qualifications) to all such problems and it can be factored out of a specific problem at hand, thus reducing its complexity. A prime example – in mathematics and physics – would be the theory of spherical harmonics. Their role in the group theory of the rotation groups is that of being a representation space for the entire set of finite-dimensional irreducible representations of the rotation group SO(3). For this topic, see Rotation group SO(3) § Spherical harmonics. The main articles listed in each subsection are referred to for more detail. Lie group The rotation matrices for each form a group, the special orthogonal group, . This algebraic structure is coupled with a topological structure inherited from in such a way that the operations of multiplication and taking the inverse are analytic functions of the matrix entries. Thus is for each a Lie group. It is compact and connected, but not simply connected. It is also a semi-simple group, in fact a simple group with the exception SO(4). The relevance of this is that all theorems and all machinery from the theory of analytic manifolds (analytic manifolds are in particular smooth manifolds) apply and the well-developed representation theory of compact semi-simple groups is ready for use. Lie algebra The Lie algebra of is given by and is the space of skew-symmetric matrices of dimension , see classical group, where is the Lie algebra of , the orthogonal group. For reference, the most common basis for is Exponential map Connecting the Lie algebra to the Lie group is the exponential map, which is defined using the standard matrix exponential series for For any skew-symmetric matrix , is always a rotation matrix. An important practical example is the case. In rotation group SO(3), it is shown that one can identify every with an Euler vector , where is a unit magnitude vector. By the properties of the identification , is in the null space of . Thus, is left invariant by and is hence a rotation axis. According to Rodrigues' rotation formula on matrix form, one obtains, where This is the matrix for a rotation around axis by the angle . For full detail, see exponential map SO(3). Baker–Campbell–Hausdorff formula The BCH formula provides an explicit expression for in terms of a series expansion of nested commutators of and . This general expansion unfolds as In the case, the general infinite expansion has