the algebra generated by V along with the anticommutation relation . It is an abstract version of the algebra generated by the gamma or Pauli matrices. If V = , with the standard form we denote the Clifford algebra by Cℓn(). Since by the choice of an orthonormal basis every complex vectorspace with non-degenerate form is isomorphic to this standard example, this notation is abused more generally if . If is even, is isomorphic as an algebra (in a non-unique way) to the algebra of complex matrices (by the Artin–Wedderburn theorem and the easy to prove fact that the Clifford algebra is central simple). If is odd, is isomorphic to the algebra of two copies of the complex matrices. Therefore, in either case has a unique (up to isomorphism) irreducible representation (also called simple Clifford module), commonly denoted by Δ, of dimension 2[n/2]. Since the Lie algebra is embedded as a Lie subalgebra in equipped with the Clifford algebra commutator as Lie bracket, the space Δ is also a Lie algebra representation of called a spin representation. If n is odd, this Lie algebra representation is irreducible. If n is even, it splits further into two irreducible representations called the Weyl or half-spin representations. Irreducible representations over the reals in the case when V is a real vector space are much more intricate, and the reader is referred to the Clifford algebra article for more details. Spin groups Spinors form a vector space, usually over the complex numbers, equipped with a linear group representation of the spin group that does not factor through a representation of the group of rotations (see diagram). The spin group is the group of rotations keeping track of the homotopy class. Spinors are needed to encode basic information about the topology of the group of rotations because that group is not simply connected, but the simply connected spin group is its double cover. So for every rotation there are two elements of the spin group that represent it. Geometric vectors and other tensors cannot feel the difference between these two elements, but they produce opposite signs when they affect any spinor under the representation. Thinking of the elements of the spin group as homotopy classes of one-parameter families of rotations, each rotation is represented by two distinct homotopy classes of paths to the identity. If a one-parameter family of rotations is visualized as a ribbon