The Poincare-Birkhoff-Witt Basis For A Universal Enveloping Algebra

AUTHORS:

  • Travis Scrimshaw (2013-11-03): Initial version

  • Travis Scrimshaw (2024-01-02): Adding the center

class sage.algebras.lie_algebras.poincare_birkhoff_witt.PoincareBirkhoffWittBasis(g, basis_key, prefix, **kwds)[source]

Bases: CombinatorialFreeModule

The Poincare-Birkhoff-Witt (PBW) basis of the universal enveloping algebra of a Lie algebra.

Consider a Lie algebra \(\mathfrak{g}\) with ordered basis \((b_1,\dots,b_n)\). Then the universal enveloping algebra \(U(\mathfrak{g})\) is generated by \(b_1,\dots,b_n\) and subject to the relations

\[[b_i, b_j] = \sum_{k = 1}^n c_{ij}^k b_k\]

where \(c_{ij}^k\) are the structure coefficients of \(\mathfrak{g}\). The Poincare-Birkhoff-Witt (PBW) basis is given by the monomials \(b_1^{e_1} b_2^{e_2} \cdots b_n^{e_n}\). Specifically, we can rewrite \(b_j b_i = b_i b_j + [b_j, b_i]\) where \(j > i\), and we can repeat this to sort any monomial into

\[b_{i_1} \cdots b_{i_k} = b_1^{e_1} \cdots b_n^{e_n} + LOT\]

where \(LOT\) are lower order terms. Thus the PBW basis is a filtered basis for \(U(\mathfrak{g})\).

EXAMPLES:

We construct the PBW basis of \(\mathfrak{sl}_2\):

sage: L = lie_algebras.three_dimensional_by_rank(QQ, 3, names=['E','F','H'])
sage: PBW = L.pbw_basis()

>>> from sage.all import *
>>> L = lie_algebras.three_dimensional_by_rank(QQ, Integer(3), names=['E','F','H'])
>>> PBW = L.pbw_basis()

We then do some computations; in particular, we check that \([E, F] = H\):

sage: E, F, H = PBW.algebra_generators()
sage: E * F
PBW['E']*PBW['F']
sage: F * E
PBW['E']*PBW['F'] - PBW['H']
sage: E * F - F * E
PBW['H']

>>> from sage.all import *
>>> E, F, H = PBW.algebra_generators()
>>> E * F
PBW['E']*PBW['F']
>>> F * E
PBW['E']*PBW['F'] - PBW['H']
>>> E * F - F * E
PBW['H']

Next we construct another instance of the PBW basis, but sorted in the reverse order:

sage: def neg_key(x):
....:     return -L.basis().keys().index(x)
sage: PBW2 = L.pbw_basis(prefix='PBW2', basis_key=neg_key)

>>> from sage.all import *
>>> def neg_key(x):
...     return -L.basis().keys().index(x)
>>> PBW2 = L.pbw_basis(prefix='PBW2', basis_key=neg_key)

We then check the multiplication is preserved:

sage: PBW2(E) * PBW2(F)
PBW2['F']*PBW2['E'] + PBW2['H']
sage: PBW2(E*F)
PBW2['F']*PBW2['E'] + PBW2['H']
sage: F * E + H
PBW['E']*PBW['F']

>>> from sage.all import *
>>> PBW2(E) * PBW2(F)
PBW2['F']*PBW2['E'] + PBW2['H']
>>> PBW2(E*F)
PBW2['F']*PBW2['E'] + PBW2['H']
>>> F * E + H
PBW['E']*PBW['F']

We now construct the PBW basis for Lie algebra of regular vector fields on \(\CC^{\times}\):

sage: L = lie_algebras.regular_vector_fields(QQ)
sage: PBW = L.pbw_basis()
sage: G = PBW.algebra_generators()
sage: G[2] * G[3]
PBW[2]*PBW[3]
sage: G[3] * G[2]
PBW[2]*PBW[3] + PBW[5]
sage: G[-2] * G[3] * G[2]
PBW[-2]*PBW[2]*PBW[3] + PBW[-2]*PBW[5]

>>> from sage.all import *
>>> L = lie_algebras.regular_vector_fields(QQ)
>>> PBW = L.pbw_basis()
>>> G = PBW.algebra_generators()
>>> G[Integer(2)] * G[Integer(3)]
PBW[2]*PBW[3]
>>> G[Integer(3)] * G[Integer(2)]
PBW[2]*PBW[3] + PBW[5]
>>> G[-Integer(2)] * G[Integer(3)] * G[Integer(2)]
PBW[-2]*PBW[2]*PBW[3] + PBW[-2]*PBW[5]

Todo

When the Lie algebra is finite dimensional, set the ordering of the basis elements, translate the structure coefficients, and work with fixed-length lists as the exponent vectors. This way we only will run any nontrivial sorting only once and avoid other potentially expensive comparisons between keys.

class Element[source]

Bases: IndexedFreeModuleElement

algebra_generators()[source]

Return the algebra generators of self.

EXAMPLES:

sage: L = lie_algebras.sl(QQ, 2)
sage: PBW = L.pbw_basis()
sage: PBW.algebra_generators()
Finite family {alpha[1]: PBW[alpha[1]], alphacheck[1]: PBW[alphacheck[1]], -alpha[1]: PBW[-alpha[1]]}

>>> from sage.all import *
>>> L = lie_algebras.sl(QQ, Integer(2))
>>> PBW = L.pbw_basis()
>>> PBW.algebra_generators()
Finite family {alpha[1]: PBW[alpha[1]], alphacheck[1]: PBW[alphacheck[1]], -alpha[1]: PBW[-alpha[1]]}
casimir_element(order=2, *args, **kwds)[source]

Return the Casimir element of self.

See also

casimir_element()

INPUT:

  • order – (default: 2) the order of the Casimir element

EXAMPLES:

sage: L = LieAlgebra(QQ, cartan_type=['G', 2])
sage: U = L.pbw_basis()
sage: C = U.casimir_element(); C
1/4*PBW[alpha[2]]*PBW[-alpha[2]] + 1/12*PBW[alpha[1]]*PBW[-alpha[1]]
 + 1/12*PBW[alpha[1] + alpha[2]]*PBW[-alpha[1] - alpha[2]] + 1/12*PBW[2*alpha[1] + alpha[2]]*PBW[-2*alpha[1] - alpha[2]]
 + 1/4*PBW[3*alpha[1] + alpha[2]]*PBW[-3*alpha[1] - alpha[2]]
 + 1/4*PBW[3*alpha[1] + 2*alpha[2]]*PBW[-3*alpha[1] - 2*alpha[2]]
 + 1/12*PBW[alphacheck[1]]^2 + 1/4*PBW[alphacheck[1]]*PBW[alphacheck[2]]
 + 1/4*PBW[alphacheck[2]]^2 - 5/12*PBW[alphacheck[1]] - 3/4*PBW[alphacheck[2]]
sage: all(g * C == C * g for g in U.algebra_generators())
True

>>> from sage.all import *
>>> L = LieAlgebra(QQ, cartan_type=['G', Integer(2)])
>>> U = L.pbw_basis()
>>> C = U.casimir_element(); C
1/4*PBW[alpha[2]]*PBW[-alpha[2]] + 1/12*PBW[alpha[1]]*PBW[-alpha[1]]
 + 1/12*PBW[alpha[1] + alpha[2]]*PBW[-alpha[1] - alpha[2]] + 1/12*PBW[2*alpha[1] + alpha[2]]*PBW[-2*alpha[1] - alpha[2]]
 + 1/4*PBW[3*alpha[1] + alpha[2]]*PBW[-3*alpha[1] - alpha[2]]
 + 1/4*PBW[3*alpha[1] + 2*alpha[2]]*PBW[-3*alpha[1] - 2*alpha[2]]
 + 1/12*PBW[alphacheck[1]]^2 + 1/4*PBW[alphacheck[1]]*PBW[alphacheck[2]]
 + 1/4*PBW[alphacheck[2]]^2 - 5/12*PBW[alphacheck[1]] - 3/4*PBW[alphacheck[2]]
>>> all(g * C == C * g for g in U.algebra_generators())
True
center()[source]

Return the center of self.

See also

CenterUEA

EXAMPLES:

sage: g = LieAlgebra(QQ, cartan_type=['A', 2])
sage: U = g.pbw_basis()
sage: U.center()
Center of Universal enveloping algebra of Lie algebra of ['A', 2]
 in the Chevalley basis in the Poincare-Birkhoff-Witt basis

sage: g = lie_algebras.Heisenberg(GF(3), 4)
sage: U = g.pbw_basis()
sage: U.center()
Center of Universal enveloping algebra of Heisenberg algebra of rank 4
 over Finite Field of size 3 in the Poincare-Birkhoff-Witt basis

>>> from sage.all import *
>>> g = LieAlgebra(QQ, cartan_type=['A', Integer(2)])
>>> U = g.pbw_basis()
>>> U.center()
Center of Universal enveloping algebra of Lie algebra of ['A', 2]
 in the Chevalley basis in the Poincare-Birkhoff-Witt basis

>>> g = lie_algebras.Heisenberg(GF(Integer(3)), Integer(4))
>>> U = g.pbw_basis()
>>> U.center()
Center of Universal enveloping algebra of Heisenberg algebra of rank 4
 over Finite Field of size 3 in the Poincare-Birkhoff-Witt basis
degree_on_basis(m)[source]

Return the degree of the basis element indexed by m.

EXAMPLES:

sage: L = lie_algebras.sl(QQ, 2)
sage: PBW = L.pbw_basis()
sage: E,H,F = PBW.algebra_generators()
sage: PBW.degree_on_basis(E.leading_support())
1
sage: m = ((H*F)^10).trailing_support(key=PBW._monomial_key)  # long time
sage: PBW.degree_on_basis(m)  # long time
20
sage: ((H*F*E)^4).maximal_degree()  # long time
12

>>> from sage.all import *
>>> L = lie_algebras.sl(QQ, Integer(2))
>>> PBW = L.pbw_basis()
>>> E,H,F = PBW.algebra_generators()
>>> PBW.degree_on_basis(E.leading_support())
1
>>> m = ((H*F)**Integer(10)).trailing_support(key=PBW._monomial_key)  # long time
>>> PBW.degree_on_basis(m)  # long time
20
>>> ((H*F*E)**Integer(4)).maximal_degree()  # long time
12
gens()[source]

Return the algebra generators of self.

EXAMPLES:

sage: L = lie_algebras.sl(QQ, 2)
sage: PBW = L.pbw_basis()
sage: PBW.algebra_generators()
Finite family {alpha[1]: PBW[alpha[1]], alphacheck[1]: PBW[alphacheck[1]], -alpha[1]: PBW[-alpha[1]]}

>>> from sage.all import *
>>> L = lie_algebras.sl(QQ, Integer(2))
>>> PBW = L.pbw_basis()
>>> PBW.algebra_generators()
Finite family {alpha[1]: PBW[alpha[1]], alphacheck[1]: PBW[alphacheck[1]], -alpha[1]: PBW[-alpha[1]]}
lie_algebra()[source]

Return the underlying Lie algebra of self.

EXAMPLES:

sage: L = lie_algebras.sl(QQ, 2)
sage: PBW = L.pbw_basis()
sage: PBW.lie_algebra() is L
True

>>> from sage.all import *
>>> L = lie_algebras.sl(QQ, Integer(2))
>>> PBW = L.pbw_basis()
>>> PBW.lie_algebra() is L
True
one_basis()[source]

Return the basis element indexing \(1\).

EXAMPLES:

sage: L = lie_algebras.three_dimensional_by_rank(QQ, 3, names=['E','F','H'])
sage: PBW = L.pbw_basis()
sage: ob = PBW.one_basis(); ob
1
sage: ob.parent()
Free abelian monoid indexed by {'E', 'F', 'H'}

>>> from sage.all import *
>>> L = lie_algebras.three_dimensional_by_rank(QQ, Integer(3), names=['E','F','H'])
>>> PBW = L.pbw_basis()
>>> ob = PBW.one_basis(); ob
1
>>> ob.parent()
Free abelian monoid indexed by {'E', 'F', 'H'}
product_on_basis(lhs, rhs)[source]

Return the product of the two basis elements lhs and rhs.

EXAMPLES:

sage: L = lie_algebras.three_dimensional_by_rank(QQ, 3, names=['E','F','H'])
sage: PBW = L.pbw_basis()
sage: I = PBW.indices()
sage: PBW.product_on_basis(I.gen('E'), I.gen('F'))
PBW['E']*PBW['F']
sage: PBW.product_on_basis(I.gen('E'), I.gen('H'))
PBW['E']*PBW['H']
sage: PBW.product_on_basis(I.gen('H'), I.gen('E'))
PBW['E']*PBW['H'] + 2*PBW['E']
sage: PBW.product_on_basis(I.gen('F'), I.gen('E'))
PBW['E']*PBW['F'] - PBW['H']
sage: PBW.product_on_basis(I.gen('F'), I.gen('H'))
PBW['F']*PBW['H']
sage: PBW.product_on_basis(I.gen('H'), I.gen('F'))
PBW['F']*PBW['H'] - 2*PBW['F']
sage: PBW.product_on_basis(I.gen('H')**2, I.gen('F')**2)
PBW['F']^2*PBW['H']^2 - 8*PBW['F']^2*PBW['H'] + 16*PBW['F']^2

sage: E,F,H = PBW.algebra_generators()
sage: E*F - F*E
PBW['H']
sage: H * F * E
PBW['E']*PBW['F']*PBW['H'] - PBW['H']^2
sage: E * F * H * E
PBW['E']^2*PBW['F']*PBW['H'] + 2*PBW['E']^2*PBW['F']
 - PBW['E']*PBW['H']^2 - 2*PBW['E']*PBW['H']

>>> from sage.all import *
>>> L = lie_algebras.three_dimensional_by_rank(QQ, Integer(3), names=['E','F','H'])
>>> PBW = L.pbw_basis()
>>> I = PBW.indices()
>>> PBW.product_on_basis(I.gen('E'), I.gen('F'))
PBW['E']*PBW['F']
>>> PBW.product_on_basis(I.gen('E'), I.gen('H'))
PBW['E']*PBW['H']
>>> PBW.product_on_basis(I.gen('H'), I.gen('E'))
PBW['E']*PBW['H'] + 2*PBW['E']
>>> PBW.product_on_basis(I.gen('F'), I.gen('E'))
PBW['E']*PBW['F'] - PBW['H']
>>> PBW.product_on_basis(I.gen('F'), I.gen('H'))
PBW['F']*PBW['H']
>>> PBW.product_on_basis(I.gen('H'), I.gen('F'))
PBW['F']*PBW['H'] - 2*PBW['F']
>>> PBW.product_on_basis(I.gen('H')**Integer(2), I.gen('F')**Integer(2))
PBW['F']^2*PBW['H']^2 - 8*PBW['F']^2*PBW['H'] + 16*PBW['F']^2

>>> E,F,H = PBW.algebra_generators()
>>> E*F - F*E
PBW['H']
>>> H * F * E
PBW['E']*PBW['F']*PBW['H'] - PBW['H']^2
>>> E * F * H * E
PBW['E']^2*PBW['F']*PBW['H'] + 2*PBW['E']^2*PBW['F']
 - PBW['E']*PBW['H']^2 - 2*PBW['E']*PBW['H']
class sage.algebras.lie_algebras.poincare_birkhoff_witt.PoincareBirkhoffWittBasisSemisimpleLieAlgebra(g, basis_key=None, *args, **kwds)[source]

Bases: PoincareBirkhoffWittBasis

The Poincare-Birkhoff-Witt basis of a finite dimensional triangular Kac-Moody Lie algebra (i.e., a semisimple Lie algebra).

class Element[source]

Bases: Element

transpose()[source]

Return the transpose map of self.

This is the transpose map on the Lie algebra extended as an anti-involution of self.

EXAMPLES:

sage: g = LieAlgebra(QQ, cartan_type=['D', 4])
sage: U = g.pbw_basis()
sage: e = U.e()
sage: f = U.f()
sage: elts = [e[1], e[1]*e[2], e[3]+e[4], e[1]*e[3]*e[4] + e[2],
....:         f[1], f[1]*f[2], f[3]+f[4], e[1]*e[3]*e[4] + e[2],
....:         e[1]*f[1], f[1]*e[1], (e[2]*f[2] - f[2]*e[2])^2]
sage: all((b*bp).transpose() == bp.transpose() * b.transpose()
....:     for b in elts for bp in elts)
True

>>> from sage.all import *
>>> g = LieAlgebra(QQ, cartan_type=['D', Integer(4)])
>>> U = g.pbw_basis()
>>> e = U.e()
>>> f = U.f()
>>> elts = [e[Integer(1)], e[Integer(1)]*e[Integer(2)], e[Integer(3)]+e[Integer(4)], e[Integer(1)]*e[Integer(3)]*e[Integer(4)] + e[Integer(2)],
...         f[Integer(1)], f[Integer(1)]*f[Integer(2)], f[Integer(3)]+f[Integer(4)], e[Integer(1)]*e[Integer(3)]*e[Integer(4)] + e[Integer(2)],
...         e[Integer(1)]*f[Integer(1)], f[Integer(1)]*e[Integer(1)], (e[Integer(2)]*f[Integer(2)] - f[Integer(2)]*e[Integer(2)])**Integer(2)]
>>> all((b*bp).transpose() == bp.transpose() * b.transpose()
...     for b in elts for bp in elts)
True
contravariant_form(x, y)[source]

Return the (universal) contravariant form of x and y.

Let \(\varphi \colon U(\mathfrak{g}) \to U(\mathfrak{h})\) denote the projection onto the Cartan subalgebra and \(\tau\) be the transpose map. The (universal) contravariant form is defined as

\[(x, y) = \varphi(\tau(x) y).\]

EXAMPLES:

sage: g = LieAlgebra(QQ, cartan_type=['G', 2])
sage: U = g.pbw_basis()
sage: f1, f2 = U.f()
sage: e1, e2 = U.e()
sage: U.contravariant_form(U.one(), U.one())
1
sage: U.contravariant_form(f1, f1)
PBW[alphacheck[1]]
sage: U.contravariant_form(f2, f2)
PBW[alphacheck[2]]
sage: U.contravariant_form(f1*f2, f1*f2)
PBW[alphacheck[1]]*PBW[alphacheck[2]] + 3*PBW[alphacheck[2]]
sage: U.contravariant_form(e1*e1*e2, e2*e1*e2)
0
sage: cas = U.casimir_element()
sage: ccc = U.contravariant_form(cas, cas); ccc
1/144*PBW[alphacheck[1]]^4 + 1/24*PBW[alphacheck[1]]^3*PBW[alphacheck[2]]
 + 5/48*PBW[alphacheck[1]]^2*PBW[alphacheck[2]]^2
 + 1/8*PBW[alphacheck[1]]*PBW[alphacheck[2]]^3 + 1/16*PBW[alphacheck[2]]^4
 + 5/72*PBW[alphacheck[1]]^3 + 1/3*PBW[alphacheck[1]]^2*PBW[alphacheck[2]]
 + 7/12*PBW[alphacheck[1]]*PBW[alphacheck[2]]^2 + 3/8*PBW[alphacheck[2]]^3
 + 25/144*PBW[alphacheck[1]]^2 + 5/8*PBW[alphacheck[1]]*PBW[alphacheck[2]]
 + 9/16*PBW[alphacheck[2]]^2
sage: ccc.parent() is U
True

>>> from sage.all import *
>>> g = LieAlgebra(QQ, cartan_type=['G', Integer(2)])
>>> U = g.pbw_basis()
>>> f1, f2 = U.f()
>>> e1, e2 = U.e()
>>> U.contravariant_form(U.one(), U.one())
1
>>> U.contravariant_form(f1, f1)
PBW[alphacheck[1]]
>>> U.contravariant_form(f2, f2)
PBW[alphacheck[2]]
>>> U.contravariant_form(f1*f2, f1*f2)
PBW[alphacheck[1]]*PBW[alphacheck[2]] + 3*PBW[alphacheck[2]]
>>> U.contravariant_form(e1*e1*e2, e2*e1*e2)
0
>>> cas = U.casimir_element()
>>> ccc = U.contravariant_form(cas, cas); ccc
1/144*PBW[alphacheck[1]]^4 + 1/24*PBW[alphacheck[1]]^3*PBW[alphacheck[2]]
 + 5/48*PBW[alphacheck[1]]^2*PBW[alphacheck[2]]^2
 + 1/8*PBW[alphacheck[1]]*PBW[alphacheck[2]]^3 + 1/16*PBW[alphacheck[2]]^4
 + 5/72*PBW[alphacheck[1]]^3 + 1/3*PBW[alphacheck[1]]^2*PBW[alphacheck[2]]
 + 7/12*PBW[alphacheck[1]]*PBW[alphacheck[2]]^2 + 3/8*PBW[alphacheck[2]]^3
 + 25/144*PBW[alphacheck[1]]^2 + 5/8*PBW[alphacheck[1]]*PBW[alphacheck[2]]
 + 9/16*PBW[alphacheck[2]]^2
>>> ccc.parent() is U
True
e(i=None)[source]

Return the generators \(e\) of self.

INPUT:

  • i – (optional) if specified, return just the generator \(e_i\)

EXAMPLES:

sage: U = lie_algebras.so(QQ, 5).pbw_basis()
sage: U.e()
Finite family {1: PBW[alpha[1]], 2: PBW[alpha[2]]}
sage: U.e(1)
PBW[alpha[1]]

>>> from sage.all import *
>>> U = lie_algebras.so(QQ, Integer(5)).pbw_basis()
>>> U.e()
Finite family {1: PBW[alpha[1]], 2: PBW[alpha[2]]}
>>> U.e(Integer(1))
PBW[alpha[1]]
f(i=None)[source]

Return the generators \(f\) of self.

INPUT:

  • i – (optional) if specified, return just the generator \(f_i\)

EXAMPLES:

sage: U = lie_algebras.so(QQ, 5).pbw_basis()
sage: U.f()
Finite family {1: PBW[-alpha[1]], 2: PBW[-alpha[2]]}
sage: U.f(1)
PBW[-alpha[1]]

>>> from sage.all import *
>>> U = lie_algebras.so(QQ, Integer(5)).pbw_basis()
>>> U.f()
Finite family {1: PBW[-alpha[1]], 2: PBW[-alpha[2]]}
>>> U.f(Integer(1))
PBW[-alpha[1]]
transpose()[source]

The transpose map.

EXAMPLES:

sage: g = LieAlgebra(QQ, cartan_type=['F', 4])
sage: U = g.pbw_basis()
sage: U.transpose
Generic endomorphism of Universal enveloping algebra of Lie algebra
 of ['F', 4] in the Chevalley basis in the Poincare-Birkhoff-Witt basis

>>> from sage.all import *
>>> g = LieAlgebra(QQ, cartan_type=['F', Integer(4)])
>>> U = g.pbw_basis()
>>> U.transpose
Generic endomorphism of Universal enveloping algebra of Lie algebra
 of ['F', 4] in the Chevalley basis in the Poincare-Birkhoff-Witt basis