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Baker-Campbell-Hausdorff formula
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In mathematics, the Baker-Campbell-Hausdorff formula is the solution to
for non-commuting X and Y. It links Lie Groups to Lie Algebras, by expressing the logarithm of the product of two Lie group elements as a Lie algebra element in
canonical coordinates, a significant guiding connection appreciated before the full development of the theory.
It is named for Henry Frederick Baker, John Edward Campbell, and Felix Hausdorff. It was first noted in print by Campbell (1897); elaborated by Henri Poincaré (1899) and Baker (1902);
and systematized geometrically, and linked to the Jacobi identity by Hausdorff (1906).
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Encyclopedia
In mathematics, the Baker-Campbell-Hausdorff formula is the solution to
for non-commuting X and Y. It links Lie Groups to Lie Algebras, by expressing the logarithm of the product of two Lie group elements as a Lie algebra element in
canonical coordinates, a significant guiding connection appreciated before the full development of the theory.
It is named for Henry Frederick Baker, John Edward Campbell, and Felix Hausdorff. It was first noted in print by Campbell (1897); elaborated by Henri Poincaré (1899) and Baker (1902);
and systematized geometrically, and linked to the Jacobi identity by Hausdorff (1906). The explicit combinatoric formula furnished below was introduced by Eugene Dynkin (1947).
The Baker-Campbell-Hausdorff formula: existence
The Baker-Campbell-Hausdorff formula implies that if X and Y are in some Lie algebra defined over any field of characteristic 0, then
can be written as a formal infinite sum of elements of . For many applications one does not need an explicit expression for this infinite sum but just its existence, and this can be seen as follows. The ring
- S = R⟨X,Y⟩
of all non-commuting formal power series in non-commuting variables X and Y has a ring homomorphism Δ from S to the completion of
- S⊗S,
called the coproduct, such that
- S(X) = X⊗1 + 1⊗X
and similarly for Y. This has
the following properties:
- exp is an isomorphism (of sets) from the elements of S with constant term 0 to the elements with constant term 1, with inverse log
- r=exp(s) is grouplike (this means Δ(r)=r⊗r) if and only if s is primitive (this means Δ(s)=s⊗1+1⊗s).
- The grouplike elements form a group under multiplication.
- The primitive elements are exactly the formal infinite sums of elements of the Lie algebra generated by X and Y.
The existence of the Baker-Campbell-Hausdorff formula can now be seen as follows:
The elements X and Y are primitive, so exp(X) and exp(Y) are grouplike, so their product exp(X)exp(Y) is also grouplike, so its logarithm log(exp(X)exp(Y)) is primitive, and hence can be written as an infinite sum of elements of the Lie algebra generated by X and Y.
The universal enveloping algebra of the free Lie algebra generated by X and Y is isomorphic to the algebra of all non-commuting polynomials in X and Y. In common with all universal enveloping algebras, it has a natural structure of a Hopf algebra, with a coproduct Δ. The ring S used above is just a completion of this Hopf algebra.
An explicit Baker-Campbell-Hausdorff formula
Specifically, let G be a simply-connected Lie group with Lie algebra . Let
be the exponential map.
The general formula is given by:
which uses the notation
This term is zero if or if and (Sagle & Walde 1973, pp. 134+135).
The first few terms are well-known, with all higher-order terms involving [X,Y] and commutator nestings thereof (thus in the Lie algebra):
Note the X-Y (anti-)/symmetry in alternating orders of the expansion, since .
Selected tractable cases
There is no expression in closed form for an arbitrary Lie algebra, though there are exceptional tractable cases, as well as efficient algorithms for working out the expansion in applications.
For example, note that if [X,Y] vanishes, then the above formula manifestly reduces to X + Y. If the commutator [X,Y] is a constant (central), then all but the first three terms on the right-hand side of the above vanish.
If one of the Lie algebra elements X maps the kernel of ad Y into itself, other forms of the Campbell-Baker-Hausdorff formula might serve well:
as is evident from the integral formula below. (The coefficients of the nested commutators linear in Y are normalized Bernoulli numbers, outlined below.) Thus, when the commutator happens to be [X, Y] = sY, for some non-zero s, this formula reduces to just Z = X + sY / (1 − exp(-s)), which then leads to braiding identities such as
There are numerous such well-known expressions applied routinely in physics, cf. Magnus (1954). A popular integral formula is
involving a generating function for the Bernoulli numbers,
utilized by Poincaré and Hausdorff. Recall
,
for the Bernoulli numbers, ,
, ...
Matrix Lie Group Illustration
For a matrix Lie group the Lie algebra is the tangent space of the identity I, and the commutator is simply [X, Y] = XY − YX; the exponential map is the standard exponential map of matrices,
When one solves for Z in
one obtains a simpler formula:
Note that the first, second, third, and fourth order terms are:
The Zassenhaus formula
A related combinatoric expansion, useful in dual applications is
where all exponents of order larger than t are likewise
nested commutators.
The Hadamard lemma
A standard combinatoric lemma utilized, among others, in the above explicit expansions is
This formula can be proved trivially by parametric induction: evaluation of the derivative with respect to s of ,
recursive determination of the Taylor expansion coefficients in
terms of nested commutators, and evaluation at .
See also
External links
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