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

Acta Acad. Paed.. Agriensis, Sectio Mathematicae 28 (2001) 87-113 THE CONDITION FOR GENERALIZING INVERTIBLE SUBSPACES IN CLIFFORD ALGEBRAS Nguyen Canh Luong (Hanoi, Vietnam) Abstract. Let A be a universal Clifford algebra induced by m-dimensional real linear space with basis {ei ,e2...,e

m}

. The necessary and sufficient condition for the subspaces of form Li =h'n{e

0,ei

,. ..,e,„ ,e

m+\

, • • • ,Cm+s } to be invertible is m=

2 (mod 4), s = l and e

m+

1=ei2...

m (see [2]). In this paper we improve this assertion for the subspaces of the form L—lin{eo,eA

1 ,--,

eA

m >

eA

?n + 1 f->

eA

m_^

s }, where A,C{l,2,...,m} (i = l,2,...,m + s). 1. Introduction Let V

m be ail m—dimensional (m > 1) real linear space with basis {ei

, e

2,..., e

m). Consider the 2

m— dimensional real space A with basis where e/

t-j := e

t- (i — 1,2,..., in). In the following, for each K = {ki, k

2,..., k

t] C {1, 2,..., m) we write ex = ejfcijfc

a...jfc( with e^ = e

0, and so E — {e

0, e{!)

,..., e{

m}

, e{i,2}

, • • •, e{m-i,m}

, • • •, e{i,2,..,m}

} E — {eo, ei,

. . ., e

m, e

1

2,..., e

m_

i

m,. . ., e

1

2...,„.}• The product of two elements e

A, es G E is given by (1) = (-l)»

(j4nß

)(-l)

p(j

4'

ß

)^A

S; A,B C {1, 2,. . ., m}, where jeB p{A,j) = tt{i eA:i> ]} A A B = (A\B

) U (B\A) and fjA denotes the number of elements of A.