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And a unique morphism from (Y, X, ( , ) , . . ) to the adjoint root datum of type (/,-). If (Y, X, ( , ) , . . ) and (Y', X', ( , ) ' , . . ) are two root data of type (/, •), we can define a third root datum ( Y e Y', X © X ' , ( , ) " , . . ) where (,)" = (,) © (,)', the imbedding I —> Y © Y' has as components the given imbeddings I —> Y, I —*• Y' and the imbedding I —> X © X' has as first component the given imbedding I —> X and as second component zero. ) is Y-regular, then (Y © Y', X © X', ( , ) " , .

By the assumption of (c), we have that (a) holds for z = x® F^y. By the earlier part of the proof, it follows that (a) holds for z\ = [q + \}%x F^q+1^y + v*~2qFiXF^y. Again by the assumption of (c), we have that (a) holds for It follows that (a) holds for z3 = [q + l]iX®F^+1)y; = v*~2qFiX®Flq)y. since [q + l]i ^ 0, it also holds for Thus (c) is proved. It remains to prove (b). Let x G Mm,y G Nn be such that Eiy = 0. From the definition, we have L"(x y) = x y. We must prove that T['x(x y) = T['x(x) T^iy).

For any A G X, the operator T"_e : M —> M defines an isomorphism of the X-weight space of M onto the Si(X)-weight space of M. The inverse of this isomorphism is the restriction ofT'i e. This follows immediately from the previous lemma. 5 . 3 . T H E OPERATORS Let M,N be two objects of C[. 1. Ei(x <8>y) = EiX®y + v\x ® E{y, F{(x ®y)=x®Fiy We define linear maps L", L\ : M N -> M N by + v^8FiX y. 46 5. 4. Although the sums are infinite, any vector in M N is annihilated by all but finitely many terms in the sum; hence the operators are well defined.

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Abstract Algebra - A Comprehensive Trtmt by C. Menini, F. van Oystaeyen

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