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Strong Macdonadl theorems | Strong Macdonadl theorems | ||
| − | * conjectured by Hanlon 86,90 | + | * conjectured by Hanlon 86,90 |
** The cohomology algebra H'(L[z]/z^n) is a free super-commutative algebra with generators i=1,2,\cdots, l | ** The cohomology algebra H'(L[z]/z^n) is a free super-commutative algebra with generators i=1,2,\cdots, l | ||
** N generators of cohomology degree 2m_i+1, one with z-deg 0 | ** N generators of cohomology degree 2m_i+1, one with z-deg 0 | ||
| − | * Feigin 91 | + | * Feigin 91 |
** H^0_{res}(L[z,s]) with s odd | ** H^0_{res}(L[z,s]) with s odd | ||
** L[z,s]^{*}=\oplus (z^is^jL)^{*} | ** L[z,s]^{*}=\oplus (z^is^jL)^{*} | ||
** L[z]\oplus sL[z] | ** L[z]\oplus sL[z] | ||
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==strong Macdonald theorems== | ==strong Macdonald theorems== | ||
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Let L be a reductive Lie algebra. The strong Macdonald theorems of Fishel, Grojnowski, and Teleman state that the cohomology algebras of L[z]/zN and L[z,s] (where s is an odd variable) are free skew-commutative algebras with generators in certain degrees. The theorems were originally conjectured by Hanlon and Feigin as Lie algebra cohomology extensions of Macdonald's constant term conjecture for an untwisted affine root system. I will explain how to get strong Macdonald theorems for the fixed-point subalgebras of L[z]/zN and L[z,s] under a diagram automorphism twist. These twisted strong Macdonald theorems are Lie algebra cohomology versions of Macdonald's constant term conjecture for twisted affine root systems. I will also explain how to calculate the cohomology algebra of p/zNp and p[s] when p is a parahoric in a twisted loop algebra. In this case the cohomology of p/zNp is not necessarily free, as it contains a factor which is isomorphic to the cohomology algebra of flag variety for the corresponding parabolic. | Let L be a reductive Lie algebra. The strong Macdonald theorems of Fishel, Grojnowski, and Teleman state that the cohomology algebras of L[z]/zN and L[z,s] (where s is an odd variable) are free skew-commutative algebras with generators in certain degrees. The theorems were originally conjectured by Hanlon and Feigin as Lie algebra cohomology extensions of Macdonald's constant term conjecture for an untwisted affine root system. I will explain how to get strong Macdonald theorems for the fixed-point subalgebras of L[z]/zN and L[z,s] under a diagram automorphism twist. These twisted strong Macdonald theorems are Lie algebra cohomology versions of Macdonald's constant term conjecture for twisted affine root systems. I will also explain how to calculate the cohomology algebra of p/zNp and p[s] when p is a parahoric in a twisted loop algebra. In this case the cohomology of p/zNp is not necessarily free, as it contains a factor which is isomorphic to the cohomology algebra of flag variety for the corresponding parabolic. | ||
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==related items== | ==related items== | ||
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* [[Lie algebra cohomology]] | * [[Lie algebra cohomology]] | ||
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==articles== | ==articles== | ||
| + | * Fishel, Susanna, Ian Grojnowski, and Constantin Teleman. “The Strong Macdonald Conjecture and Hodge Theory on the Loop Grassmannian.” arXiv:math/0411355, November 16, 2004. http://arxiv.org/abs/math/0411355. | ||
| + | * Fishel, S., I. Grojnowski, and C. Teleman. “The Strong Macdonald Conjecture.” arXiv:math/0107072, July 10, 2001. http://arxiv.org/abs/math/0107072. | ||
| + | * Hanlon, Phil. "Some conjectures and results concerning the homology of nilpotent Lie algebras." Advances in Mathematics 84.1 (1990): 91-134. doi:[http://dx.doi.org/10.1016/0001-8708%2890%2990037-N 10.1016/0001-8708(90)90037-N]. | ||
| + | * '''[Hanlon1986]''' Hanlon, Phil. “Cyclic Homology and the Macdonald Conjectures.” Inventiones Mathematicae 86, no. 1 (February 1986): 131–59. doi:[http://dx.doi.org/10.1007/BF01391498 10.1007/BF01391498]. | ||
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[[분류:개인노트]] | [[분류:개인노트]] | ||
[[분류:math and physics]] | [[분류:math and physics]] | ||
[[분류:math]] | [[분류:math]] | ||
| + | [[분류:migrate]] | ||
2020년 12월 28일 (월) 05:17 기준 최신판
introduction
Strong Macdonadl theorems
- conjectured by Hanlon 86,90
- The cohomology algebra H'(L[z]/z^n) is a free super-commutative algebra with generators i=1,2,\cdots, l
- N generators of cohomology degree 2m_i+1, one with z-deg 0
- Feigin 91
- H^0_{res}(L[z,s]) with s odd
- L[z,s]^{*}=\oplus (z^is^jL)^{*}
- L[z]\oplus sL[z]
strong Macdonald theorems
Let L be a reductive Lie algebra. The strong Macdonald theorems of Fishel, Grojnowski, and Teleman state that the cohomology algebras of L[z]/zN and L[z,s] (where s is an odd variable) are free skew-commutative algebras with generators in certain degrees. The theorems were originally conjectured by Hanlon and Feigin as Lie algebra cohomology extensions of Macdonald's constant term conjecture for an untwisted affine root system. I will explain how to get strong Macdonald theorems for the fixed-point subalgebras of L[z]/zN and L[z,s] under a diagram automorphism twist. These twisted strong Macdonald theorems are Lie algebra cohomology versions of Macdonald's constant term conjecture for twisted affine root systems. I will also explain how to calculate the cohomology algebra of p/zNp and p[s] when p is a parahoric in a twisted loop algebra. In this case the cohomology of p/zNp is not necessarily free, as it contains a factor which is isomorphic to the cohomology algebra of flag variety for the corresponding parabolic.
articles
- Fishel, Susanna, Ian Grojnowski, and Constantin Teleman. “The Strong Macdonald Conjecture and Hodge Theory on the Loop Grassmannian.” arXiv:math/0411355, November 16, 2004. http://arxiv.org/abs/math/0411355.
- Fishel, S., I. Grojnowski, and C. Teleman. “The Strong Macdonald Conjecture.” arXiv:math/0107072, July 10, 2001. http://arxiv.org/abs/math/0107072.
- Hanlon, Phil. "Some conjectures and results concerning the homology of nilpotent Lie algebras." Advances in Mathematics 84.1 (1990): 91-134. doi:10.1016/0001-8708(90)90037-N.
- [Hanlon1986] Hanlon, Phil. “Cyclic Homology and the Macdonald Conjectures.” Inventiones Mathematicae 86, no. 1 (February 1986): 131–59. doi:10.1007/BF01391498.