Baker–Campbell–Hausdorff formula

In mathematics, the Baker–Campbell–Hausdorff formula gives a value of that solves the equation 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 in Lie algebraic terms, that is, as a formal series in and and iterated commutators thereof. The first few terms of this series are: where "" indicates terms involving higher commutators of and . If and are sufficiently small elements of the Lie algebra of a Lie group , the series is convergent. Meanwhile, every element sufficiently close to the identity in can be expressed as for a small in . Thus, we can say that near the identity the group multiplication in —written as —can be expressed in purely Lie algebraic terms. The Baker–Campbell–Hausdorff formula can be used to give comparatively simple proofs of deep results in the Lie group–Lie algebra correspondence.