lieAlgebraBasis -- computes an enhanced basis for a simple Lie algebra

Usage:

lieAlgebraBasis("A",2)
Inputs:
- t, a string,
- m, an integer,
- Method, a string,
- Check, a Boolean value,
Optional inputs:
- Check
- Method
Outputs:
- LAB, an instance of the type LieAlgebraBasis,

Description

See LieAlgebraBasis for more details and references.

The optional argument "Method" may be set to "Lusztig" for the Lusztig canonical basis, or "FH" for the basis described in Fulton-Harris, {\it Representation Theory: A First Course. The default is "FH" for types A, B, C, D, and G.

The optional argument "Check" (default: true) runs a suite of tests on the basis constructed. See the unexported function "checkLieAlgebraBasis" in the package code for more details.

The user may either input the type and rank, or the simple Lie algebra.

i1 : LAB=lieAlgebraBasis("A",2);

i2 : peek LAB

o2 = LieAlgebraBasis{BasisElements => {| 1 0  0 |, | 0 0 0  |, | 0 1 0 |, | 0 0 0 |, | 0 0 1 |, | 0 0 0 |, | 0 0 0 |, | 0 0 0 |}                                    }
                                       | 0 -1 0 |  | 0 1 0  |  | 0 0 0 |  | 0 0 1 |  | 0 0 0 |  | 1 0 0 |  | 0 0 0 |  | 0 0 0 |
                                       | 0 0  0 |  | 0 0 -1 |  | 0 0 0 |  | 0 0 0 |  | 0 0 0 |  | 0 0 0 |  | 0 1 0 |  | 1 0 0 |
                     Bracket => FunctionClosure[../../../../../Macaulay2/packages/LieAlgebraRepresentations/LieAlgebraBases/lieAlgebraBasisTypeAFH.m2:168:10-168:26]
                     DualBasis => {| 2/3 0    0    |, | 1/3 0   0    |, | 0 0 0 |, | 0 0 0 |, | 0 0 0 |, | 0 1 0 |, | 0 0 0 |, | 0 0 1 |}
                                   | 0   -1/3 0    |  | 0   1/3 0    |  | 1 0 0 |  | 0 0 0 |  | 0 0 0 |  | 0 0 0 |  | 0 0 1 |  | 0 0 0 |
                                   | 0   0    -1/3 |  | 0   0   -2/3 |  | 0 0 0 |  | 0 1 0 |  | 1 0 0 |  | 0 0 0 |  | 0 0 0 |  | 0 0 0 |
                     Labels => {H_1, H_2, E_(1,2), E_(2,3), E_(1,3), E_(2,1), E_(3,2), E_(3,1)}
                     LieAlgebra => 𝔞
                                    2
                     LoweringOperatorIndices => {5, 6, 7}
                     RaisingOperatorIndices => {2, 3, 4}
                     Weights => {{0, 0}, {0, 0}, {2, -1}, {-1, 2}, {1, 1}, {-2, 1}, {1, -2}, {-1, -1}}
                     WriteInBasis => writeInslnBasis

i3 : sl3=simpleLieAlgebra("A",2);

i4 : lieAlgebraBasis(sl3)===LAB

o4 = false

LAB#"BasisElements" LABLusztig = lieAlgebraBasis("A",2,"Method"=>Lusztig); LABLusztig#"BasisElements"

Ways to use lieAlgebraBasis:

lieAlgebraBasis(LieAlgebra)
lieAlgebraBasis(String,ZZ)

For the programmer

The object lieAlgebraBasis is a method function with options.

The source of this document is in LieAlgebraRepresentations/documentation.m2:1418:0.