|
Query[Gradation] - check if a list of subspaces defines a gradation of a Lie algebra
Calling Sequences
Query([g0, g1, ..., gN], "Gradation")
Parameters
g0, g1, - a list of independent vectors defining subspaces of a Lie algebra g
|
|
Description
|
|
|
•
|
A collection of subspaces g0, g1, ... gN of a Lie algebra g defines a gradation of g if g = g0 + g1 + ... + gN (vector space direct sum) with [gi, gj] in g(i + j) for i + j <= N and [gi, gj] = 0 for i + j > N.
|
|
•
|
Query([g0, g1, g2, ... gN], "Gradation") returns true if the subspaces g0, g1, g2, ..., gN define a gradation of the Lie algebra g.
|
|
•
|
The command Query is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form Query(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Query(...).
|
|
|
|
Examples
|
|
|
>
|
 
|
Example 1.
The Lie algebra of 4 x 4 Upper triangular matrices is a 10 dimensional Lie algebra which is naturally graded - g0 consists of the matrices with only non-zero elements on the diagonal, g1 consists of the matrices with non-elements immediately above the diagonal (the super diagonal) and so on.
We use Query to verify this. First we use the program MatrixAlgebras to generate the Lie algebra data structure for the Lie algebra of upper triangular matrices. Here eij denotes the matrix with a 1 in the i-th row and j-th column.
|
>
|
|
|
![L1 := [[e1, e2] = e2, [e1, e3] = e3, [e1, e4] = e4, [e2, e5] = e2, [e2, e6] = e3, [e2, e7] = e4, [e3, e8] = e3, [e3, e9] = e4, [e4, e10] = e4, [e5, e6] = e6, [e5, e7] = e7, [e6, e8] = e6, [e6, e9] = e7, [e7, e10] = e7, [e8, e9] = e9, [e9, e10] = e9], [e11, e12, e13, e14, e22, e23, e24, e33, e34, e44], [epsilon11, epsilon12, epsilon13, epsilon14, epsilon22, epsilon23, epsilon24, epsilon33, epsilon34, epsilon44]](/support/helpjp/helpview.aspx?si=7166/file07650/math95.png)
| (2.1) |
|
>
|
|
To display the Lie algebra multiplication table, we need to increase the value of the interface parameter rtablesize.
|
upper_gl4R >
|
|
|
| (2.2) |
|
upper_gl4R >
|
|
|
| (2.3) |
Now define the 4 subspaces which will define our gradation.
|
upper_gl4R >
|
![Query([g0, g1, g2, g3], "Gradation")](/support/helpjp/helpview.aspx?si=7166/file07650/math134.png)
|
|
| (2.4) |
|
|
