|
#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Willem de Graaf.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## This file contains functions to construct semisimple Lie algebras of type
## $A_n$, $B_n$, $C_n$, $D_n$, $E_6$, $E_7$, $E_8$, $F_4$, $G_2$,
## as s.c. algebras. Also there are the restricted Lie algebras
## of types W,H,K,S.
##
## The algorithm used for types $A-G$ is the one described in
## Kac, Infinite Dimensional Lie Algebras, and de Graaf, Lie Algebras:
## Theory and Algorithms.
##
##
##############################################################################
##
#F AddendumSCTable( <T>, <i>, <j>, <k>, <val> )
##
## This function adds the structure constant c_{ij}^k to the table 'T'.
## If 'T[i][j]' contains already some constants, then 'k' and 'val' have
## to be inserted at the right position.
##
BindGlobal( "AddendumSCTable", function( T, i, j, k, val )
local pos,m,r,inds,cfs;
pos:= Position( T[i][j][1], k );
if pos = fail then
if T[i][j][1] = [] then
SetEntrySCTable( T, i, j, [ val, k ] );
else
m:=T[i][j][1][1];
r:=1;
inds:=[];
cfs:=[];
while m<k do
Add(inds,m);
Add(cfs,T[i][j][2][r]);
r:=r+1;
if r > Length(T[i][j][1]) then
m:= k;
else
m:= T[i][j][1][r];
fi;
od;
Add(inds,k);
Add(cfs,val);
while r <= Length(T[i][j][1]) do
Add(inds,T[i][j][1][r]);
Add(cfs,T[i][j][2][r]);
r:=r+1;
od;
T[i][j]:= [inds,cfs];
T[j][i]:= [inds,-cfs];
fi;
else
cfs:= ShallowCopy( T[i][j][2] );
cfs[pos]:= cfs[pos]+val;
T[i][j]:= [T[i][j][1], cfs];
cfs:= ShallowCopy( T[j][i][2] );
cfs[pos]:= cfs[pos]-val;
T[j][i]:= [T[j][i][1], cfs];
fi;
end );
BindGlobal( "SimpleLieAlgebraTypeA_G", function( type, n, F )
local T, # The table of the Lie algebra constructed.
i,j,k,l, # Loop variables.
lst, # A list.
R, # Positive roots
cc, # List of coefficients.
lenR, # length of 'R'
Rij, # The sum of two roots from 'R'.
eps, # The so-called "epsilon"-function.
epsmat, # A matrix used to calculate the eps-function.
dim, # The dimension of the Lie algebra.
C, # Cartan matrix
L, # Lie algebra, result
vectors, # vectors spanning a Cartan subalgebra
CSA, # List of indices of the basis vectors of a Cartan
# subalgebra.
e,
inds, # List of indices.
r,r1,r2, # Roots.
roots, # List of roots.
primes, # List of lists of corresponding roots.
B, # Basis of a vector space.
cfs, # List of coefficient lists.
d, # Order of the diagram automorphism.
found, # Boolean.
a,
q,
perm, # Permutation representing the diagram automorphism.
shorts,
posR, # Positive roots.
CartanMatrixToPositiveRoots; # Function for determining the
# positive roots.
CartanMatrixToPositiveRoots:= function( C )
local rank, posr, ready, ind, le, i, a, j, ej, r, b,
q;
rank:= Length( C );
# `posr' will be a list of the positive roots. We start with the
# simple roots, which are simply unit vectors.
posr:= IdentityMat( rank );
ready:= false;
ind:= 1;
le:= rank;
while ind <= le do
# We loop over those elements of `posR' that have been found in
# the previous round, i.e., those at positions ranging from
# `ind' to `le'.
le:= Length( posr );
for i in [ind..le] do
a:= posr[i];
# We determine whether a+ej is a root (where ej is the j-th
# simple root.
for j in [1..rank] do
ej:= posr[j];
# We determine the maximum number `r' such that a-r*ej is
# a root.
r:= -1;
b:= ShallowCopy( a );
while b in posr do
b:= b-ej;
r:=r+1;
od;
q:= r-LinearCombination( TransposedMat( C )[j], a );
if q>0 and (not a+ej in posr ) then
Add( posr, a+ej );
fi;
od;
od;
ind:= le+1;
le:= Length( posr );
od;
return posr;
end;
# The following function is the so-called epsilon function.
eps:= function( a, b, epm )
local rk;
rk:= Length( epm );
return Product( [1..rk],i ->
Product( [1..rk], j ->
epm[i][j] ^ ( a[i]*b[j] ) ) );
end;
if type in [ "A", "D", "E" ] then
# We are in the simply-laced case. Here we construct the root
# system and the matrix of the epsilon function. Then we can
# fill the multiplication table directly.
C:= 2*IdentityMat( n );
if type = "A" then
for i in [1..n-1] do
C[i][i+1]:= -1;
C[i+1][i]:= -1;
od;
elif type = "D" then
if n < 4 then
Error("<n> must be >= 4");
fi;
for i in [1..n-2] do
C[i][i+1]:= -1;
C[i+1][i]:= -1;
od;
C[n-2][n]:=-1;
C[n][n-2]:= -1;
else
C:= [
[ 2, 0, -1, 0, 0, 0, 0, 0 ], [ 0, 2, 0, -1, 0, 0, 0, 0 ],
[ -1, 0, 2, -1, 0, 0, 0, 0 ], [ 0, -1, -1, 2, -1, 0, 0, 0 ],
[ 0, 0, 0, -1, 2, -1, 0, 0 ], [ 0, 0, 0, 0, -1, 2, -1, 0 ],
[ 0, 0, 0, 0, 0, -1, 2, -1 ], [ 0, 0, 0, 0, 0, 0, -1, 2 ] ];
if n = 6 then
C:= C{ [ 1 .. 6 ] }{ [ 1 .. 6 ] };
elif n = 7 then
C:= C{ [ 1 .. 7 ] }{ [ 1 .. 7 ] };
elif n < 6 or 8 < n then
Error( "<n> must be one of 6, 7, 8" );
fi;
fi;
R:= CartanMatrixToPositiveRoots( C );
# We conctruct `epsmat', which satisfies
# /
# |-1 if i=j,
# epsmat[i][j] = |-1 if i and j are connected, and i>j
# | 1 if i and j are not connected or i<j.
# \
# (where `connected' means connected in the Dynkin diagram.
epsmat:= [];
for i in [ 1 .. n ] do
epsmat[i]:= [];
for j in [ 1 .. i-1 ] do
epsmat[i][j]:= 1;
od;
epsmat[i][i]:= -1;
for j in [ i+1 .. n ] do
epsmat[i][j]:= (-1)^C[i][j];
od;
od;
lenR:= Length( R );
dim:= 2*lenR + n;
posR:= List( R, r -> Zero(F)*r );
# Initialize the s.c. table
T:= EmptySCTable( dim, Zero(F), "antisymmetric" );
lst:= [ 1 .. n ] + 2 * lenR;
for i in [1..lenR] do
for j in [i..lenR] do
Rij:= R[i]+R[j];
if Rij in R then
k:= Position(R,Rij);
e:= eps(R[i],R[j],epsmat)*One(F);
SetEntrySCTable( T, i, j, [ e, k ] );
SetEntrySCTable( T, i+lenR, j+lenR, [ -e, k+lenR ] );
fi;
if i = j and T[i][j+lenR] = [[],[]] then
# We form the product x_{\alpha_i}*x_{-\alpha_i}, which
# will be an element of the Cartan subalgebra.
inds:= Filtered( [1..n], x -> R[i][x] <> 0 );
T[i][j+lenR]:= [ lst{inds}, R[i]{inds}*One(F) ];
T[j+lenR][i]:= [ lst{inds}, -R[i]{inds}*One(F) ];
fi;
od;
od;
for i in [1..lenR] do
for j in [1..lenR] do
Rij:= R[i]-R[j];
if Rij in R then
k:= Position(R,Rij);
SetEntrySCTable( T, i, j+lenR,
[-One(F)*eps(R[i],-R[j],epsmat),k] );
elif -Rij in R then
k:= Position(R,-Rij);
SetEntrySCTable( T, i, j+lenR,
[One(F)*eps(R[i],-R[j],epsmat),k+lenR] );
fi;
od;
for j in [1..n] do
# We take care of the comutation relations of the form
# [h_j,x_{\beta_i}]= < \beta_i, \alpha_j > x_{\beta_i}.
cc:= LinearCombination( R[i], C[j] );
if cc <> 0*cc then
posR[i][j]:= One(F)*cc;
T[2*lenR+j][i]:=[[i],[One(F)*cc]];
T[i][2*lenR+j]:=[[i],[-One(F)*cc]];
T[2*lenR+j][i+lenR]:=[[i+lenR],[-One(F)*cc]];
T[i+lenR][2*lenR+j]:=[[i+lenR],[One(F)*cc]];
fi;
od;
od;
L:= LieAlgebraByStructureConstants( F, T );
# A Cartan subalgebra is spanned by the last 'n' basis elements.
CSA:= [ dim-n+1 .. dim ];
vectors:= BasisVectors( CanonicalBasis( L ) ){ CSA };
SetCartanSubalgebra( L, SubalgebraNC( L, vectors, "basis" ) );
SetIsRestrictedLieAlgebra( L, Characteristic( F ) > 0 );
elif type in [ "B", "C", "F", "G" ] then
# Now we are in the non simply laced case. In each case we construct
# a simply laced root system, which has a diagram automorphism.
# We take an epsilon function which is invariant under the diagram
# automorphism. Furthermore, the permutation `perm' will represent
# the diagram aotomorphism as acting on the roots (so that
# Permuted( r, perm ) is the result of applying the diagram
# automorphism to the root r).
if type = "B" then
# In this case we construct D_{n+1}.
if n <= 1 then
Error( "<n> must be >= 2");
fi;
C:= 2*IdentityMat( n+1 );
for i in [1..n-1] do
C[i][i+1]:= -1;
C[i+1][i]:= -1;
od;
C[n-1][n+1]:=-1;
C[n+1][n-1]:= -1;
R:= CartanMatrixToPositiveRoots( C );
epsmat:= NullMat( n+1, n+1 ) + 1;
for i in [ 1 .. n-1 ] do
epsmat[i+1][i]:= -1;
epsmat[i][i]:= -1;
od;
epsmat[n+1][n-1]:= -1;
epsmat[n][n]:= -1;
epsmat[n+1][n+1]:= -1;
perm:= (n,n+1);
d:= 2;
elif type = "C" then
# In this case we construct A_{2n-1}.
if n < 2 then
Error( "<n> must be >= 3");
fi;
C:= 2*IdentityMat( 2*n-1 );
for i in [1..2*n-2] do
C[i][i+1]:= -1;
C[i+1][i]:= -1;
od;
R:= CartanMatrixToPositiveRoots( C );
epsmat:= NullMat( 2*n-1, 2*n-1 ) + 1;
for i in [ 1 .. n-1 ] do
epsmat[i][i+1]:= -1;
epsmat[i][i]:= -1;
od;
for i in [n..2*n-2] do
epsmat[i+1][i]:= -1;
epsmat[i][i]:= -1;
od;
epsmat[2*n-1][2*n-1]:= -1;
perm:= ();
for i in [1..n-1] do
perm:= perm*(i,2*n-i);
od;
d:= 2;
elif type = "F" then
# In this case we construct E_6.
if n <> 4 then
Error( "<n> must be equal to 4");
fi;
C:= IdentityMat( 6 );
C[1][3]:=-1; C[2][4]:=-1; C[3][4]:=-1; C[4][5]:=-1; C[5][6]:=-1;
C:= C+TransposedMat( C );
R:= CartanMatrixToPositiveRoots( C );
epsmat:= NullMat( 6, 6 ) + 1;
for i in [1..6] do epsmat[i][i]:= -1; od;
epsmat[1][3]:=-1; epsmat[3][4]:=-1; epsmat[5][4]:=-1;
epsmat[6][5]:=-1; epsmat[2][4]:=-1;
perm:= (1,6)*(3,5);
d:= 2;
elif type = "G" then
# In this case we conctruct D_4.
if n <> 2 then
Error( "<n> must be equal to 2");
fi;
C:= IdentityMat( 4 );
C[1][2]:=-1; C[2][3]:=-1; C[2][4]:=-1;
C:= C+TransposedMat( C );
R:= CartanMatrixToPositiveRoots( C );
epsmat:= NullMat( 4, 4 ) + 1;
for i in [1..4] do epsmat[i][i]:= -1; od;
epsmat[1][2]:=-1; epsmat[4][2]:=-1; epsmat[3][2]:=-1;
perm:= (1,3,4);
d:= 3;
fi;
# Now `roots' will be the list of positive roots of the resulting Lie
# algebra. They are formed from the roots in `R' by applying the
# diagram automorphism. If a r\in R is invariant under the
# automorphism, then it is added to `roots' (and its prime is
# the root itself). Otherwise we add \frac{1}{d}(r+\phi(r)+\cdots
# + \phi^{d-1}(r)), where \phi is the diagram automorphism.
# In this case the prime of the root are all \phi^i(r).
if d = 2 then
roots:= [ ];
primes:= [ ];
for r in R do
r1:= Permuted( r, perm );
if r = r1 then
Add( roots, r );
Add( primes, [ r ] );
else
if not (r+r1)/2 in roots then
Add( roots, (r+r1)/2 );
Add( primes, [ r, r1 ] );
fi;
fi;
od;
B:= Basis( VectorSpace( Rationals, roots{[1..n]} ),roots{[1..n]});
cfs:= List( roots, x -> Coefficients( B, x ) );
elif d = 3 then
roots:= [ ];
primes:= [ ];
for r in R do
r1:= Permuted( r, perm );
if r = r1 then
Add( roots, r );
Add( primes, [ r ] );
else
r2:= (r+r1+Permuted(r1,perm))/3;
if not r2 in roots then
Add( roots, r2 );
Add( primes, [ r, r1, Permuted( r1, perm ) ] );
fi;
fi;
od;
B:= Basis( VectorSpace( Rationals, roots{[1..n]} ),roots{[1..n]});
cfs:= List( roots, x -> Coefficients( B, x ) );
fi;
# `shorts' will be a list of indices indicating where the
# short simple roots are. The coefficients on those places
# in `cfs' need to be divided by `d'.
shorts:= Filtered( [1..n], ii -> Length( primes[ii] ) > 1 );
for i in [1..Length(cfs)] do
for j in shorts do
cfs[i][j]:= cfs[i][j]/d;
od;
od;
Append( R, -R );
lenR:= Length( roots );
dim:= 2*lenR + n;
posR:= List( [1..lenR], ii -> List( [1..n], jj -> Zero( F ) ) );
# Initialize the s.c. table
T:= EmptySCTable( dim, Zero(F), "antisymmetric" );
lst:= [ 1 .. n ] + 2 * lenR;
for i in [1..lenR] do
for j in [i..lenR] do
Rij:= roots[i]+roots[j];
if Rij in roots then
# We look for `r' in `primes[i]' and `r1' in `primes[j]'
# such that `r+r1' lies in `R'.
found:= false;
for k in [1..Length(primes[i])] do
if found then break; fi;
r:= primes[i][k];
for l in [1..Length(primes[j])] do
r1:= primes[j][l];
if r+r1 in R then
found := true; break;
fi;
od;
od;
# `q' will be the maximal integer such that `roots[i]-
# roots[j]' is a root.
k:= Position( roots, Rij );
q:=0; a:= roots[i] - roots[j];
while a in roots or -a in roots do
q:=q+1;
a:= a-roots[j];
od;
e:= eps(r,r1,epsmat)*(q+1)*One(F);
SetEntrySCTable( T, i, j, [ e, k ] );
SetEntrySCTable( T, i+lenR, j+lenR, [ -e, k+lenR ] );
fi;
if i = j and T[i][j+lenR] = [[],[]] then
# We form the product x_{\alpha_i}*x_{-\alpha_i}, which
# will be an element of the Cartan subalgebra.
inds:= Filtered( [1..n], x -> cfs[i][x] <> 0 );
if Length( primes[i] ) = 1 then
T[i][j+lenR]:= [ lst{inds}, cfs[i]{inds}*One(F) ];
T[j+lenR][i]:= [ lst{inds}, -cfs[i]{inds}*One(F) ];
else
T[i][j+lenR]:= [ lst{inds}, cfs[i]{inds}*d*One(F) ];
T[j+lenR][i]:= [ lst{inds}, -cfs[i]{inds}*d*One(F) ];
fi;
fi;
od;
od;
for i in [1..lenR] do
for j in [1..lenR] do
Rij:= roots[i]-roots[j];
if Rij in roots then
found:= false;
for k in [1..Length(primes[i])] do
if found then break; fi;
r:= primes[i][k];
for l in [1..Length(primes[j])] do
r1:= primes[j][l];
if r-r1 in R then
found := true; break;
fi;
od;
od;
k:= Position( roots, Rij );
q:=0; a:= roots[i] + roots[j];
while a in roots or -a in roots do
q:=q+1;
a:= a+roots[j];
od;
SetEntrySCTable( T, i, j+lenR,
[-One(F)*(q+1)*eps(r,-r1,epsmat),k] );
elif -Rij in roots then
found:= false;
for k in [1..Length(primes[i])] do
if found then break; fi;
r:= primes[i][k];
for l in [1..Length(primes[j])] do
r1:= primes[j][l];
if r-r1 in R then
found := true; break;
fi;
od;
od;
k:= Position( roots, -Rij );
q:=0; a:= roots[i] + roots[j];
while a in roots or -a in roots do
q:=q+1;
a:= a+roots[j];
od;
SetEntrySCTable( T, i, j+lenR,
[One(F)*(q+1)*eps(r,-r1,epsmat),k+lenR] );
fi;
od;
for j in [1..n] do
# Now we take care of the relations [h,x_{\beta}]....
cc:= LinearCombination( roots[i], C[j] );
if Length( primes[j] ) > 1 then
# i.e., `roots[j]' is "short".
cc:= d*cc;
fi;
if cc <> 0*cc then
posR[i][j]:= One(F)*cc;
T[2*lenR+j][i]:=[[i],[One(F)*cc]];
T[i][2*lenR+j]:=[[i],[-One(F)*cc]];
T[2*lenR+j][i+lenR]:=[[i+lenR],[-One(F)*cc]];
T[i+lenR][2*lenR+j]:=[[i+lenR],[One(F)*cc]];
fi;
od;
od;
L:= LieAlgebraByStructureConstants( F, T );
# A Cartan subalgebra is spanned by the last 'n' basis elements.
CSA:= [ dim-n+1 .. dim ];
vectors:= BasisVectors( CanonicalBasis( L ) ){ CSA };
SetCartanSubalgebra( L, SubalgebraNC( L, vectors, "basis" ) );
SetIsRestrictedLieAlgebra( L, Characteristic( F ) > 0 );
fi;
R:= Objectify( NewType( NewFamily( "RootSystemFam", IsObject ),
IsAttributeStoringRep and IsRootSystemFromLieAlgebra ),
rec() );
SetUnderlyingLieAlgebra( R, L );
SetPositiveRoots( R, posR );
SetNegativeRoots( R, -posR );
SetSimpleSystem( R, posR{[1..n]} );
SetCanonicalGenerators( R, [ CanonicalBasis( L ){[1..n]},
CanonicalBasis( L ){[lenR+1..lenR+n]},
vectors ] );
SetPositiveRootVectors( R, CanonicalBasis(L){[1..lenR]} );
SetNegativeRootVectors( R, CanonicalBasis(L){[lenR+1..2*lenR]} );
SetChevalleyBasis( L, [ PositiveRootVectors( R ),
NegativeRootVectors( R ),
vectors ] );
if not ( Characteristic( F ) in [ 2, 3 ] ) then
C:= 2*IdentityMat( n );
for i in [1..n] do
for j in [1..n] do
if i <> j then
q:= 0;
r:= posR[i]+posR[j];
while r in posR do
q:=q+1;
r:= r+posR[j];
od;
C[i][j]:= -q;
fi;
od;
od;
SetCartanMatrix( R, C );
SetSemiSimpleType( L, Concatenation( type, String( n ) ) );
fi;
SetRootSystem( L, R );
if Characteristic( F ) = 0 then
SetIsSimpleAlgebra( L, true );
fi;
return L;
end );
##############################################################################
##
#F SimpleLieAlgebraTypeW( <n>, <F> )
##
## The Witt Lie algebra is constructed.
##
## The Witt algebra can be constructed as a subalgebra of the derivation
## algebra of a certain polynomial algebra.
## (see e.g. R. Farnsteiner and H. Strade,
## Modular Lie Algebras and Their Representations, Dekker, New York, 1988.)
## It is determined by a prime p and list of integers
## n=(n_1...n_m). It is spanned by the elements
##
## x^{\alpha}D_j
##
## where \alpha=(i_1..i_m) is a multi index such that 0 <= i_k < p^{n_k}-1
## and 1 <= j <=m. The Lie multiplication is given by
##
## [x^{\alpha}D_i,x^{\beta}D_j]={(\alpha+\beta-\epsilon_i)\choose (\alpha)}*
## x^{\alpha+\beta-\epsilon_i}D_j-{(\alpha+\beta-\epsilon_j)\choose(\beta)}*
## x^{\alpha+\beta-\epsilon_j}D_i.
##
## (We refer to the above mentioned book for the notation.)
##
BindGlobal( "SimpleLieAlgebraTypeW", function( n, F )
local p, # The characteristic of 'F'.
pn,
dim, # The dimension of the resulting Lie algebra.
eltlist, # A list of basis elements of the Lie algebra.
i,j,k, # Loop variables.
u,noa, # Integers.
a, # A list of integers.
T, # Multiplication table.
x1,x2, # Elements from 'eltlist'.
ex, # Multi index.
no, # Integer (position in a list).
cf, # Coefficient (element from 'F').
L; # The Lie algebra.
if not IsList( n ) then
Error( "<n> must be a list of nonnegative integers" );
fi;
p:= Characteristic( F );
if p = 0 then
Error( "<F> must be a field of nonzero characteristic" );
fi;
pn:=p^Sum( n );
dim:= Length( n )*pn;
eltlist:=[];
# First we construct a list of basis elements. A basis element is given by
# a multi index and an integer u such that 1 <= u <=m.
for i in [0..dim-1] do
# calculate the multi-index a and the derivation D_u belonging to i
u:= EuclideanQuotient( i, pn )+1;
noa:= i mod pn;
# Now we calculate the multi index belonging to noa.
# The relation between multi index and number is given as follows:
# if (i_1...i_m) is the multi index then to that index belongs a number
# noa given by
#
# noa = i_1 + p^n[1]( i_2 + p^n[2]( i_3 + .......))
#
a:=[];
for k in [1..Length( n )-1] do
a[k]:= noa mod p^n[k];
noa:= (noa-a[k])/(p^n[k]);
od;
Add( a, noa );
eltlist[i+1]:=[a,u];
od;
# Initialising the table.
T:=EmptySCTable( dim, Zero( F ), "antisymmetric" );
# Filling the table.
for i in [1..dim] do
for j in [i+1..dim] do
# We calculate [x_i,x_j]. This product is a sum of two elements.
x1:= eltlist[i];
x2:= eltlist[j];
if x2[1][x1[2]] > 0 then
ex:= ShallowCopy( x1[1]+x2[1] );
ex[x1[2]]:=ex[x1[2]]-1;
cf:=One(F);
for k in [1..Length( n )] do
cf:= Binomial( ex[k], x1[1][k] ) * cf;
od;
if not IsZero(cf) then
no:=Position(eltlist,[ex,x2[2]]);
AddendumSCTable( T, i, j, no, cf );
fi;
fi;
if x1[1][x2[2]] > 0 then
ex:= ShallowCopy( x1[1]+x2[1] );
ex[x2[2]]:=ex[x2[2]]-1;
cf:=One(F);
for k in [1..Length( n )] do
cf:= Binomial( ex[k], x2[1][k] ) * cf;
od;
if not IsZero(cf) then
no:=Position(eltlist,[ex,x1[2]]);
AddendumSCTable( T, i, j, no, -cf );
fi;
fi;
od;
od;
L:= LieAlgebraByStructureConstants( F, T );
SetIsRestrictedLieAlgebra( L, ForAll( n, x -> x=1 ) );
# We also return the list of basis elements of 'L', because this is needed
# in the functions for the Lie algebras of type 'S' and 'H'.
return [ L, eltlist ];
end );
##############################################################################
##
#F SimpleLieAlgebraTypeS( <n>, <F> )
##
## The "special" Lie algebra is constructed as a subalgebra of the
## Witt Lie algebra. It is spanned by all elements x\in W such that
## div(x)=0, where W is the Witt algebra.
## We refer to the book cited in the comments to the function
## 'SimpleLieAlgebraTypeW' for the details.
##
BindGlobal( "SimpleLieAlgebraTypeS", function( n, F )
local dim, # The dimension of the Witt algebra.
i,j, # Loop variables.
WW, # The output of 'SimpleLieAlgebraTypeW'.
eqs, # The equation system for a basis of the Lie algebra.
divlist, # A list of elements of the Witt algebra.
x, # Element from 'divlist'.
dones, # A list of the elements of 'divlist' that have already
# been processed.
eq, # An equation (to be added to 'eqs').
bas, # Basis vectors of the solution space.
L; # The Lie algebra.
WW:=SimpleLieAlgebraTypeW( n, F );
dim:= Dimension( WW[1] );
divlist:= WW[2];
for i in [1..dim] do
#Apply the operator "div" to the elements of divlist.
divlist[i][1][divlist[i][2]]:=divlist[i][1][divlist[i][2]]-1;
od;
# At some positions of 'divlist' there will be the same element. An equation
# will then be a vector of 1's and 0's such that a 1 appears at every
# position where there is a copy of a particular element. After this we
# do not need to consider this element again, so we add it to 'dones'.
eqs:=[]; dones:=[]; i:=1;
while i <= dim do
eq:=List([1..dim],x->Zero(F));
x:=divlist[i];
if not x in dones then
Add(dones,x);
if x[1][x[2]]>=0 then
eq[i]:= One( F );
for j in [i+1..dim] do
if divlist[j][1]=x[1] then
eq[j]:=One( F );
fi;
od;
Add(eqs,eq);
fi;
fi;
i:=i+1;
od;
bas:= NullspaceMat( TransposedMat( eqs ) );
bas:= List( bas, v -> LinearCombination( Basis( WW[1] ), v ) );
L:= LieDerivedSubalgebra( Subalgebra( WW[1], bas, "basis" ) );
SetIsRestrictedLieAlgebra( L, ForAll( n, x -> x=1 ) );
return L;
end );
##############################################################################
##
#F SimpleLieAlgebraTypeH( <n>, <F> )
##
## Just like the special algebra, the Hamiltonian algebra is constructed as
## a subalgebra of the Witt Lie algebra. It is spanned by the image of
## a linear map D_H which maps a special kind of polynomial algebra into
## the Witt algebra. Again we refer to the book cited in the notes to
## 'SimpleLieAlgebraTypeW' for the details.
##
BindGlobal( "SimpleLieAlgebraTypeH", function( n, F )
local p, # Chracteristic of 'F'.
m, # The length of 'n'.
i,j, # Loop variables.
noa, # Integer.
a, # List of integers "belonging" to 'noa'.
x1,x2, # Multi indices.
mons, # List of multi indices (or monomials).
WW, # The output of 'SimpleLieAlgebraTypeW'.
cf, # List of coefficients of an element of the Witt algebra.
pos, # Position in a list.
sp, # Vector space.
bas, # Basis vectors of the Lie algebra.
L; # The Lie algebra.
p:= Characteristic( F );
if p = 0 then
Error( "<F> must be a field of nonzero characteristic" );
fi;
if not IsList( n ) then
Error( "<n> must be a list of nonnegative integers" );
fi;
m:= Length( n );
if m mod 2 <> 0 then
Error( "<n> must be a list of even length" );
fi;
# 'mons' will be a list of multi indices [i1...1m] such that
# ik < p^n[k] for 1 <= k <= m. The encoding is the same as in
# 'SimpleLieAlgebraTypeW'. The last (or "maximal") element is not taken
# in the list. 'mons' will correspond to the monomials that span the
# algebra which is mapped into the Witt algebra by the map D_H.
mons:= [];
for i in [0..p^Sum( n ) - 2 ] do
a:= [ ];
noa:= i;
for j in [1..m-1] do
a[j]:= noa mod p^n[j];
noa:= (noa-a[j])/(p^n[j]);
od;
a[m]:= noa;
Add(mons,a);
od;
WW:= SimpleLieAlgebraTypeW( n, F );
for i in [1..Length(mons)] do
# The map D_H is applied to the element 'mons[i]'.
x1:= mons[i];
cf:= List( WW[2], e -> Zero(F) );
for j in [1..m/2] do
if x1[j] > 0 then
x2:= ShallowCopy( x1 );
x2[j]:= x2[j] - 1;
pos:= Position( WW[2], [x2,j+m/2] );
cf[pos]:= One( F );
fi;
if x1[j+m/2] > 0 then
x2:= ShallowCopy( x1 );
x2[j+m/2]:= x2[j+m/2] - 1;
pos:= Position( WW[2], [x2,j] );
cf[pos]:= -One( F );
fi;
od;
if not IsZero(cf) then
if IsBound( sp ) then
if not IsContainedInSpan( sp, cf ) then
CloseMutableBasis( sp, cf );
fi;
else
sp:= MutableBasis( F, [ cf ] );
fi;
fi;
od;
bas:= BasisVectors( sp );
bas:= List( bas, x -> LinearCombination( Basis(WW[1]), x ) );
L:= Subalgebra( WW[1], bas, "basis" );
SetIsRestrictedLieAlgebra( L, ForAll( n, x -> x=1 ) );
return L;
end );
##############################################################################
##
#F SimpleLieAlgebraTypeK( <n>, <F> )
##
## The kontact algebra has the same underlying vector space as a
## particular kind of polynomial algebra. On this space a Lie bracket
## is defined. We refer to the book cited in the comments to the function
## 'SimpleLieAlgebraTypeW' for the details.
##
BindGlobal( "SimpleLieAlgebraTypeK", function( n, F )
local p, # The characteristic of 'F'.
m, # The length of 'n'.
pn, # The dimension of the resulting Lie algebra.
eltlist, # List of basis elements of the Lie algebra.
i,j,k, # Loop variables.
noa, # Integer.
a, # The multi index "belonging" to 'noa'.
T,S, # Tables of structure constants.
x1,x2,y1,y2, # Elements from 'eltlist'.
r, # Integer.
pos, # Position in a list.
coef, # Function calculating a product of binomials.
v, # A value.
vals, # A list of values.
ii, # List of indices.
cc, # List of coefficients.
L; # The Lie algebra.
coef:= function( a, b, F )
# Here 'a' and 'b' are two multi indices. This function calculates
# the product of the binomial coefficients 'a[i] \choose b[i]'.
local cf,i;
cf:= One( F );
for i in [1..Length(a)] do
cf:= Binomial( a[i], b[i] ) * cf;
od;
return cf;
end;
p:= Characteristic( F );
if p = 0 then
Error( "<F> must be a field of nonzero characteristic" );
fi;
if not IsList( n ) then
Error( "<n> must be a list of nonnegative integers" );
fi;
m:= Length( n );
if m mod 2 <> 1 or m = 1 then
Error( "<n> must be a list of odd length >= 3" );
fi;
pn:= p^Sum( n );
r:= ( m - 1 )/2;
eltlist:=[];
# First we construct a list of basis elements.
for i in [0..pn-1] do
noa:= i;
a:=[];
for k in [1..m-1] do
a[k]:= noa mod p^n[k];
noa:= (noa-a[k])/(p^n[k]);
od;
a[m]:= noa;
eltlist[i+1]:=a;
od;
# Initialising the table.
T:= EmptySCTable( pn, Zero(F), "antisymmetric" );
for i in [1..pn] do
for j in [i+1..pn] do
# We calculate [x_i,x_j]. The coefficients of this element w.r.t. the basis
# contained in 'eltlist' will be stored in the vector 'vals'.
# The formula for the commutator is quite complicated, and this leads to
# many if-statements. (These if-statements are largely due to the fact that
# D_i(x^a)=0 if a[i]=0, so that we have to check that this element is not 0.)
x1:= eltlist[i];
x2:= eltlist[j];
vals:= List([1..pn],i->Zero( F ) );
for k in [1..r] do
if x1[k] > 0 then
if x2[k+r] > 0 then
y1:= ShallowCopy(x1); y2:= ShallowCopy(x2);
y1[k]:=y1[k]-1; y2[k+r]:=y2[k+r]-1;
v:=coef( y1+y2, y1, F );
if not IsZero(v) then
pos:= Position( eltlist, y1+y2 );
vals[pos]:= vals[pos] + v;
fi;
fi;
if x2[ m ] > 0 then
y1:= ShallowCopy(x1); y2:= ShallowCopy(x2);
y1[k]:=y1[k]-1; y2[ m ]:=y2[ m ]-1;
v:=coef(x1+y2,y1,F)*(x2[k]+1);
if not IsZero(v) then
pos:= Position( eltlist, x1+y2 );
vals[pos]:= vals[pos]-v;
fi;
fi;
fi;
if x1[ m ] > 0 then
if x2[k+r] > 0 then
y1:= ShallowCopy(x1); y2:= ShallowCopy(x2);
y1[m]:=y1[m]-1; y2[k+r]:=y2[k+r]-1;
v:=coef( y1+x2, y2, F )*(x1[k+r]+1);
if not IsZero(v) then
pos:= Position( eltlist, y1+x2 );
vals[pos]:= vals[pos] + v;
fi;
fi;
if x2[ m ] > 0 then
y1:= ShallowCopy(x1); y2:= ShallowCopy(x2);
y1[m]:=y1[m]-1; y2[ m ]:=y2[ m ]-1;
y1[k+r]:=y1[k+r]+1; y2[k]:=y2[k]+1;
v:=coef(y1+y2,y1,F)*y1[k+r]*y2[k];
if not IsZero(v) then
pos:= Position( eltlist, y1+y2 );
vals[pos]:= vals[pos]-v;
fi;
y1:= ShallowCopy(x1); y2:= ShallowCopy(x2);
y1[m]:=y1[m]-1; y2[ m ]:=y2[ m ]-1;
y1[k]:=y1[k]+1; y2[k+r]:=y2[k+r]+1;
v:=coef(y1+y2,y1,F)*y1[k]*y2[k+r];
if not IsZero(v) then
pos:= Position( eltlist, y1+y2 );
vals[pos]:= vals[pos]+v;
fi;
fi;
if x2[k] > 0 then
y1:= ShallowCopy(x1); y2:= ShallowCopy(x2);
y1[m]:=y1[m]-1; y2[k]:=y2[k]-1;
v:=coef( y1+x2, y2, F )*(x1[k]+1);
if not IsZero(v) then
pos:= Position( eltlist, y1+x2 );
vals[pos]:= vals[pos] + v;
fi;
fi;
fi;
if x1[k+r] > 0 then
if x2[k] > 0 then
y1:= ShallowCopy(x1); y2:= ShallowCopy(x2);
y1[k+r]:=y1[k+r]-1; y2[k]:=y2[k]-1;
v:=coef( y1+y2, y1, F );
if not IsZero(v) then
pos:= Position( eltlist, y1+y2 );
vals[pos]:= vals[pos] - v;
fi;
fi;
if x2[ m ] > 0 then
y1:= ShallowCopy(x1); y2:= ShallowCopy(x2);
y1[k+r]:=y1[k+r]-1; y2[ m ]:=y2[ m ]-1;
v:=coef(x1+y2,y1,F)*(x2[k+r]+1);
if not IsZero(v) then
pos:= Position( eltlist, x1+y2 );
vals[pos]:= vals[pos]-v;
fi;
fi;
fi;
if x1[m]>0 then
y1:= ShallowCopy(x1);
y1[m]:=y1[m]-1;
v:=coef(y1+x2,x2,F);
if not IsZero(v) then
pos:= Position( eltlist, y1+x2 );
vals[pos]:= vals[pos]-2*v;
fi;
fi;
if x2[m]>0 then
y2:= ShallowCopy(x2);
y2[m]:=y2[m]-1;
v:= coef(x1+y2,x1,F);
if not IsZero(v) then
pos:= Position( eltlist, x1+y2 );
vals[pos]:= vals[pos]+2*v;
fi;
fi;
od;
# We convert 'vals' to multiplication table format.
ii:=[]; cc:=[];
for k in [1..Length(vals)] do
if not IsZero(vals[k]) then
Add(ii,k);
Add(cc,vals[k]);
fi;
od;
T[i][j]:=[ii,cc];
T[j][i]:=[ii,-cc];
od;
od;
if (m + 3) mod p = 0 then
# In this case the kontact algebra is somewhat smaller.
S:= EmptySCTable( pn-1, Zero(F), "antisymmetric" );
for i in [1..pn-1] do
for j in [1..pn-1] do
S[i][j]:=T[i][j];
od;
od;
T:=S;
fi;
L:= LieAlgebraByStructureConstants( F, T );
SetIsRestrictedLieAlgebra( L, ForAll( n, x -> x=1 ) );
return L;
end );
##############################################################################
##
#F SimpleLieAlgebraTypeM( <n>, <F> )
##
## The Melikyan Lie algebra is constructed.
##
## The code is due to Erik Postma.
##
## The Melikyan Lie algebra is most conveniently constructed by
## viewing it as the direct sum of a Witt type Lie algebra and two
## of its modules. This is the presentation described by
## M.I. Kuznetsov, The Melikyan algebras as Lie algebras of the
## type G2, Comm. Algebra 19 (1991).
##
## The Melikyan Lie algebra is parametrized by two positive
## integers, n1 and n2, and can only be defined over fields of
## characteristic 5. It can be decomposed into a 2*5^(n1 + n2)-dimensional
## subalgebra isomorphic to W(n1, n2), having a basis of monomials
## X1^i1 X2^i2 dXk where 0 <= i1 < 5^n1, 0 <= i2 < 5^n2, k in {1, 2}; a
## 5^(n1 + n2)-dimensional module of this subalgebra which we call O,
## having a basis of elements we call X1^i1 X2^i2 (where i1 and i2 are
## within the same boundaries); and a 2*5^(n1 + n2)-dimensional
## module which we call Wtilde, having a basis of elements we
## call X1^i1 X2^i2 dXk^tilde (again with i1 and i2 within the same
## boundaries, and with k in {1, 2}).
##
## The multiplication is described in the above paper and in the code
## below. We use lists of symbolic descriptions for the basis
## elements: [i1, i2] for X1^i1 X2^i2 and [[i1, i2], k] for either
## X1^i1 X2^i2 dXk or X1^i1 X2^i2 dXk^tilde. All valid such
## symbolic descriptions can be found in two lists, OBasis and
## WBasis, respectively. In the basis of the full algebra, we first
## put the elements of W as ordered in WBasis, then the elements of O
## as ordered in OBasis, and finally the elements of Wtilde, again as
## ordered in WBasis. Throughout the function below, we describe
## basis elements using either these symbolic descriptions, or the
## positions in this basis.
BindGlobal( "SimpleLieAlgebraTypeM", function (n, F)
local n1, n2, # The parameters.
one, zero, # Shortcuts to the field elements.
dimO, dimW, # Dimensions of the O and W spaces.
OBasis, # A representation of a basis for O.
posO, # Function to find the position of a given
# OBasis element in the basis.
OProduct, # The regular product of two elements of OBasis.
WBasis, # A representation of a basis for W.
div, # The divergence function for elements of WBasis.
posW, # Function to find the position of a WBasis
# element in the basis.
WOProduct, # The action of W on O.
WProduct, # The regular product of two elements of WBasis.
WBracket, # The commutator of two elements of WBasis
# w.r.t. WProduct.
degrees, # The list of degrees of different components.
GradingFunction, # The function giving the grading components.
tildify, clean, # Utility functions.
table, i, w1, j, w2, result, term, prod, x2, x1, d;
# Temporary results and counters.
if not (IsList (n) and Length (n) = 2 and n [1] > 0 and n [2] > 0)
then
Error ("<n> must be a list of two positive integers");
fi;
if Characteristic (F) <> 5 then
Error ("<F> must be a field of characteristic 5");
fi;
n1 := n [1];
n2 := n [2];
dimO := 5^(n1 + n2);
dimW := 2*dimO;
one := One (F);
zero := Zero (F);
# The element [a, b] of OBasis represents the element
# X1^a X2^b / (a! b!)
# of the truncated polynomial ring.
OBasis := Cartesian ([0 .. 5^n1 - 1], [0 .. 5^n2 - 1]);
# The position of an OBasis element in the basis.
posO := function (o)
return o [2] + 5^n2 * o [1] + 1;
end;
# Given two OBasis elements x1 and x2, returns a list with a
# coefficient coeff and the position pos of a basis element, such
# that
# x1 * x2 = coeff * OBasis [pos]
OProduct := function (x1, x2)
local pow;
pow := ShallowCopy (x1 + x2);
if pow [1] < 5^n1 and pow [2] < 5^n2 then
return [Binomial (pow [1], x1 [1]) *
(Binomial (pow [2], x1 [2]) * one),
posO (pow)];
else
return [zero, 1];
fi;
end;
# The element [[a, b], c] of WBasis represents the element
# O dXc
# where O is the element of OBasis represented by [a, b].
WBasis := Cartesian (OBasis, [1, 2]);
# The divergence: f dX1 + g dX2 -> dX1 (f) + dX2 (g), maps WBasis
# elements to OBasis elements. Note: if the result is 0, we return
# that instead of the OBasis element.
div := function (abc)
local ab, pos;
if abc [1] [abc [2]] = 0 then
return 0;
fi;
pos := abc [2];
ab := ShallowCopy (abc [1]);
ab [pos] := ab [pos] - 1;
return ab;
end;
# The position of the WBasis element [OBasis (o), c] in the basis,
# where o is the number of an OBasis element.
posW := function (o, c)
return 2 * o + c - 2;
end;
# Given a WBasis element [[a1, b1], c1] and an OBasis element [a2,
# b2], representing the usual monomials, this function computes
# p = X1^a1 X2^a2 (dXc1 X1^a2 X2^b2),
# and returns a list [pos, coeff] with the position in OBasis of
# the basis element this is a multiple of, and its coefficient; so
# that
# p = coeff * OBasis [pos].
WOProduct := function (w1, x2)
local pow;
if x2 [w1 [2]] > 0 then
pow := ShallowCopy (x2);
pow [w1 [2]] := pow [w1 [2]] - 1;
return OProduct (w1 [1], pow);
else
return [zero, 1];
fi;
end;
# Given two WBasis elements [[a1, b1], c1] and [[a2, b2], c2],
# representing the usual monomials, this
# function computes
# p = X1^a1 X2^a2 (dXc1 (X1^a2 X2^b2)) dXc2,
# and returns a list [pos, coeff] with the position in WBasis of
# the basis element this is a multiple of, and its coefficient; so
# that
# p = coeff * WBasis [pos].
WProduct := function (x1, x2)
local prod;
prod := WOProduct (x1, x2 [1]);
if prod [1] <> zero then
return [prod [1], posW (prod [2], x2 [2])];
else
return [zero, 1];
fi;
end;
# The bracket on W is defined as mapping x1, x2 to their
# commutator, where the multiplication is as above. This function
# returns a list ls of, alternatingly, coefficients and positions,
# such that the bracket of x1 and x2 is equal to
# ls [1] * WBasis [ls [2]] + ls [3] * WBasis [ls [4]].
# However, if any coefficient is 0, the corresponding list
# elements are omitted. So the list returned has length 4, 2 or 0.
WBracket := function (x1, x2)
local result, prod;
prod := WProduct (x1, x2);
if prod [1] <> zero then
result := prod;
else
result := [];
fi;
prod := WProduct (x2, x1);
if prod [1] <> zero then
Append (result, [- prod [1], prod [2]]);
fi;
return result;
end;
# The order of the basis elements is: first the basis elements of
# W, then of O, then of Wtilde. Definitions of W, Wtilde and O can
# be found in H. Strade, Simple Lie Algebras over Fields of
# Positive Characteristic, Walter de Gruyter - Berlin/New York 2004.
# This is the realization found in M.I. Kuznetsov, The Melikian
# algebras as Lie algebras of the type G2, Comm. Algebra 19
# (1991), 1281-1312.
# tildify adds cst to each even position in ls. It is useful for
# mapping a result of WBracket from W to Wtilde, or an OBasis
# element to the correct position in the full basis.
tildify := function (ls, cst)
local i;
i := 2;
while IsBound (ls [i]) do
ls [i] := ls [i] + cst;
i := i + 2;
od;
end;
# clean is a function that 'cleans' a list before submission to
# SetEntrySCTable. That is, if any positions are the same, the
# coefficients are added.
clean := function (ls)
local ps, i;
ps := rec ();
i := 2;
while IsBound (ls [i]) do
if IsBound (ps.(ls [i])) then
ls [ps.(ls [i]) - 1] := ls [ps.(ls [i]) - 1] + ls [i - 1];
Unbind (ls [i - 1]);
Unbind (ls [i]);
else
ps.(ls [i]) := i;
fi;
i := i + 2;
od;
return Compacted (ls);
end;
table := EmptySCTable (dimO + 2 * dimW, Zero (F), "antisymmetric");
for i in [1 .. dimW] do
w1 := WBasis [i];
for j in [1 .. dimW] do
w2 := WBasis [j];
if i < j then
# Compute the product for w1 and w2 in W.
# This is simply [w1, w2].
SetEntrySCTable (table, i, j, clean (WBracket (w1, w2)));
# Compute the product for w1 and w2 in WTilde.
# This is f1g2 - f2g1 if w1 = f1d1 + f2d2, w2 = g1d1 +
# g2d2.
if w1 [2] <> w2 [2] then
prod := OProduct (w1 [1], w2 [1]);
if prod [1] <> zero then
SetEntrySCTable (table, i + dimW + dimO,
j + dimW + dimO,
[(3 - 2 * w1 [2]) * # This is the coefficient
# plus or minus one.
prod [1], prod [2] + dimW]);
fi;
fi;
fi;
# Compute the product for w1 in W, w2 in WTilde.
# This is defined as [w1, w2]^tilde + 2 div(w1) w2^tilde
# [w1, w2]^tilde:
result := WBracket (w1, w2);
tildify (result, dimW + dimO);
# 2 div(w1) w2^tilde:
d := div (w1);
if d <> 0 then
term := OProduct (d, w2 [1]);
if term [1] <> zero then
Append (result, [2 * term [1],
posW (term [2], w2 [2]) + dimW + dimO]);
fi;
fi;
SetEntrySCTable (table, i, j + dimW + dimO, clean (result));
od;
for j in [1 .. dimO] do
x2 := OBasis [j];
# Compute the product for w1 in W, x2 in O.
# This is w1 (x2) - 2 div (w1) x2.
# w1 (x2):
result := WOProduct (w1, x2);
# - 2 div (w1) x2:
d := div (w1);
if d <> 0 then
term := OProduct (d, x2);
if term [1] <> zero then
Append (result, [-2 * term [1], term [2]]);
fi;
fi;
tildify (result, dimW);
SetEntrySCTable (table, i, j + dimW, clean (result));
# Compute the product for w1 in Wtilde, x2 in O.
# This is - x2 w1^un-tilde.
# We put it in the table as the product of x2 and w1, so
# that we don't have to bother with the minus sign.
result := OProduct (x2, w1 [1]);
SetEntrySCTable (table, j + dimW, i + dimW + dimO,
[result [1], posW (result [2], w1 [2])]);
od;
od;
for i in [1 .. dimO] do
x1 := OBasis [i];
for j in [i + 1 .. dimO] do
x2 := OBasis [j];
# Compute the product for x1 and x2 in O.
# This is 2 (x2 dX2(x1) - x1 dX2(x2))dX1^tilde + 2 (x1
# dX1(x2) - x2 dX1(x1)) dX2^tilde.
# 2 x2 dX2(x1) dX1:
result := WProduct ([x2, 2], [x1, 1]);
result [1] := 2 * result [1];
# - 2 x1 dX2(x2) dX1:
term := WProduct ([x1, 2], [x2, 1]);
Append (result, [- 2 * term [1], term [2]]);
# 2 x1 dX1(x2) dX2:
term := WProduct ([x1, 1], [x2, 2]);
Append (result, [2 * term [1], term [2]]);
# - 2 x2 dX1(x1) dX2:
term := WProduct ([x2, 1], [x1, 2]);
Append (result, [- 2 * term [1], term [2]]);
tildify (result, dimW + dimO);
SetEntrySCTable (table, i + dimW, j + dimW,
clean (result));
od;
od;
result := LieAlgebraByStructureConstants (F, table);
SetIsRestrictedLieAlgebra (result, n1 = 1 and n2 = 1);
degrees := Concatenation (List (WBasis, lst ->
lst [1] * [[2, 1], [1, 2]] +
\[\]([[-2, -1], [-1, -2]], lst [2])),
List (OBasis, lst ->
lst * [[2, 1], [1, 2]] + [-1, -1]),
List (WBasis, lst ->
lst [1] * [[2, 1], [1, 2]] +
\[\]([[-1, 0], [0, -1]], lst [2])));
GradingFunction := d -> Subspace (result,
Basis(result) {Positions (degrees, d)});
SetGrading (result, rec(
source :=
FreeLeftModule(Integers, [[1, 0], [0, 1]], "basis"),
hom_components := GradingFunction,
non_zero_hom_components := Set (degrees)));
# GradingFunction := function (d)
# local degsum, r, oposns;
# r := d[1] + d[2] mod 3;
# if r = 0 then
#
# degsum := (d [1] + d [2] - r) / 3 + 1;
# oposns := List ([Maximum (0, degsum - 5^n2 + 1) ..
# Minimum (degsum, 5^n1 - 1)],
# i -> posO ([i, degsum - i]));
# if r = 0 then
# return SubspaceNC (result,
# Basis (result) {Concatenation (
# List (oposns, p -> posW (p, 1)),
# List (oposns, p -> posW (p, 2)))},
# "basis");
# elif r = 1 then
# return SubspaceNC (result,
# Basis (result) {oposns + dimW},
# "basis");
# else # r = 2
# return SubspaceNC (result,
# Basis (result) {3 * dimO + Concatenation (
# List (oposns, p -> posW (p, 1)),
# List (oposns, p -> posW (p, 2)))},
# "basis");
# fi;
# end;
# SetGrading (result,
# rec (min_degree := -3,
# max_degree := 3 * (5^n1 + 5^n2) - 7,
# source := Integers,
# hom_components := GradingFunction));
return result;
end );
##############################################################################
##
#F SimpleLieAlgebra( <type>, <n>, <F> )
##
InstallGlobalFunction( SimpleLieAlgebra, function( type, n, F )
local A;
# Check the arguments.
if not ( IsString( type ) and ( IsInt( n ) or IsList( n ) ) and
IsRing( F ) ) then
Error( "<type> must be a string, <n> an integer, <F> a ring" );
fi;
if type in [ "A","B","C","D","E","F","G" ] then
A := SimpleLieAlgebraTypeA_G( type, n, F );
elif type = "W" then
A := SimpleLieAlgebraTypeW( n, F )[1];
elif type = "S" then
A := SimpleLieAlgebraTypeS( n, F );
elif type = "H" then
A := SimpleLieAlgebraTypeH( n, F );
elif type = "K" then
A := SimpleLieAlgebraTypeK( n, F );
elif type = "M" then
A := SimpleLieAlgebraTypeM( n, F );
else
Error( "<type> must be one of \"A\", \"B\", \"C\", \"D\", \"E\", ",
"\"F\", \"G\", \"H\", \"K\", \"M\", \"S\", \"W\" " );
fi;
# store the pth power images in the family (LB)
if IsRestrictedLieAlgebra(A) then
FamilyObj(Representative(A))!.pMapping := PthPowerImages(Basis(A));
fi;
return A;
end );
[ Dauer der Verarbeitung: 0.68 Sekunden
(vorverarbeitet)
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