The Baker-Campbell-Hausdorff formula¶

AUTHORS:

Eero Hakavuori (2018-09-23): initial version

sage.algebras.lie_algebras.bch.bch_iterator(X=None, Y=None)[source]¶

A generator function which returns successive terms of the Baker-Campbell-Hausdorff formula.

INPUT:

X – (optional) an element of a Lie algebra
Y – (optional) an element of a Lie algebra

The BCH formula is an expression for \(\log(\exp(X)\exp(Y))\) as a sum of Lie brackets of X and Y with rational coefficients. In arbitrary Lie algebras, the infinite sum is only guaranteed to converge for X and Y close to zero.

If the elements X and Y are not given, then the iterator will return successive terms of the abstract BCH formula, i.e., the BCH formula for the generators of the free Lie algebra on 2 generators.

If the Lie algebra containing X and Y is not nilpotent, the iterator will output infinitely many elements. If the Lie algebra is nilpotent, the number of elements outputted is equal to the nilpotency step.

EXAMPLES:

The terms of the abstract BCH formula up to fifth order brackets:

Sage

sage: from sage.algebras.lie_algebras.bch import bch_iterator
sage: bch = bch_iterator()
sage: next(bch)
X + Y
sage: next(bch)
1/2*[X, Y]
sage: next(bch)
1/12*[X, [X, Y]] + 1/12*[[X, Y], Y]
sage: next(bch)
1/24*[X, [[X, Y], Y]]
sage: next(bch)
-1/720*[X, [X, [X, [X, Y]]]] + 1/180*[X, [X, [[X, Y], Y]]]
+ 1/360*[[X, [X, Y]], [X, Y]] + 1/180*[X, [[[X, Y], Y], Y]]
+ 1/120*[[X, Y], [[X, Y], Y]] - 1/720*[[[[X, Y], Y], Y], Y]

Python

>>> from sage.all import *
>>> from sage.algebras.lie_algebras.bch import bch_iterator
>>> bch = bch_iterator()
>>> next(bch)
X + Y
>>> next(bch)
1/2*[X, Y]
>>> next(bch)
1/12*[X, [X, Y]] + 1/12*[[X, Y], Y]
>>> next(bch)
1/24*[X, [[X, Y], Y]]
>>> next(bch)
-1/720*[X, [X, [X, [X, Y]]]] + 1/180*[X, [X, [[X, Y], Y]]]
+ 1/360*[[X, [X, Y]], [X, Y]] + 1/180*[X, [[[X, Y], Y], Y]]
+ 1/120*[[X, Y], [[X, Y], Y]] - 1/720*[[[[X, Y], Y], Y], Y]

For nilpotent Lie algebras the BCH formula only has finitely many terms:

Sage

sage: L = LieAlgebra(QQ, 2, step=3)
sage: L.inject_variables()
Defining X_1, X_2, X_12, X_112, X_122
sage: [Z for Z in bch_iterator(X_1, X_2)]
[X_1 + X_2, 1/2*X_12, 1/12*X_112 + 1/12*X_122]
sage: [Z for Z in bch_iterator(X_1 + X_2, X_12)]
[X_1 + X_2 + X_12, 1/2*X_112 - 1/2*X_122, 0]

Python

>>> from sage.all import *
>>> L = LieAlgebra(QQ, Integer(2), step=Integer(3))
>>> L.inject_variables()
Defining X_1, X_2, X_12, X_112, X_122
>>> [Z for Z in bch_iterator(X_1, X_2)]
[X_1 + X_2, 1/2*X_12, 1/12*X_112 + 1/12*X_122]
>>> [Z for Z in bch_iterator(X_1 + X_2, X_12)]
[X_1 + X_2 + X_12, 1/2*X_112 - 1/2*X_122, 0]

The elements X and Y don’t need to be elements of the same Lie algebra if there is a coercion from one to the other:

Sage

sage: L = LieAlgebra(QQ, 3, step=2)
sage: L.inject_variables()
Defining X_1, X_2, X_3, X_12, X_13, X_23
sage: S = L.subalgebra(X_1, X_2)
sage: bch1 = [Z for Z in bch_iterator(S(X_1), S(X_2))]; bch1
[X_1 + X_2, 1/2*X_12]
sage: bch1[0].parent() == S
True
sage: bch2 = [Z for Z in bch_iterator(S(X_1), X_3)]; bch2
[X_1 + X_3, 1/2*X_13]
sage: bch2[0].parent() == L
True

Python

>>> from sage.all import *
>>> L = LieAlgebra(QQ, Integer(3), step=Integer(2))
>>> L.inject_variables()
Defining X_1, X_2, X_3, X_12, X_13, X_23
>>> S = L.subalgebra(X_1, X_2)
>>> bch1 = [Z for Z in bch_iterator(S(X_1), S(X_2))]; bch1
[X_1 + X_2, 1/2*X_12]
>>> bch1[Integer(0)].parent() == S
True
>>> bch2 = [Z for Z in bch_iterator(S(X_1), X_3)]; bch2
[X_1 + X_3, 1/2*X_13]
>>> bch2[Integer(0)].parent() == L
True

The BCH formula requires a coercion from the rationals:

Sage

sage: L.<X,Y,Z> = LieAlgebra(ZZ, 2, step=2)
sage: bch = bch_iterator(X, Y); next(bch)
Traceback (most recent call last):
...
TypeError: the BCH formula is not well defined since Integer Ring has no coercion from Rational Field

Python

>>> from sage.all import *
>>> L = LieAlgebra(ZZ, Integer(2), step=Integer(2), names=('X', 'Y', 'Z',)); (X, Y, Z,) = L._first_ngens(3)
>>> bch = bch_iterator(X, Y); next(bch)
Traceback (most recent call last):
...
TypeError: the BCH formula is not well defined since Integer Ring has no coercion from Rational Field

ALGORITHM:

The BCH formula \(\log(\exp(X)\exp(Y)) = \sum_k Z_k\) is computed starting from \(Z_1 = X + Y\), by the recursion

\[(m+1)Z_{m+1} = \frac{1}{2}[X - Y, Z_m] + \sum_{2\leq 2p \leq m}\frac{B_{2p}}{(2p)!}\sum_{k_1+\cdots+k_{2p}=m} [Z_{k_1}, [\cdots [Z_{k_{2p}}, X + Y]\cdots],\]

where \(B_{2p}\) are the Bernoulli numbers, see Lemma 2.15.3. in [Var1984].

Warning

The time needed to compute each successive term increases exponentially. For example on one machine iterating through \(Z_{11},...,Z_{18}\) for a free Lie algebra, computing each successive term took 4-5 times longer, going from 0.1s for \(Z_{11}\) to 21 minutes for \(Z_{18}\).