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Lectures On Quadratic Jordan Algebras by Nathan Jacobson
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Lectures On Quadratic Jordan Algebras written by Nathan Jacobson
In these lectures we shall detailded and self-contained exposition of McCrimmon’s structure theiry including his recently developed theory of radicals and absolute zero divisors which constitute an important addition even to the classical linear theory. In our treatment we restrict attention to algebras with unit. This effects a sybstantial simplication. However, it should be noted that McCrimmon has also given an axiomatization for quadratic Jordan algebras withour unit and has developted the structure theory also for these. Perhaps the reader should be warned at the outsetthat he may find two (hopefully no more) parts of the exposition somewhat heavynamely, the derivation of the long list of identities in §1.3 and the proof of Osborn’s thorem on algebras of capacity two. The first of these could have been avoided by proving a general theorem in identities due to Macdonald. However, time did not permit this. The simplification of the proof of Osborn’s theorem remains an open problem. We shall see at the end of our exposition that this difficulty evaporates in the important special case of finite dimensional quadratic Jordan algebras over an algebraically closed field.
Book Detail :-
Title: Lectures On Quadratic Jordan Algebras
Author(s): Nathan Jacobson
Publisher: American Mathematical Society
Series: Tata Institute of Fundamental Research Lectures on Mathematics
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About Author :-
Author Nathan Jacobson (1910–1999) was an American mathematician. Nachman Arbiser in Warsaw, Jacobson emigrated to America with his family in 1918. Recognized as one of the leading algebraists of his generation, he wrote more than a dozen standard textbooks. He graduated from the University of Alabama in 1930 and was awarded a doctorate in mathematics from Princeton University in 1934. While working on his thesis, Non-commutative polynomials and cyclic algebras, he was advised by Joseph Wedderburn. Jacobson taught and researched at Bryn Mawr College (1935–1936), the University of Chicago (1936–1937), the University of North Carolina at Chapel Hill (1937–1943), and Johns Hopkins University (1943–1947) before joining Yale University in 1947. He remained at Yale until his retirement.
Nathan Jacobson is from Yale University, Department of Mathematics, New Haven Connecticut.
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Book Contents :-
Lectures On Quadratic Jordan Algebras written by Nathan Jacobson cover the following topics.
0 Artinian Semi-simple Rings with Involution
1. Determination of the semi-simple artinian rings
1. Special Jordan and quadratic Jordan algebras
2. Definition of Jordan and quadratic Jordan algebras
3. Basic identities
4. Category isomorphism for Q/2 Characteristic two case.
5. Inner and outer ideals. Difference algebras
6. Special universal envelopes
7. Quadratic Jordan algebras of quadratic forms
8. The exceptional quadratic Jordan algebra
9. Quadratic Jordan algebras defined by certian cubic forms
2-Pierce Decompositions. Standard Quadratic
1. Idempotents. Pierce decompositions
2. Standard quadratic Jordan matrix algebras
3. Connectedness and strong connectedness of
4. Strong coordinatization theorem
1. The radical of a quadratic Jordan algebra
2. Properties of the radical
3. Absolute zero divisors
4. Minimal inner ideals
5. Axioms for the structure theory
7. First structure theorem
8. A theorem on alternative algebras with involution
9. Simple quadratic Jordan algebras of capacity two
10. Second structure theorem
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