Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2001. Sectio Mathematicae (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 28)
NGUYEN CANH LUONG, The condition for generalizing invertible subspaces in Clifford algebras
The condition for generalizing invertible snbspaces in Clifford algebras 91 e A t + e A i = 0, i E {1, 2,..., ?77 ). Hence either fl^ = 4p t- + 1 or tjA l = 4p,- + 2 ( P i E IN), i G {l,2,...,m}. So . = -e 0 (i = 1, 2,..., m). Let- m = 0 (mod 4) or rn = 3 (mod 4). Choosing a = e 0 + and b = eo — eA m_i w c find at = e 0 + e^ m+ 1 - ex m+ 1 - e J4 m+ 1e Am+ 1 = e 0 - e^ .. . e^.e^ .. .e A m = e 0 - [(-l) m(-l) 2L i^ i leo] = eo - (-1) í1 íT ± Üco = e 0 - e 0 = 0. Hence the non-zero numbers a and b are not invertible. Let m = 1 (mod 4). Choosing a = e A l + e Am+ 1 and b = e A l — e Am+ 1 we get ab = (e A l + e Am+ 1 )(e A } - e Am+ 1 ) = e A le A l - e A le Am+ 1 + e Am+ 1e A l - e Am+ 1e Am+ 1 m(m+l) =-e 0 - e A le A le A 2 .. ,e A m + e A le A 2 .. .e A me A l - (-1) 2 e 0 = . .. + (-l ) m~ 1e A le A le A 2 .. ,e A m = e A 2 .. .e^ - .. .e j4 m = 0. Hence a and b are not invertible. So rn = 2 (mod 4). The theorem is proved. References [1] F. BRACKX, R. DELANGHE AND F. SOMMEN, Clifford. Analysis , Pitman Advanced Publishing Program, Boston-London-Melbourne, 1982. [2] N. C. LUONG AND N. V. MAU, On Invertibility of Linear Subspaces Generating Clifford Algebras, Vietnam Journal of Mathematics, 25:2(1997) 133-140. [3] NGUYEN CANH LUONG, Remark on the Maximal Dimension of Invertible Subspaces in the Clifford Algebras, Proceeding of the Fifth Vietnamese Conference, 145-150. Nguyen Canh Luong Department of Mathematics University of Technology of Hanoi Hanoi, F105-Nha C14, 1 Dai Co Viet Vietnam e-mail: ncluong@hn.vnn.vn
