Research On The Related Properties Of Clifford Algebras Cl₂ And Cl_1,2

Clifford algebra Clp,q is a class of 2p+q-dimensional real associative algebra,also known as geometric algebra,which has a wide range of applications in geometry,quantum theory,general relativity,and other fields.In this thesis,we focus on the ring isomorphism,the Moore-Penrose inverse,similarity,pseudo-similarity of Clifford algebra Cl2 and Cl1,2,and the properties of the finite ring Cl2/Zp.In order to study the properties of Clifford algebra Cl2,we first introduce the ring isomorphism of Cl2 and a ring of real matrices of 4 order,and we represent the elements in Cl2 by real matrices of 4 order.Then,according to the concept of Moore-Penrose inverse of real matrices,we introduce the concept of the Moore-Penrose inverse of the elements in Cl2 by such a ring isomorphism,and we obtain the Moore-Penrose inverse of the elements in Cl2.Finally,by discussing the reversible solutions of the equations ax=xb and ax=(?)b,we obtain some necessary and sufficient conditions for two numbers in Cl2 to be similar and pseudo-similar.In order to study the properties of Clifford algebra we first introduce the ring isomorphism of Clifford algebra Cl1,2 and a ring of real matrices of 8 order,and we represent the elements in Cll,2 by real matrices of 8 order.Secondly,by the same solution of equation ax=0 and a’ax=0 and the same solution of equation xa’=0 and xa’a=0,we obtain the Moore-Penrose inverse of elements in Clifford algebra Cl1,2.Then,using the Moore-Penrose inverse,we solve the linear equation axb=d in Cl1,2.In the end,we obtain some necessary and sufficient conditions for two numbers in Cl1,2 to be similar.In order to study the the properties of the finite ring Cl2/Zp,we first give the concept of the finite ring Cl2/Zp.Then,by the identity a2-2a0a+Ia=0,we obtain some necessary and sufficient conditions for the elements of the finite ring Cl2/Zp to be idempotent elements when p is an odd prime number,and the idempotent elements of the finite ring Cl2/Zp when P=2.Then,we obtain some necessary and sufficient conditions for the elements of the finite ring Cl2/Zp to be zero divisors and nilpotent elements respectively when P is a prime number.Finally,we obtain the ring isomorphism between Cl2/Zp and a ring of matrices of order 2 over Z_p.