Real Clifford Algebras And Subgroups Of Their Unit Groups

Clifford algebra is an associative algebra that was introduced by the English math-ematician W. K. Clifford (1845-1879) to generalize Hamilton’s quaternion to higher di-mension. Clifford algebras have intuitive geometric interpretations and been applied to physics, the black hole theory, cosmology, quantum orbit, quantum field theory, robot, computer vision, etc. The study of Clifford algebras culminates in the 8-periodicity theory of Cartan and Atiyah, Bott and Shapiro. In this dissertation, by using the 8-periodicity theory we mainly study the real Clifford algebra Clp,q as tensor products over the center and its matrix representations, matrix representations of the unit group, properties of the Clifford group and lattice ordered semigroup structure in the generating space—the lattice ordered semigroup in Minkowski space of dimension n.The dissertation is organized as follows.In Chapter 1, we provide some background, a brief survey related to our research and statements of our main results.In Chapter 2, we provide preliminaries.In Chapter 3, we first study the real Clifford algebra Clp,q as tensor products over center and its matrix representations. We present a unified tensor product expression of the real Clifford algebra Clp,q based on the 8-periodicity theory, and give matrix representations via its tensor products.Theorem 3.1.3.For any nonnegative integers p,q we have where p+q≡ε mod 2,k=((p+g)-ε)/2,p-|q-ε|≡i mod 8,δ=[i/4] the integral part of i/4.Corollary 3.1.1. where p+q≡ε mod 2,k=((p+q)-ε)/2,p-|q-ε|≡i mod 8,δ=[i/4].Then we present simpler tensor products and matrix representations of the real Clifford algebra Cl0,2k+1.Theorem 3.2.1.Let k be a nonnegative integer.Then where 2k+1≡α mod 8,δ=[1-{α/3}],in whic {α/3} denotes the decimal portion of α/3.Theorem 3.2.2.Let k be a nonnegative integer.Then where 2k+1≡α mod 8 and δ=[1-{α/3}].We conclude Chapter 3 with a discussion about the structure of tensor product factors of the real Clifford algebra Clp,q.In Chapter 4,we first classify matrix representations of the real Clifford algebra Clp,q.Theorem 4.1.1.Let k be a nonnegative integer.For any F∈End(Clp,q),a restric-tion f of a map F on Cen(Clp,q)satisfies where p+q≡εmod 2,k=((p+q)-ε)/2,p-|q-ε|≡i mod 8,δ=[i/4].Then we give faithful and non-faithful matrix representations of Clp,q (p+q=3) respectively,so we can obtain all Inatrix represnations of the real Clifford algebra Clp,q.In Chapter 5,based on tensor products and representations of the real Clifford algebra given in Chapter 3-4,we discuss matrix representations of unit group Clp,q*.Theorem 5.1.1. where p+q≡ε mod 2,k=((p+g)-ε)/2,p-|q-ε|≡i mod 8,δ=[i/4].We characterize units of the real Clifford algebra Clp,q(p+q=3) and give matrix representations of the unit group in terms of three involutions of the real Clifford algebra Clp,q.Theorem 5.2.2.Let Z(Clp,q)be the zero-divisor set of Clp,q(1)Cl0,3*={α+βe123∈Cl0,3|α,β∈<e1,e2>,α≠β}, Cl2,1*={α+βe123∈Cl2,1|α,β∈<e1,e2>,(α±β)(α±β)≠0}, Cl3,0*={α+βe123∈Cl3,0|α,β∈<e1,e2>,aa≠0}..(2)Z(Cl0,3)={α+βe123∈Cl0,3|α,β∈<e1,e2>,α=±β}, Z(Cl2,1)={α+βe123∈Cl2,1|α,β∈<e1,e2>,(α±β)(α±β)=0}, Z(Cl3,0)={α+βe123∈Cl3,0|α,β∈<e1,e2>,aa=0}.We also give relations of three subsets of the Clifford group by use of relations of the Clifford group rp,q and a basis of Rp,q.Theorem 5.3.1.For any nonnegative integers p,q letΓ1={a∈Clp,q*| axa-1∈Rp,q,aa∈R,(?)x∈Rp,q},Γ2={a∈Clp,q*| qxa∈Rp,q,aa∈R,(?)x∈RP,q}Γ3＝{a∈Clp,q*| aeia∈Rp,q,i=1,...,p+q,aa∈R} Then Γ1=Γ2=Γ3.Finally,we discuss lattice ordered semigroups in Rn-1,1 as the generating space of the real Clifford algebra Cln-1,1.