bracket.at

LieAlgebraElement type

set_type LieAlgebraElement=(StructureConstantTable t,(ratvec,ratvec) X)

Fields: t, X

data type for an element of a reductive Lie algebra Q
N=number of roots
n=rank
X=ratvec of size N <-> \sum_{i=0}^{N-1} x_i X_i where X_i are the root vectors
H=ratvec of size n <-> X_* in atlas coordinates
arbitrary element of \g: (X,H)

root_datum

also defined in basic.at, W_reps.at, K_Nilpotent.at, sommers.at, parabolics.at, K.at, nilpotent_orbits.at, W_classes.at, extended_misc.at, character_tables.at, springer_tables.at, disconnected.at, cyclotomic_Lie_algebra.at, exceptional.at

root_part

also defined in cyclotomic_Lie_algebra.at

semisimple_part

also defined in center.at, cyclotomic_Lie_algebra.at

H

also in this file at line 151; also defined in groups.at, cyclotomic_Lie_algebra.at, exceptional.at

=

also in this file at line 22; also defined in basic.at, combinatorics.at, modules.at, parabolics.at, synthetic.at, hash.at, tits.at, K_highest_weights.at, nilpotent_orbits.at, cyclotomic.at, polynomial.at, W_reps_type_BC.at, hodgeParamLaurentPol.at, hodgeParamPol.at, laurentPolynomial.at, extended.at, disconnected.at, cyclotomic_Lie_algebra.at, cyclotomicMat.at

null

ss_

ss_?

also defined in all_finite_order.at, cyclotomic_Lie_algebra.at, cyclotomicMat.at

=

also in this file at line 19; also defined in basic.at, combinatorics.at, modules.at, parabolics.at, synthetic.at, hash.at, tits.at, K_highest_weights.at, nilpotent_orbits.at, cyclotomic.at, polynomial.at, W_reps_type_BC.at, hodgeParamLaurentPol.at, hodgeParamPol.at, laurentPolynomial.at, extended.at, disconnected.at, cyclotomic_Lie_algebra.at, cyclotomicMat.at

+

also defined in basic.at, extParamPol.at, modules.at, ratmat.at, cyclotomic.at, polynomial.at, hodgeParamLaurentPol.at, hodgeParamPol.at, laurentPolynomial.at, cyclotomic_Lie_algebra.at, cyclotomicMat.at, hodge_functions.at, lazy_lists.at

*

also defined in basic.at, extParamPol.at, complex.at, modules.at, hodge_tensor.at, tits_centralizer.at, stable.at, synthetic.at, Weylgroup.at, group_operations.at, ratmat.at, tits.at, twisted_root_datum.at, cyclotomic.at, polynomial.at, hodgeParamLaurentPol.at, hodgeParamPol.at, laurentPolynomial.at, cyclotomic_Lie_algebra.at, cyclotomicMat.at, hodge_functions.at, sparse.at, lazy_lists.at

coordinates

(t,v,w)-> v##w

also in this file at line 63, line 69; also defined in cyclotomic_Lie_algebra.at

lie_algebra_element

element of Lie algebra coming from ratvec of size n+N=dim(\g)

also defined in cyclotomic_Lie_algebra.at

lie_algebra_element_semisimple

also defined in cyclotomic_Lie_algebra.at

lie_algebra_element_root_vectors

also defined in cyclotomic_Lie_algebra.at

basis

1\le i\le dim\g -> (t,(v,w)) where either v or w has exactly one entry 1

show

also defined in modules.at, K_Nilpotent.at, good_W_representatives.at, arthur_parameters.at, sub_cells.at, associated_variety_annihilator.at, geck_generic.at, L_packet.at, ratmat.at, Levi_subgroups.at, nilpotent_orbits.at, subgroups.at, poset.at, cyclotomic.at, polynomial.at, character_tables.at, springer_tables.at, hodgeParamLaurentPol.at, hodgeParamPol.at, laurentPolynomial.at, weak_packets.at, cyclotomic_Lie_algebra.at, cyclotomicMat.at, exceptional.at, nilpotent_posets.at, lazy_lists.at

format

nice string representation of LieAlgebraElement

also defined in cyclotomic.at, polynomial.at, extended_misc.at, cyclotomicMat.at

root_part_array

also defined in cyclotomic_Lie_algebra.at

show_long

also defined in character_tables.at, springer_tables.at, hodgeParamLaurentPol.at, hodgeParamPol.at, weak_packets.at, cyclotomic_Lie_algebra.at, exceptional.at

coordinates 2 overloads

coordinates of X_alpha: [0,...,1,..0] (N-tuple)

also in this file at line 31; also defined in cyclotomic_Lie_algebra.at

support

also defined in parabolics.at, nilpotent_orbits.at, W_reps_type_BC.at, cyclotomic_Lie_algebra.at

bracket_root_term

[X,Y] =[(X.root_term,X.H),(Y.root_term,Y.H)]=
[X.root_term,Y.root_term] + other terms
this function returns just the root term of this bracket
(not the torus part which may also be nonzero)
this is computed from the structure constant table

also defined in cyclotomic_Lie_algebra.at

bracket_torus_term

torus term, coming from [X_alpha,X_-alpha]

also defined in cyclotomic_Lie_algebra.at

bracket_off_diagonal_terms

[X,Y]=(X.root_terms,X.H),(Y.root_terms,Y.H)] -> [X.H,Y.root_terms] + [X.H,Y.root_terms],
which gives a term \sum_1^N a_i X_i

also defined in cyclotomic_Lie_algebra.at

bracket

put the three bracket calculations together

also defined in cyclotomic_Lie_algebra.at

X_alpha 2 overloads

root -> (v,null(rank)) where v=e(N,i)=1 in i^th spot, i=# of root in rd.roots

set X_alpha(StructureConstantTable t,int i)=LieAlgebraElement:(t,(e(#t.root_datum.roots,i),null(#t.root_datum.simple_roots)))

also defined in cyclotomic_Lie_algebra.at

H

H(t,h)=(t,null(N),h)

also in this file at line 18; also defined in groups.at, cyclotomic_Lie_algebra.at, exceptional.at

X 2 overloads

(t,alpha,h)=(t,v,h) where v=root_term(t,alpha)

alternative format

also defined in cyclotomic_Lie_algebra.at, lazy_lists.at

ad

also defined in cyclotomic_Lie_algebra.at

solve_ad

solve ad(X)(Y)=Z for Y

two_eigenspace 2 overloads

2-eigenspace of ad(H): as list of roots

2-eigenspace of ad(H): as list of LieAlgebraElements

JM_triple 5 overloads

complete Jacobson Morozov triple: given H, find X,Y so that [H,X]=2X, [H,Y]=-2Y, [X,Y]=H

also defined in tits_centralizer.at

JM_triple_signs 4 overloads

different idea: find a triple where all coefficients are \pm 1

also in this file at line 259, line 261, line 263

JM_triples_signs 2 overloads

JM_triple_signs 3 overloads

also in this file at line 211, line 222, line 224, line 226

JM_triples 2 overloads

compute all triples for rd

also defined in tits_centralizer.at

jacobi_formula

general Jacobi formula [[X,Y],Z]+[[Z,X],Y]+[Y,X],X]

also defined in structure_constants.at, cyclotomic_Lie_algebra.at

jacobi_identities

test all Jacobi identities for the Lie algebra
see jacobi_identity in structure_constants.at

also in this file at line 297; also defined in cyclotomic_Lie_algebra.at

jacobi_identities_long

also in this file at line 298; also defined in cyclotomic_Lie_algebra.at

jacobi_identities

also in this file at line 283; also defined in cyclotomic_Lie_algebra.at

jacobi_identities_long

also in this file at line 289; also defined in cyclotomic_Lie_algebra.at

centralizer

also defined in good_W_representatives.at, nilpotent_centralizer.at, K_highest_weights.at, subgroups.at, conjugate.at, cyclotomic_Lie_algebra.at, weyl_character_formula.at

embed

embed X for L into G
use this to promote triple (H,X,Y) from L up to G

L must be an equal rank subgroup of rd

also defined in cyclotomic.at, character_table_reps.at, cyclotomic_Lie_algebra.at, cyclotomicMat.at