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

88 Nguyen Canh Luong Each element a = c

lA^A £ A is called a Clifford number. The product of A two Clifford numbers a = aA^A\ b — bßeß is defined by the formula A B ab = S QAbBtACB.4 B It is easy to check that in this way A is turned into a linear associative noncommutative algebra over R. It is called the Clifford algebra over V

m. It follows at once from the multiplication rule (1) that e® is identity element, which is denoted by eo and in particular aej + ejei = 0 for i ^ j; ej = -1 (i, j = 1,2,..., m) and e>k

xk

2...k

t = e

fc

le

fc

2 .. . e

fc

(; 1 < ki < k

2 < ... < fc* < m. The involution for basic vectors is given by ek

1k

2...k

i — 1) 2 £k

1k

2...kf For any a = Yl

aA

eA £ we write ä = ^a^e^. For any Clifford number a = A A Y^ a

Ae

A, we write |a| = (]C aÍ) 3 • A A 2. Result and Proof We use the following definitions. (i) An element a £ is said to be invertible if there exists an element a-1 such that aa~

l = a1a = eo; a.1 is said to be the inverse of a. (ii) A subspace X C A is said to be invertible if every non-zero element in A' is invertible in A. (iii

) L(ui

, U2, • • ., u

n) = lin{

ui, t/2,. . . u

n } , ui £ A (z = 1,2,..., n). It is well-known (see [1]) that for any special Clifford number of the form m a = ai

ei / 0 we have a1 = A. So L(eo, ei

, ..., e

m) is invertible, and if m = 2 t=o |a

|" m+l (mod 4) (see [2]), then every a — ^ c^e; ^ 0, where e

m+

i — ei2...

m is invertible i- 0 and a1 = -j

——. So L(eo, ei, . .., e

m, e

m

+i) is invertible.