The universal cover of SO(n)

Given a vector space V and a non-singular quadratic form Q on V , the orthogonal group O(Q) is the subgroup of GL(V ) that preserves Q. The special orthogonal group SO(Q) is given by O(Q) ∩ SL(V ). If Q is the standard inner product on R then SO(Q) is denoted SO(n). This is a connected, compact Lie group. For n > 2, the group is semi-simple. In the case that n is odd, SO(n) corresponds to the Dynkin diagram Bn. In the case that n is even, SO(n) corresponds to the Dynkin diagram Dn. The special orthogonal group SO(n) is unique among the classical groups in not being simply connected. To study the representations of its Lie algebra so(n), it is necessary to understand the universal cover of SO(n). The project of this paper is to compute the root system of SO(n), to show that SO(n) is not simply connected by determining its fundamental group, and to explicitly construct its universal cover when n > 2. The universal cover, Spin(n), is constructed as a particular subgroup of the Clifford algebra of Q. Clifford algebras are defined and discussed in section 4.1 to prepare for the construction of Spin(n) in section 4.2.