The Baker-Campbell-Hausdorff formula¶
AUTHORS:
Eero Hakavuori (2018-09-23): initial version
- sage.algebras.lie_algebras.bch.bch_iterator(X=None, Y=None)¶
A generator function which returns successive terms of the Baker-Campbell-Hausdorff formula.
INPUT:
X
– (optional) an element of a Lie algebraY
– (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
andY
with rational coefficients. In arbitrary Lie algebras, the infinite sum is only guaranteed to converge forX
andY
close to zero.If the elements
X
andY
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
andY
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: 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]
For nilpotent Lie algebras the BCH formula only has finitely many terms:
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]
The elements
X
andY
don’t need to be elements of the same Lie algebra if there is a coercion from one to the other: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
The BCH formula requires a coercion from the rationals:
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
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 \(Z_{18}\).