LieAlgebraBasis -- class for an enhanced Lie algebra basis

Description

This class represents a specific kind of basis of a Lie algebra. We assume that the basis is adapted to the decomposition of $\mathfrak{g}$ into its root spaces, i.e. $\mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\Phi^{+}} \mathfrak{g}_{\alpha} \oplus \bigoplus_{\Phi^{+}} \mathfrak{g}_{-\alpha}$.

This class also stores additional information about the basis in addition to the basis elements themselves. For instance, it records the weights of the basis elements and their dual elements with respect to the Killing form.

Currently, the package computes three types of bases for simple Lie algebras.

1. the Lusztig canonical basis, as detailed in Geck and Lang, "Canonical structure constants for simple Lie algebras", arXiv:2404.07652v1. This basis is available for any simple Lie algebra.

2. natural bases of the matrix Lie algebras $sl_n$, $sp(2n), $so(m)$, as described by Fulton and Harris in Representation Theory: A First Course, Sections 15.1, 16.1, and 18.1, respectively.

3. the basis of $g_2$ described by Fulton and Harris in Representation Theory: A First Course, Section 22.1.

The default option is to return the Fulton-Harris basis in types A, B, C, D, and G.

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

o1 = Enhanced basis of 𝔞
                        2

o1 : LieAlgebraBasis

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

Functions and methods returning an object of class LieAlgebraBasis:

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

Methods that use an object of class LieAlgebraBasis:

adjointRepresentation(LieAlgebraBasis) -- see adjointRepresentation -- creates the adjoint representation of a Lie algebra
isLieAlgebraRepresentation(LieAlgebraBasis,List) -- see isLieAlgebraRepresentation -- checks whether a list of matrices defines a Lie algebra representation
lieAlgebraRepresentation(LieAlgebraModule,LieAlgebraBasis,List) -- see lieAlgebraRepresentation -- create a LieAlgebraRepresentation
net(LieAlgebraBasis) (missing documentation)
trivialRepresentation(LieAlgebraBasis) -- see trivialRepresentation -- creates the trivial representation of a Lie algebra
universalEnvelopingAlgebra(LieAlgebraBasis) -- see universalEnvelopingAlgebra -- computes the universal enveloping algebra of a Lie algebra
uNminus(LieAlgebraBasis) -- see uNminus -- computes the universal enveloping algebra of the Lie algebra $N^{-}$

For the programmer

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

The source of this document is in LieAlgebraRepresentations/documentation.m2:1378:0.