Catarina Santa-Clara's Algebras, Rings And Their Representations: Proceedings Of PDF

By Catarina Santa-Clara

ISBN-10: 9812565981

ISBN-13: 9789812565983

ISBN-10: 9812774556

ISBN-13: 9789812774552

Surveying the main influential advancements within the box, this complaints reports the newest examine on algebras and their representations, commutative and non-commutative earrings, modules, conformal algebras, and torsion theories. the amount collects stimulating discussions from world-renowned names together with Tsit-Yuen Lam, Larry Levy, Barbara Osofsky, and Patrick Smith.

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Extra info for Algebras, Rings And Their Representations: Proceedings Of The International Conference on Algebras, Modules and Rings, Lisbon, Portugal, 14-18 July 2003

Example text

Grsemiprime, with zero annihilator). 2. 1. Let K be a field of characteristic not two and let T be a vector space over K. We say that T is a triple system if it is endowed with a trilinear map (•,-,•) : T x T x T - > T , called the triple product of T. A triple system T is called a Lie triple system if its triple product, denoted by [•,-,•], satisfies (1) [x,x,y]=0 (2) [x, y, z] + [y, z, x] + [z, x, y] = 0 (Jacobi identity) (3) [x,y,[a,b,c]]- [a,b,[x,y,c]} = [[x,y,a],b,c] + [a,[x,y,b],c] for any x, y, z, a,b,c€ T.

4) e = —1, 6' = — 1. In this case (x,y,z) = zxy. We observe t h a t any of the above four cases is a p e r m u t a t i o n of T with the triple product (x, y, z) = xyz and then we complete assertion 1 of the theorem. Assertion 2 is an easy consequence of the following well known facts. T h e isometries of Ht are of the form x H-> va(x) with \v\ = 1 and a being either an automorphism or an antiautomorphism of the algebra H, and all these automorphisms can be written a s m qxq~l where q e S3 and the antiautomorphism a s m qxq^1 with q as above and where — is the conjugation of EL • Acknowledgment T h e authors are grateful to the referee for his valuable suggestions.

5) [Jo,T]c[[T,/],T]c[T,/,T]c/. (6) We also have By the Jacobi identity and (5) [[Jo, T], /] c [[T, / ] , Jo] + [[/, Jo], T] c [[T, / ] , Jo] C Jo- (7) By applying (7) and (5) we get [[Jo,T],/,/] = [[[J 0 ,T],/],/] c [Jo,/] = 0. (8) By (6) and (8), [Jo,T] is an ideal of / . By applying again (6) and (8), [[Jo,T],/,[J o ,T]] = 0. 7. We then have J = J\. Thus, [J,/] C Jo = 0 and so [J,I,I] = 0, hence [J, / , J] = 0. By applying again the semiprimeness of / , we obtain J = • 0. By arguing as in [2], we can state the following.

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Algebras, Rings And Their Representations: Proceedings Of The International Conference on Algebras, Modules and Rings, Lisbon, Portugal, 14-18 July 2003 by Catarina Santa-Clara


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