr +𝔥 form a basis for 𝔤/𝔥.
The program QuotientAlgebra(h, m) creates a Lie algebra data structure for the quotient algebra 𝔤/𝔥. using the vectors in the complement m as the representative basis elements for 𝔤/𝔥.
A Lie algebra data structure contains the structure constants in a standard format used by the LieAlgebras package (see LieAlgebraData). The command DGsetup is then used to initialize a Lie algebra -- that is, to define the basis elements for the Lie algebra and its dual and to store the structure constants for the Lie algebra in memory.
With the optional keyword present, QuotientAlgebra(h, m, "Matrix") returns the Lie algebra data structure for 𝔤/𝔥 and the matrix representation of the canonical projection map π : 𝔤 →𝔤/𝔥 defined by x→ x + 𝔥.
The command QuotientAlgebra is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form QuotientAlgebra(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-QuotientAlgebra(...).
First initialize a Lie algebra and display the multiplication table.
L1 ≔ _DG⁡LieAlgebra,Alg1,5,1,5,1,2,2,3,1,1,2,5,2,1,2,5,3,1,3,5,3,1,4,5,4,2
L1 ≔ e1,e5=2⁢e1,e2,e3=e1,e2,e5=e2+e3,e3,e5=e3,e4,e5=2⁢e4
Check that h1= e1, e3 is an ideal and find the quotient algebra (call it Alg2) using the complementary vectors e2, e4, e5
h1 ≔ e1,e3:m1 ≔ e2,e4,e5:
L2 ≔ QuotientAlgebra⁡h1,m1,Alg2
L2 ≔ e1,e3=e1,e2,e3=2⁢e2
Rerun QuotientAlgebra with the keyword argument "Matrix".
L2,A ≔ QuotientAlgebra⁡h1,m1,Alg2,Matrix
We use the DifferentialGeometrycommand Transformation to convert the matrix A into a transformation Π from Alg1 to the quotient algebra Alg2.
PI ≔ Transformation⁡Alg1,Alg2,A
Π ≔ e1,0⁢f1,e2,f1,e3,0⁢f1,e4,f2,e5,f3
We can check that Π is a Lie algebra homomorphism.
We see that Π sends e1 to 0, e2 to f1 and so on.
We can verify that [e1 , e3 is a basis for the kernel of Π and that the image of Π is spanned by f1, f2, f3 (so that Π is surjective).
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