Download e-book for iPad: Automorphic Forms and Lie Superalgebras (Algebra and by Urmie Ray

By Urmie Ray

ISBN-10: 1402050097

ISBN-13: 9781402050091

ISBN-10: 1402050100

ISBN-13: 9781402050107

This publication presents the reader with the instruments to appreciate the continued category and development venture of Lie superalgebras. It offers the cloth in as uncomplicated phrases as attainable. assurance particularly information Borcherds-Kac-Moody superalgebras. The booklet examines the hyperlink among the above category of Lie superalgebras and automorphic shape and explains their development from lattice vertex algebras. it's also all priceless history info.

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Additional resources for Automorphic Forms and Lie Superalgebras (Algebra and Applications)

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0 0     0 −1 2 . . 0 0   . , .. ..  ..  .. . .     0 0 0 . . 2 −1  0 0 0 . . −1 1   0 2 0 −1 . . 0 0   0 2 −1 . . 0   0   −1 −1 2 . . 0 , ..  .. ..  .. .  . .   .   −2 0 0 0 ... 2 0 0 0 ..     ,   −1  0 0 0 . . −1 0    0 −1 0 0 0 −1 0  −1 2 −2 0   α > 1,  , −1 2 −3 , 0 −2 4 −2 0 −3 6 0 0 −2 4 where in all cases i ∈ S if and only if aii = 0 or 1. The converse is left as an exercise for the reader. 8) imply that the only finite dimensional simple Kac-Moody Lie superalgebras with non-trivial odd part are of type B(m, 1) whereas all finite dimensional semisimple Lie algebras are Kac-Moody Lie algebras.

Is the sum of the above three Lie sub-superalgebras. To Hint: Show first that G show that the sum is direct, construct for each λ ∈ H ∗ (the dual of H), an ˜ on the tensor algebra T (V ) of a vector space with a basis indexed action of G ˜− , by I and consider the action of 0 = f + h + e on 1 ∈ T (V ), where f ∈ N ˜ e ∈ N+ , and [e, f ] = h. ˜+ (resp. N ˜− ) is freely generated (ii) Deduce that the Lie sub-superalgebra N by the vectors ei (resp. fi ), i ∈ I. 28 are well defined. 2] or [Borc6]. 6.

Suppose that R = 0. For all i, j, [ei , fj ] = 0 if i = j and [h, ei ] = (h, hi )0 ei , [h, fi ] = −(h, hi )0 fi . 30 2 Borcherds-Kac-Moody Lie Superalgebras Proof. Let i = j. We first show that [ei , fj ] = 0. Without loss of generality, we may assume that ei ∈ Ls , fj ∈ L−t , s ≥ t > 0. 12, ei ∈ Vs . So for all x ∈ Ls−t , ([ei , fj ], x)0 = −(ei , [ej , x])0 . (1) Since 0 ≤ s − t < s, x ∈ Ms . We show that ([ei , fj ], x)0 = 0 (2) for all x ∈ Ls−t . If t < s, then fj ∈ Ms and so [ej , x] ∈ Ms .

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Automorphic Forms and Lie Superalgebras (Algebra and Applications) by Urmie Ray


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