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Baker–Campbell–Hausdorff formula - Wikipedia

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In mathematics, the Baker–Campbell–Hausdorff formula gives the value of 𝑍 that solves the equation 𝑒 𝑋 𝑒 𝑌 = 𝑒 𝑍 for possibly noncommutative X and Y in the Lie algebra of a Lie group. There are various ways of writing the formula, but all ultimately yield an expression for 𝑍 in Lie algebraic terms, that is, as a formal series (not necessarily convergent) in 𝑋 and 𝑌 and iterated commutators thereof. The first few terms of this series are: 𝑍 = 𝑋 + 𝑌 + 1 2 [ 𝑋 , 𝑌 ] + 1 12 [ 𝑋 , [ 𝑋 , 𝑌 ] ] + 1 12 [ 𝑌 , [ 𝑌 , 𝑋 ] ] + ⋯ , where " ⋯ " indicates terms involving higher commutators of 𝑋 and 𝑌 . If 𝑋 and 𝑌 are sufficiently small elements of the Lie algebra 𝑔 of a Lie group 𝐺 , the series is convergent. Meanwhile, every element 𝑔 sufficiently close to the identity in 𝐺 can be expressed as 𝑔 = 𝑒 𝑋 for a small 𝑋 in 𝑔 . Thus, we can say that near the identity the group multiplication in 𝐺 —written as 𝑒 𝑋 𝑒 𝑌 = 𝑒 𝑍 —can be express

Baker–Campbell–Hausdorff formula - Wikipedia Jump to content From Wikipedia, the free encyclopedia Formula in Lie theory In mathematics , the Baker–Campbell–Hausdorff formula gives the value of Z {\displaystyle Z} that solves the equation e X e Y = e Z {\displaystyle e^{X}e^{Y}=e^{Z}} for possibly noncommutative X and Y in the Lie algebra of a Lie group . There are various ways of writing the formula, but all ultimately yield an expression for Z {\displaystyle Z} in Lie algebraic terms, that is, as a formal series (not necessarily convergent) in X {\displaystyle X} and Y {\displaystyle Y} and

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