In the theory of Lie groups, the exponential map is a map from the Lie algebra g of a Lie group G into G. In case G is a matrix Lie group, the exponential map reduces to the matrix exponential. The exponential map, denoted exp:g → G, is analytic and has as such a derivative d/dtexp(X(t)):Tg → TG, where X(t) is a C1 path in the Lie algebra, and a closely related differential dexp:Tg → TG.
The formula for dexp was first proved by Friedrich Schur (1891). It was later elaborated by Henri Poincaré (1899) in the context of the problem of expressing Lie group multiplication using Lie algebraic terms. It is also sometimes known as Duhamel's formula.
The formula is important both in pure and applied mathematics. It enters into proofs of theorems such as the Baker–Campbell–Hausdorff formula, and it is used frequently in physics for example in quantum field theory, as in the Magnus expansion in perturbation theory, and in lattice gauge theory.
Throughout, the notations exp(X) and eX will be used interchangeably to denote the exponential given an argument, except when, where as noted, the notations have dedicated distinct meanings. The calculus-style notation is preferred here for better readability in equations. On the other hand, the exp-style is sometimes more convenient for inline equations, and is necessary on the rare occasions when there is a real distinction to be made.
The derivative of the exponential map is given by
- X = X(t) is a C1 (continuously differentiable) path in the Lie algebra with derivative X′(t) = dX(t)/dt. The argument t is omitted where not needed.
- adX is the linear transformation of the Lie algebra given by adX(Y) = [X, Y]. It is the adjoint action of a Lie algebra on itself.
- The fraction 1 − exp(−adX)/adX is given by the power series
- When G is a matrix Lie group, all occurrences of the exponential are given by their power series expansion.
- When G is not a matrix Lie group, 1 − exp(−adX)/adX is still given by its power series (2), while the other two occurrences of exp in the formula, which now are the exponential map in Lie theory, refer to the time-one flow of the left invariant vector field X, i.e. element of the Lie algebra as defined in the general case, on the Lie group G viewed as an analytic manifold. This still amounts to exactly the same formula as in the matrix case. Left multiplication of an element of the algebra g by an element exp(X(t)) of the Lie group is interpreted as applying the differential of the left translation dLexp(X(t)).
- The formula applies to the case where exp is considered as a map on matrix space over ℝ or C, see matrix exponential. When G = GL(n, C) or GL(n, R), the notions coincide precisely.
To compute the differential dexp of exp at X, dexpX: TgX → TGexp(X), the standard recipe
is employed. With Z(t) = X + tY the result
follows immediately from (1). In particular, dexp0:Tg0 → TGexp(0) = TGe is the identity because TgX ≃ g (since g is a vector space) and TGe ≃ g.
The proof given below assumes a matrix Lie group. This means that the exponential mapping from the Lie algebra to the matrix Lie group is given by the usual power series, i.e. matrix exponentiation. The conclusion of the proof still holds in the general case, provided each occurrence of exp is correctly interpreted. See comments on the general case below.
The outline of proof makes use of the technique of differentiation with respect to s of the parametrized expression
to obtain a first order differential equation for Γ which can then be solved by direct integration in s. The solution is then eX Γ(1, t).
Using the product rule twice one finds,
Then one observes that
by (4) above. Integration yields
Using the formal power series to expand the exponential, integrating term by term, and finally recognizing (2),
Comments on the general case
The formula in the general case is given by
which formally reduces to
Here the exp-notation is used for the exponential mapping of the Lie algebra and the calculus-style notation in the fraction indicates the usual formal series expansion. For more information and two full proofs in the general case, see the freely available Sternberg (2004) reference.
A direct formal argument
An immediate way to see what the answer must be, provided it exists is the following. Existence needs to be proved separately in each case. By direct differentiation of the standard limit definition of the exponential, and exchanging the order of differentiation and limit,
where each factor owes its place to the non-commutativity of X(t) and X´(t).
Dividing the unit interval into N sections Δs = Δk/N (Δk = 1 since the sum indices are integers) and letting N → ∞, Δk → dk, k/N → s, Σ → ∫ yields
Local behavior of the exponential map
The inverse function theorem together with the derivative of the exponential map provides information about the local behavior of exp. Any Ck, 0 ≤ k ≤ ∞, ω map f between vector spaces (here first considering matrix Lie groups) has a Ck inverse such that f is a Ck bijection in an open set around a point x in the domain provided dfx is invertible. From (3) it follows that this will happen precisely when
is invertible. This, in turn, happens when the eigenvalues of this operator are all nonzero. The eigenvalues of 1 − exp(−adX)/adX are related to those of adX as follows. If g is an analytic function of a complex variable expressed in a power series such that g(U) for a matrix U converges, then the eigenvalues of g(U) will be g(λij), where λij are the eigenvalues of U, the double subscript is made clear below.[nb 3] In the present case with g(U) = 1 − exp(−U)/U and U = adX, the eigenvalues of 1 − exp(−adX)/adX are
where the λij are the eigenvalues of adX. Putting 1 − exp(−λij)/λij = 0 one sees that dexp is invertible precisely when
The eigenvalues of adX are, in turn, related to those of X. Let the eigenvalues of X be λi. Fix an ordered basis ei of the underlying vector space V such that X is lower triangular. Then
with the remaining terms multiples of en with n > i. Let Eij be the corresponding basis for matrix space, i.e. (Eij)kl = δikδjl. Order this basis such that Eij < Enm if i − j < n − m. One checks that the action of adX is given by
with the remaining terms multiples of Emn > Eij. This means that adX is lower triangular with its eigenvalues λij = λi − λj on the diagonal. The conclusion is that dexpX is invertible, hence exp is a local bianalytical bijection around X, when the eigenvalues of X satisfy[nb 4]
In particular, in the case of matrix Lie groups, it follows, since dexp0 is invertible, by the inverse function theorem that exp is a bi-analytic bijection in a neighborhood of 0 ∈ g in matrix space. Furthermore, exp, is a bi-analytic bijection from a neighborhood of 0 ∈ g in g to a neighborhood of e ∈ G. The same conclusion holds for general Lie groups using the manifold version of the inverse function theorem.
Derivation of a Baker–Campbell–Hausdorff formula
If Z(t) is defined such that
an expression for Z(1) = log( exp X exp Y ), the Baker–Campbell–Hausdorff formula, can be derived from the above formula,
Its left-hand side is easy to see to equal Y. Thus,
However, using the relationship between Ad and ad given by (4), it is straightforward to further see that
Putting this into the form of an integral in t from 0 to 1 yields,
an integral formula for Z(1) that is more tractable in practice than the explicit Dynkin's series formula due to the simplicity of the series expansion of ψ. Note this expression consists of X+Y and nested commutators thereof with X or Y. A textbook proof along these lines can be found in Hall (2015) and Miller (1972).
Derivation of Dynkin's series formula
Dynkin's formula mentioned may also be derived analogously, starting from the parametric extension
so that, using the above general formula,
the last step by virtue of the Mercator series expansion, it follows that
and, thus, integrating,
It is at this point evident that the qualitative statement of the BCH formula holds, namely Z lies in the Lie algebra generated by X, Y and is expressible as a series in repeated brackets (A). For each k, terms for each partition thereof are organized inside the integral ∫dt tk−1. The resulting Dynkin's formula is then
For a similar proof with detailed series expansions, see Rossmann (2002).
Change the summation index in (5) to k = n − 1 and expand
in a power series. To handle the series expansions simply, consider first Z = log(eXeY). The log-series and the exp-series are given by
respectively. Combining these one obtains
where Sk is the set of all sequences s = (i1, j1, …, ik, jk) of length 2k subject to the conditions in (99).
or, with a switch of notation, see An explicit Baker–Campbell–Hausdorff formula,
Note that the summation index for the rightmost eadtX in the second term in (97) is denoted ik + 1, but is not an element of a sequence s ∈ Sk. Now integrate Z = Z(1) = ∫dZ/dtdt, using Z(0) = 0,
Write this as
This amounts to
where using the simple observation that [T, T] = 0 for all T. That is, in (100), the leading term vanishes unless jk + 1 equals 0 or 1, corresponding to the first and second terms in the equation before it. In case jk + 1 = 0, ik + 1 must equal 1, else the term vanishes for the same reason (ik + 1 = 0 is not allowed). Finally, shift the index, k → k − 1,
This is Dynkin's formula. The striking similarity with (99) is not accidental: It reflects the Dynkin–Specht–Wever map, underpinning the original, different, derivation of the formula. Namely, if
is expressible as a bracket series, then necessarily
- Adjoint representation (ad)
- Adjoint representation (Ad)
- Baker-Campbell-Hausdorff formula
- Exponential map
- Matrix exponential
- Matrix logarithm
- Magnus expansion
- A proof of the identity can be found in here. The relationship is simply that between a representation of a Lie group and that of its Lie algebra according to the Lie correspondence, since both Ad and ad are representations with ad = dAd.
It holds that
- This is seen by choosing a basis for the underlying vector space such that U is triangular, the eigenvalues being the diagonal elements. Then Uk is triangular with diagonal elements λik. It follows that the eigenvalues of U are f(λi). See Rossmann 2002, Lemma 6 in section 1.2.
- Matrices whose eigenvalues λ satisfy |Im λ| < π are, under the exponential, in bijection with matrices whose eigenvalues μ are not on the negative real line or zero. The λ and μ are related by the complex exponential. See Rossmann (2002) Remark 2c section 1.2.
- Schmid 1982
- Rossmann 2002 Appendix on analytic functions.
- Schur 1891
- Poincaré 1899
- Suzuki 1985 harvnb error: no target: CITEREFSuzuki1985 (help)
- Rossmann 2002 Theorem 5 Section 1.2
- Hall 2015 Proposition 3.35
- See also Tuynman 1995 from which Hall's proof is taken.
- Sternberg 2004 This is equation (1.11).
- Rossman 2002 harvnb error: no target: CITEREFRossman2002 (help) Proposition 7, section 1.2.
- Hall 2015 Corollary 3.44.
- Sternberg 2004 Section 1.6.
- Hall 2015Section 5.5.
- Sternberg 2004 Section 1.2.
- Dynkin 1947
- Rossmann 2002 Chapter 2.
- Hall 2015 Chapter 5.
- Sternberg 2004 Chapter 1.12.2.
- Dynkin, Eugene Borisovich (1947), "Вычисление коэффициентов в формуле Campbell–Hausdorff" [Calculation of the coefficients in the Campbell–Hausdorff formula], Doklady Akademii Nauk SSSR (in Russian), 57: 323–326 ; translation from Google books.
- Hall, Brian C. (2015), Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer, ISBN 978-3319134666
- Miller, Wllard (1972), Symmetry Groups and their Applications, Academic Press, ISBN 0-12-497460-0
- Poincaré, H. (1899), "Sur les groupes continus", Cambridge Philos. Trans., 18: 220–55
- Rossmann, Wulf (2002), Lie Groups – An Introduction Through Linear Groups, Oxford Graduate Texts in Mathematics, Oxford Science Publications, ISBN 0-19-859683-9
- Schur, F. (1891), "Zur Theorie der endlichen Transformationsgruppen", Abh. Math. Sem. Univ. Hamburg, 4: 15–32
- Suzuki, Masuo (1985). "Decomposition formulas of exponential operators and Lie exponentials with some applications to quantum mechanics and statistical physics". Journal of Mathematical Physics. 26 (4): 601–612. Bibcode:1985JMP....26..601S. doi:10.1063/1.526596.
- Tuynman (1995), "The derivation of the exponential map of matrices", Amer. Math. Monthly, 102 (9): 818–819, doi:10.2307/2974511, JSTOR 2974511
- Veltman, M, 't Hooft, G & de Wit, B (2007). "Lie Groups in Physics", online lectures.
- Wilcox, R. M. (1967). "Exponential Operators and Parameter Differentiation in Quantum Physics". Journal of Mathematical Physics. 8 (4): 962–982. Bibcode:1967JMP.....8..962W. doi:10.1063/1.1705306.