Program Manual
Maple Program
SimpleLieAlgebra.mpl (v2.21)
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Version note
In the version 2.21, various corrections are made due to the change of the structure of the weight system data produced by proc:WeightSystem. Now, the procedure can treat larger dimensional representations. New procedures, proc:mkSOSLembM, proc:SRWStoDWS and proc:DWStoSRWS, are added. Proc:mkembM is renamed as proc:mkRSembM.
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Common abbreviation
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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
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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
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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
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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
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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
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StrConstWB( dt[, sw] ) ⇒ NN=table(N[wv1,wv2]): Weyl Basic CCR coeff. - sw=1 ⇒ with an output display, sw=0 ⇒ no output display
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StrConst( NNtable ) ⇒ shows the list of non-vanishing structure constants Weight system
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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.
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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
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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
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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
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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
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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]
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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
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mkSOSLembM( d ) ⇒table[embM[H],embM[S],embM[D]]Constructing an embedding matrix for the canonical embedding SO(d) → SL(d)
GUT Symmetry breaking chain list
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SBpattern( dt, dts[, sw] ) ⇒ the SB chain list: [ [[SBlist,[[],Unbrknlist],...] - dt = the Dynkin type of the initial algebra
- dts = a list of Dynkin types for the final algebra
- sw=0 ⇒ output = the SB chain list, no display,
- sw=1 ⇒ output = the SB chain list + its monitor display.
