By Hall B.C.

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Step 2 : φ(sX) = sφ(X) for all s ∈ R. This is immediate, since if φ(etX ) = etZ , then φ(etsX ) = etsZ . Step 3 : φ(X + Y ) = φ(X) + φ(Y ). By Steps 1 and 2, e e etφ(X+Y ) = eφ[t(X+Y )] = φ et(X+Y ) . By the Lie product formula, and the fact that φ is a continuous homomorphism: =φ = lim m→∞ lim m→∞ m etX/m etY /m φ etX/m φ(etY /m) m . But then we have e etφ(X+Y ) = lim m→∞ e e etφ(X)/m etφ(Y )/m m = et(φ(X)+φ(Y )) . e e Differentiating this result at t = 0 gives the desired result. Step 4 : φ AXA−1 = φ(A)φ(X)φ(A)−1 .

Then Y X eX+Y = lim em em m→∞ m . This theorem has a big brother, called the Trotter product formula, which gives the same result in the case where X and Y are suitable unbounded operators on an infinite-dimensional Hilbert space. The Trotter formula is described, for example, in M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 8. Proof. ) Cm ≤ const. m2 . Since e e domain of the logarithm for all sufficiently large m. 7 Cm ≤ const. the logarithm gives + Y m + Cm 2 ≤ const. m2 .

21. The physics literature does not always distinguish clearly between a matrix Lie group and its Lie algebra. Before examining general properties of the Lie algebra, let us compute the Lie algebras of the matrix Lie groups introduced in the previous chapter. 2. The general linear groups. 3, etX is invertible. Thus the Lie algebra of GL(n; C) is the space of all n × n complex matrices. This Lie algebra is denoted gl(n; C). If X is any n × n real matrix, then etX will be invertible and real. On the other d etX will also be real.

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