2000 Fiscal Year Final Research Report Summary
Representation theory of infinite-dimensional Lie algebras and superalgebras and its mathematical applications
| Project/Area Number |
10440009
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| Research Category |
Grant-in-Aid for Scientific Research (B).
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | KYUSHU UNIVERSITY |
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Principal Investigator |
WAKIMOTO Minoru Faculty of Mathematics, Prof., 大学院・数理学研究院, 教授 (00028218)
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| Co-Investigator(Kenkyū-buntansha) |
SATO Eiichi Faculty of Mathematics, Prof., 大学院・数理学研究院, 教授 (10112278)
TAGAWA Hiroyuki Wakayama Unive., Faculty of Education, Ass. Prof., 教育学部, 助教授 (80283943)
YAMADA Mieko Kanazawa Univ., Faculty of Science, Prof., 理学部, 教授 (70130226)
KAGEI Yoshiyuki Faculty of Mathematics, Ass. Prof., 大学院・数理学研究院, 助教授 (80243913)
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| Project Period (FY) |
1998 – 2000
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| Keywords | affine superalgebra / integrable representation / Appell's elliptic function / asymptotics / character formula / W-algebra / N=2 superconformal algebra / admissible representation |
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| Research Abstract |
Under this Grant-in-Aid, I made joint research with Professor Victor G.Kac, and obtained the following results :
1. "Integrable representations" for affine superalgebras are never easy concept, but should be treated carefully. We found that integrable representations consist of two kinds, namely principal-integrable representations and subprincipal-integrable representations, and gave the explicit and complete list of all highest weights for principal and subprincipal integrable modules.
2. We gave an explicit construction of fundamental sl(m|n)^- and osp(m|n)^-modules by using free bosons and free fermions. Using this explicit construction, we calculated the characters and obtained three kinds of character formulas --- Weyl-Kac type, theta-function type and quasi-particle type. From these character formulas, we found that the characters of fundamental sl(m|1)^-modules are Appell's elliptic functions which were discovered by Appell in 1880's but have been forgotten over one hundred years
… More
. These functions are not modular functions, but we succeeded to compute their asymptotics.
3. The trivial representation of an affine superalgebra sl(2|2)^ is a representation of critical level, since its dual Coxeter is equal to 0. So there was no known denominator identity for such superalgebras. We obtained explicitly the denominator formula for sl(2|2)^ by using Riemann's theta relations.
4. It is known by the theory of Frenkel-Kac-Wakimoto (1994) that the W-algebra of an usual affine Lie algebra and its representations are constructed in terms of the quantized Drinfeld-Sokolov reduction. But, for affine superalgebras, an immediate extension of this method fails to give a right W-algebra, and the construction of the W-algebra associated to affine superalgebras has long been a problem. We succeeded to resolve the difficulty by tensoring the factor, which arises from the algebraic variety, with the usual BRST-complex. The W-algebra of an affine superalgebra sl (2|1)^ obtained by this method is the direct sum of the centerless Virasoro algebra and the N=2 superconformal algebra. This theory enables us to make a detail investigation on representations of the N=2 superconformal algebra by means of admissible representations of sl(2|1)^. Actually we found that, other than the usual minimal series representations, there exist curious series of N=2 representations whose characters are half-modular functions. This research is now in progress very intensively. Less
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Research Products
(18 results)
