Page 148 - Textos de Matemática Vol. 39

138 ANA PAULA SANTANA AND HELENA ALBUQUERQUE
Proof. Given X = [xij] ∈ Mn,φ(K), we have:
IX=(n φ(q−1,q,q−1)E )(n x E )=
φn q=1 qq m,p=1 mp mp q,p=1φ(q−1,q,q−1)xqpEqqEqp =
n φ(q−1,q,q−1)φ(q,q−1,qp−1)x E . q,p=1 φ(q−1 ,q,p−1 ) qp qp
But by the definition of the cocycle φ,
φ(q−1, q, q−1)φ(q, q−1, q) = 1 and
φ(q, q−1, q)φ(q−1, q, p−1) = φ(q, q−1, qp−1).
So we conclude that IφX = X. In a similar way we can prove that
XIφ =X. We will now define the deformed elementary matrices Pijφ and Eijφ(α),
for α ∈ K, i,j = 1,...,n, i ̸= j, as follows:
(1) Pijφ is the matrix obtained from Iφ by interchanging the ith and jth rows. That is,
Lemma 2.3. The following holds for any X = [xij ] ∈ Mn,φ(K) :
(1) PijφX is obtained from X by interchanging the deformed ith and jth
n q=1,q̸=i,j
φ(q−1,q,q−1)Eqq +φ(j−1,j,j−1)Eij +φ(i−1,i,i−1)Eji. (2) Eijφ(α)=Iφ +αEij.
Pijφ =
rows: the (im) entry of PijφX is the (jm) entry of X multiplied by φ(i,j−1,jm−1), and the (jm) entry of P X is the (im) entry of X
φ(j,j−1 ,jm−1 ) ijφ multiplied by φ(j,i−1,im−1).
φ(i,i−1 ,im−1 )
(2) Eijφ(α)X is obtained from X by adding α times the deformed jth row of
X totheithrow:the(im)entryofE (α)X isx +αx φ(i,j−1,jm−1). ijφ im jm φ(j−1,j,m−1)
Proof.
+
(1) PijφX=(n φ(q−1,q,q−1)Eqq +φ(j−1,j,j−1)Eij+
n m=1
q,m=1,q̸=i,j φ(j−1,j,j−1)xjmEijEjm +
n m=1
φ(i−1,i,i−1)ximEjiEim.
q=1,q̸=i,j n
φ(i−1, i, i−1)Eji)( xp,mEpm) =
p,m=1
n
φ(q−1, q, q−1)xqmEqqEqm+