Maple Program

SimpleLieAlgebra.mpl (v2.22)

Version note

In this version 2.22, minor bugs in the preloaded list of embedding matrixes are fixed.
Common abbreviation
dt = Dynkin type (e.g. A4, E6)
wv = a weight vector in the simple root basis (e.g. [2/5,4/5,11/5,8/5])
rv = a root vector in the simple root basis (e.g. [1,1,0,0])
dw = Dynkin label for the weight (e.g. [0,-1,2,-1] for A4)
hdw = the highest Dynkin label for an irrep (e.g. [1,0,0,1] for A4)
basis = "S" or "D": "S"= the simple root basis and "D"= the dynkin basis
Metric
Kmetric( dt ) ⇒ Killing metric matrix in the simple root basis
Cmatrix( dt ) ⇒ Cartan matrix
Gmetric( dt ) ⇒ Killing metric matrix in the Dynkin basis
Inner products
IPDB( dw1, dw2, dt ) ⇒ Inner product of Dynkin weights
RIP( wv1, wv2, dt ) ⇒ Inner product of weights in the simple root basis
RCP( wv, rv, dt ) ⇒ Cartan product ❬sw,rv❭
Basis change
DBtoSRB( dw, dt ) ⇒ the coordinate trf of a weight: Dynkin basis to the Simple root basis
SRBtoDB( wv, dt ) ⇒ the coordinate trf of a weight: Simple root basis to the Dynkin basis
Root system
HighestRoot( dt ) ⇒ the highest root in the simple root basis
Rlevel( wv ) ⇒ the level of a root wv
RootSystem( dt[, basis, sw] ) ⇒ the root system in the specified basis.
sw=1 ⇒ with an output display, sw=0 ⇒ no output display

Structure constant
StrConstWB( dt[, sw] ) ⇒ NN=table(N[wv1,wv2]): Weyl Basic CCR coeff.
sw=1 ⇒ with an output display, sw=0 ⇒ no output display
StrConst( NNtable ) ⇒ shows the list of non-vanishing structure constants
Weight system
WeightSystem( dt, hdw, [, basis, sw] ) ⇒ WStable: a table including the weight system DWS (in the Dynkin)or SRWS (in the simple root basis as an entry of the table type)
WStable = table("dt"=dt, "hdw"=hdw, "dim"=dim, "hl"=heighest level, "SRWS"=SRWS::table(or "DWS"=DWS::table))
sw=0 ⇒ output=WStable. No monitor output.
sw=1 ⇒ output=WStable. Display the weight system.
sw=2 ⇒ output= the flat weight list.

printWS(WStable) ⇒ shows a formated display of a weight system.
findHighestWeight( dw, dt ) ⇒ hdw: the highest weight of the irrep to which a given weight dw belongs to in Dynkin basis.
SRWStoDWS(SRWS) ⇒ converts the WStable in the Simple Root Basis created by proc:WeightSystem to that in the Dynkin Basis.
DWStoSRWS(DWS) ⇒ convert the WStable in the Dynkin Basis created by proc:WeightSystem to that in the Simple Root Basis.
Weyl transformation
WeylTrf( dt, rv[, basis] ) ⇒ Weyl trf matrix in the basis wrt the root vector rv in the simple root basis
Irr. decomposition of a product of irreps
ProductOfRep( dt, hdw1, hdw2[, sw, basis] ) ⇒ the irreps list
sw=0 (default) ⇒ output = Irreps list only
sw=1 ⇒ output = Irreps list with a formated display
sw=2 ⇒ output = Irreps list with display of the composite weight system in the Simple Root Basis (basis="S") or the Dynkin Basis(basis="D",default)

Maximal quasi-semisimple subalgebras
(quasi-semisimple = semisimple+U(1) factors)
MaxSubGlist( dt ) ⇒ [[the list of regular subalgs],[the list of special subalgs]]
MaxSubGlist0( dt ) ⇒ [ the flat list of all maximal subalgs].

Irr. decomposition of an irrep of a simple algebra L wrt a subalgebra H
SubGrdm( dt, hwd, dts, embMD[,sw] ) ⇒ DWS/WL
dts = a list of Dynkin types representing a subalgebra H: e.g. SU4xSO9xU1 ⇒ [A3,B4, U1]
embMD = a projection matrix of the Dynkin labels; Dynkin weight for L, dw ⇒ Dynkin weights for H, dw'=embMD dw.
Projection matrices for the maximal quasi-semisimple subalgebras of low-rank simple algebra are preregistered in this program (see the procedure embMlist below), but for the other cases, you have to construct the projection matrices by yourself when you use the procedure SubGrdm.
sw=0 ⇒ output=Irrep list. No monitor output
sw=1 ⇒ output=Irrep list with monitor display
sw=2 ⇒ output=Irrep list with monitor display of the projected Dynkin weight system
sw=3 ⇒ output=Irrep list + monitor display of the projeced Dynkin weight system together with the original weight system

In the output of SubGrdm, each list in the derived irreps has the structure
[[U(1) charges], highest level] = [[hdw's], [dims], multiplicity ]
Embedding of algebras
MaySubG( dts1, dts0 ) ⇒ maplist : [dts0[1]=[dts1[1],..], dts0[2]=[dts1[3],..]]
dts0 = a target list of simple algebras: e.g. [B3,A1]
dts1 = a list of may be subalgebras: e.g. [A2, A1]

embMlist( dt ) ⇒ the list of indices Z for the projection matrices preregisted in this program: embMD = embM[D]Z.

mkRSembM( dt, nodepos, type ) ⇒ table(["subalgebra"=dts,"embM"=embM])
dts= the list of the Dynkin types of the resulting subalgebra. e.g. [A4,U1] for D5
embM = table([D=embM[D],S=embM[S],H=embM[H]])
embM[D]=projection matrix for Dynkin labels,
embM[S]=matrix specifying the pull back of simple roots,
embM[H]=matrix specifying the embedding of the Cartan subalgebra
nodepos= a node position to remove from the (extended) Dynkin diagram
type=1 => remove one node from the Dynkin diagram and add U1
type=2 => use the extended Dynkin diagram

mkSOSLembM( d ) ⇒table[embM[H],embM[S],embM[D]]
Constructing an embedding matrix for the canonical embedding SO(d) → SL(d)