Derivative of the exponential map

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. d d t e X ( t ) = e X 1 − e − a d X a d X d X ( t ) d t . {displaystyle {frac {d}{dt}}e^{X(t)}=e^{X}{frac {1-e^{-mathrm {ad} _{X}}}{mathrm {ad} _{X}}}{frac {dX(t)}{dt}}.}               (1) 1 − e − a d X a d X = ∑ k = 0 ∞ ( − 1 ) k ( k + 1 ) ! ( a d X ) k . {displaystyle {frac {1-e^{-mathrm {ad} _{X}}}{mathrm {ad} _{X}}}=sum _{k=0}^{infty }{frac {(-1)^{k}}{(k+1)!}}(mathrm {ad} _{X})^{k}.}     (2) d exp X ⁡ Y = e X 1 − e − a d X a d X Y {displaystyle dexp _{X}Y=e^{X}{frac {1-e^{-mathrm {ad} _{X}}}{mathrm {ad} _{X}}}Y}     (3) A d e X = e a d X ,     X ∈ g   . {displaystyle mathrm {Ad} _{e^{X}}=e^{mathrm {ad} _{X}},~~Xin {mathfrak {g}}~.}               (4)By direct differentiation of the standard limit definition of the exponential, and exchanging the order of differentiation and limit, Z ′ = ∑ n = 1 ∞ ( − ) n − 1 n ( e a d Z − 1 ) n − 1   ( X + e t a d X Y )   , {displaystyle Z'=sum limits _{n=1}^{infty }{frac {(-)^{n-1}}{n}}(e^{mathrm {ad} _{Z}}-1)^{n-1}~(X+e^{t,mathrm {ad} _{X}}Y)~,}     (5) Z = ∑ k = 1 ∞ ( − 1 ) k − 1 k ∑ s ∈ S k 1 i 1 + j 1 + ⋯ + i k + j k [ X ( i 1 ) Y ( j 1 ) ⋯ X ( i k ) Y ( j k ) ] i 1 ! j 1 ! ⋯ i k ! j k ! , i r , j r ≥ 0 , i r + j r > 0 , 1 ≤ r ≤ k . {displaystyle Z=sum _{k=1}^{infty }{frac {(-1)^{k-1}}{k}}sum _{sin S_{k}}{frac {1}{i_{1}+j_{1}+cdots +i_{k}+j_{k}}}{frac {}{i_{1}!j_{1}!cdots i_{k}!j_{k}!}},quad i_{r},j_{r}geq 0,quad i_{r}+j_{r}>0,quad 1leq rleq k.} Change the summation index in (5) to k = n − 1 and expand d Z d t = ∑ k = 0 ∞ ( − 1 ) k k + 1 { ( e a d t X e a d t Y − 1 ) k X + ( e a d t X e a d t Y − 1 ) k e a d t X Y } {displaystyle {frac {dZ}{dt}}=sum _{k=0}^{infty }{frac {(-1)^{k}}{k+1}}left{(e^{mathrm {ad} _{tX}}e^{mathrm {ad} _{tY}}-1)^{k}X+(e^{mathrm {ad} _{tX}}e^{mathrm {ad} _{tY}}-1)^{k}e^{mathrm {ad} _{tX}}Y ight}}     (97) log ⁡ ( e X e Y ) = ∑ k = 1 ∞ ( − 1 ) k + 1 k ( e X e Y − I ) k = ∑ k = 1 ∞ ( − 1 ) k + 1 k ( ∑ i = 0 ∞ X i i ! ∑ j = 0 ∞ Y j j ! − I ) k = ∑ k = 1 ∞ ( − 1 ) k + 1 k ( ∑ i , j ≥ 0 , i + j > 1 ∞ X i Y j i ! j ! ) k . {displaystyle log(e^{X}e^{Y})=sum _{k=1}^{infty }{frac {(-1)^{k+1}}{k}}{(e^{X}e^{Y}-I)}^{k}=sum _{k=1}^{infty }{frac {(-1)^{k+1}}{k}}left({sum _{i=0}^{infty }{frac {X^{i}}{i!}}sum _{j=0}^{infty }{frac {Y^{j}}{j!}}-I} ight)^{k}=sum _{k=1}^{infty }{frac {(-1)^{k+1}}{k}}left(sum _{i,jgeq 0,i+j>1}^{infty }{frac {X^{i}Y^{j}}{i!j!}} ight)^{k}.}     (98) Z = log ⁡ ( e X e Y ) = ∑ k = 1 ∞ ( − 1 ) k + 1 k ∑ s ∈ S k X i 1 Y j 1 ⋯ X i k Y j k i 1 ! j 1 ! ⋯ i k ! j k ! , i r , j r ≥ 0 , i r + j r > 0 , 1 ≤ r ≤ k , {displaystyle Z=log(e^{X}e^{Y})=sum _{k=1}^{infty }{frac {(-1)^{k+1}}{k}}sum _{sin S_{k}}{frac {X^{i_{1}}Y^{j_{1}}cdots X^{i_{k}}Y^{j_{k}}}{i_{1}!j_{1}!cdots i_{k}!j_{k}!}},quad i_{r},j_{r}geq 0,quad i_{r}+j_{r}>0,quad 1leq rleq k,}         (99) Z = ∑ k = 0 ∞ ( − 1 ) k k + 1 ∑ s ∈ S k + 1 1 i 1 + j 1 + ⋯ + i k + j k + i k + 1 + j k + 1 [ X ( i 1 ) Y ( j 1 ) ⋯ X ( i k ) Y ( j k ) X ( i k + 1 ) Y ( j k + 1 ) ] i 1 ! j 1 ! ⋯ i k ! j k ! i k + 1 ! j k + 1 ! , i r , j r ≥ 0 , i r + j r > 0 , 1 ≤ r ≤ k + 1 , {displaystyle Z=sum _{k=0}^{infty }{frac {(-1)^{k}}{k+1}}sum _{sin S_{k+1}}{frac {1}{i_{1}+j_{1}+cdots +i_{k}+j_{k}+i_{k+1}+j_{k+1}}}{frac {}{i_{1}!j_{1}!cdots i_{k}!j_{k}!i_{k+1}!j_{k+1}!}},quad i_{r},j_{r}geq 0,quad i_{r}+j_{r}>0,quad 1leq rleq k+1,}     (100) Z = log ⁡ e X e Y = ∑ k = 1 ∞ ( − 1 ) k − 1 k ∑ s ∈ S k 1 i 1 + j 1 + ⋯ + i k + j k [ X ( i 1 ) Y ( j 1 ) ⋯ X ( i k ) Y ( j k ) ] i 1 ! j 1 ! ⋯ i k ! j k ! ,   i r , j r ≥ 0 ,   i r + j r > 0 ,   1 ≤ r ≤ k . {displaystyle Z=log e^{X}e^{Y}=sum _{k=1}^{infty }{frac {(-1)^{k-1}}{k}}sum _{sin S_{k}}{frac {1}{i_{1}+j_{1}+cdots +i_{k}+j_{k}}}{frac {}{i_{1}!j_{1}!cdots i_{k}!j_{k}!}},~i_{r},j_{r}geq 0,~i_{r}+j_{r}>0,~1leq rleq k.} X i 1 Y j 1 ⋯ X i k Y j k = [ X ( i 1 ) Y ( j 1 ) ⋯ X ( i k ) Y ( j k ) ] i 1 + j 1 + ⋯ + i k + j k . {displaystyle X^{i_{1}}Y^{j_{1}}cdots X^{i_{k}}Y^{j_{k}}={frac {}{i_{1}+j_{1}+cdots +i_{k}+j_{k}}}.}     (B) 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 derived from the power series of the exponential map of a linear endomorphism, as in matrix exponentiation 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.