Pub Date : 2010-02-16DOI: 10.1215/21562261-1214384
B. Feigin, E. Feigin, M. Jimbo, T. Miwa, E. Mukhin
We construct a family of irreducible representations of the quantum continuous $gl_infty$ whose characters coincide with the characters of representations in the minimal models of the $W_n$ algebras of $gl_n$ type. In particular, we obtain a simple combinatorial model for all representations of the $W_n$-algebras appearing in the minimal models in terms of $n$ interrelating partitions.
{"title":"Quantum continuous $mathfrak{gl}_{infty}$: Tensor products of Fock modules and $mathcal{W}_{n}$-characters","authors":"B. Feigin, E. Feigin, M. Jimbo, T. Miwa, E. Mukhin","doi":"10.1215/21562261-1214384","DOIUrl":"https://doi.org/10.1215/21562261-1214384","url":null,"abstract":"We construct a family of irreducible representations of the quantum continuous $gl_infty$ whose characters coincide with the characters of representations in the minimal models of the $W_n$ algebras of $gl_n$ type. In particular, we obtain a simple combinatorial model for all representations of the $W_n$-algebras appearing in the minimal models in terms of $n$ interrelating partitions.","PeriodicalId":50142,"journal":{"name":"Journal of Mathematics of Kyoto University","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2010-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1215/21562261-1214384","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66024837","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We define a notion of Gorenstein flat dimension for unbounded complexes over left GF-closed rings. Over Gorenstein rings we introduce a notion of Gorenstein coho- mology for complexes; we also define a generalized Tate cohomol- ogy for complexes over Gorenstein rings, and we show that there is a close connection between the absolute, the Gorenstein and the generalized Tate cohomology.
{"title":"Gorenstein flat dimension of complexes","authors":"A. Iacob","doi":"10.1215/KJM/1265899484","DOIUrl":"https://doi.org/10.1215/KJM/1265899484","url":null,"abstract":"We define a notion of Gorenstein flat dimension for unbounded complexes over left GF-closed rings. Over Gorenstein rings we introduce a notion of Gorenstein coho- mology for complexes; we also define a generalized Tate cohomol- ogy for complexes over Gorenstein rings, and we show that there is a close connection between the absolute, the Gorenstein and the generalized Tate cohomology.","PeriodicalId":50142,"journal":{"name":"Journal of Mathematics of Kyoto University","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2010-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1215/KJM/1265899484","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66113826","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that the Klein cubic threefold $F$ is the only smooth cubic threefold which has an automorphism of order $11$. We compute the period lattice of the intermediate Jacobian of $F$ and study its Fano surface $S$. We compute also the set of fibrations of $S$ onto a curve of positive genus and the intersection between the fibres of these fibrations. These fibres generate an index $2$ sub-group of the Neron-Severi group and we obtain a set of generators of this group. The Neron-Severi group of $S$ has rank $25=h^{1,1}$ and discriminant $11^{10}$.
{"title":"The Fano surface of the Klein cubic threefold","authors":"X. Roulleau","doi":"10.1215/KJM/1248983032","DOIUrl":"https://doi.org/10.1215/KJM/1248983032","url":null,"abstract":"We prove that the Klein cubic threefold $F$ is the only smooth cubic threefold which has an automorphism of order $11$. We compute the period lattice of the intermediate Jacobian of $F$ and study its Fano surface $S$. We compute also the set of fibrations of $S$ onto a curve of positive genus and the intersection between the fibres of these fibrations. These fibres generate an index $2$ sub-group of the Neron-Severi group and we obtain a set of generators of this group. The Neron-Severi group of $S$ has rank $25=h^{1,1}$ and discriminant $11^{10}$.","PeriodicalId":50142,"journal":{"name":"Journal of Mathematics of Kyoto University","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2010-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66087227","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2010-01-01DOI: 10.1215/0023608X-2009-003
B. Hayati, M. Amini
Let B be a Banach algebra with bounded approximate identity, and let M(B) be its multiplier algebra. If there exists a continuous linear injection B∗ → M(B) such that, for every b ∈ B and every u, v ∈ B∗, 〈u, vb〉B = 〈v, bu〉B , then M(B) is a dual Banach algebra and the following are equivalent: (i) B is amenable; (ii) M(B) is Connes amenable; (iii) M(B) has a normal, virtual diagonal.
设B是一个有界近似恒等式的巴拿赫代数,M(B)是它的乘数代数。如果存在一个连续线性注入B∗→M(B),使得对于每个B∈B和每个u, v∈B∗,< u, vb > B = < v,但> B,则M(B)是对偶Banach代数,并且下列是等价的:(ii) M(B)是否符合Connes的规定;(iii) M(B)有一条法向虚对角线。
{"title":"Connes-amenability of multiplier Banach algebras","authors":"B. Hayati, M. Amini","doi":"10.1215/0023608X-2009-003","DOIUrl":"https://doi.org/10.1215/0023608X-2009-003","url":null,"abstract":"Let B be a Banach algebra with bounded approximate identity, and let M(B) be its multiplier algebra. If there exists a continuous linear injection B∗ → M(B) such that, for every b ∈ B and every u, v ∈ B∗, 〈u, vb〉B = 〈v, bu〉B , then M(B) is a dual Banach algebra and the following are equivalent: (i) B is amenable; (ii) M(B) is Connes amenable; (iii) M(B) has a normal, virtual diagonal.","PeriodicalId":50142,"journal":{"name":"Journal of Mathematics of Kyoto University","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2010-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1215/0023608X-2009-003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66040264","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2010-01-01DOI: 10.1215/0023608X-2009-019
B. Belaïdi, A. Farissi
In this article, we discuss the growth of solutions of the second-order nonhomogeneous linear differential equation where a, b are complex constants and A j ( z ) (cid:2)≡ 0 ( j = 0 , 1) , and F (cid:2)≡ 0 are entire functions such that max { ρ ( A j ) ( j = 0 , 1) ,ρ ( F ) } < 1 . We also investigate the relationship between small functions and differential polynomials g f ( z ) = d 2 f (cid:2)(cid:2) + d 1 f (cid:2) + d 0 f , where d 0 ( z ) ,d 1 ( z ) ,d 2 ( z ) are entire functions that are not all equal to zero with ρ ( d j ) < 1 ( j = 0 , 1 , 2) generated by solutions of the above equation.
在本文中,我们讨论了二阶非齐次线性微分方程解的增长,其中a, b是复常数,且a j (z) (cid:2)≡0 (j = 0,1), F (cid:2)≡0是使得max {ρ (a j) (j = 0,1),ρ (F)} < 1的整函数。我们也调查之间的关系小函数和微分多项式g f d (z) = 2 f (cid: 2) (cid: 2) + d 1 f f (cid: 2) + d 0, 0 d (z), 2 d (z), d (z)是整个函数并非都是等于零,ρ(d j) < 1 (j = 0, 1, 2)由上述方程的解决方案。
{"title":"Relation between differential polynomials and small functions","authors":"B. Belaïdi, A. Farissi","doi":"10.1215/0023608X-2009-019","DOIUrl":"https://doi.org/10.1215/0023608X-2009-019","url":null,"abstract":"In this article, we discuss the growth of solutions of the second-order nonhomogeneous linear differential equation where a, b are complex constants and A j ( z ) (cid:2)≡ 0 ( j = 0 , 1) , and F (cid:2)≡ 0 are entire functions such that max { ρ ( A j ) ( j = 0 , 1) ,ρ ( F ) } < 1 . We also investigate the relationship between small functions and differential polynomials g f ( z ) = d 2 f (cid:2)(cid:2) + d 1 f (cid:2) + d 0 f , where d 0 ( z ) ,d 1 ( z ) ,d 2 ( z ) are entire functions that are not all equal to zero with ρ ( d j ) < 1 ( j = 0 , 1 , 2) generated by solutions of the above equation.","PeriodicalId":50142,"journal":{"name":"Journal of Mathematics of Kyoto University","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2010-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1215/0023608X-2009-019","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66040583","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give a complete description of the integral cohomology ring of the flag manifold E 8 /T, where E 8 denotes the compact exceptional Lie group of rank 8 and T its maximal torus, by the method due to Borel and Toda. This completes the computation of the integral cohomology rings of the flag manifolds for all compact connected simple Lie groups.
{"title":"The integral cohomology ring of $E_7/T$","authors":"Masaki Nakagawa","doi":"10.3792/pjaa.86.64","DOIUrl":"https://doi.org/10.3792/pjaa.86.64","url":null,"abstract":"We give a complete description of the integral cohomology ring of the flag manifold E 8 /T, where E 8 denotes the compact exceptional Lie group of rank 8 and T its maximal torus, by the method due to Borel and Toda. This completes the computation of the integral cohomology rings of the flag manifolds for all compact connected simple Lie groups.","PeriodicalId":50142,"journal":{"name":"Journal of Mathematics of Kyoto University","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2009-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.3792/pjaa.86.64","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70207733","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On Samelson products in Sp(n)","authors":"Tomoaki Nagao","doi":"10.1215/KJM/1248983038","DOIUrl":"https://doi.org/10.1215/KJM/1248983038","url":null,"abstract":"","PeriodicalId":50142,"journal":{"name":"Journal of Mathematics of Kyoto University","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2009-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66087731","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2009-06-10DOI: 10.1215/0023608X-2009-001
Masaki Izumi, R. Srinivasan
Toeplitz CAR flows are a class of E_0-semigroups including the first type III example constructed by R. T. Powers. We show that the Toeplitz CAR flows contain uncountably many mutually non cocycle conjugate E_0-semigroups of type III. We also generalize the type III criterion for Toeplitz CAR flows employed by Powers (and later refined by W. Arveson), and show that Toeplitz CAR flows are always either of type I or type III.
Toeplitz CAR流是一类e_0 -半群,包括R. T. Powers构造的第一个III型例子。我们证明了Toeplitz CAR流包含不可数的互非循环共轭e_0半群。我们还推广了power使用的Toeplitz CAR流的III型准则(后来由W. Arveson改进),并表明Toeplitz CAR流总是I型或III型。
{"title":"Toeplitz CAR flows and type I factorizations","authors":"Masaki Izumi, R. Srinivasan","doi":"10.1215/0023608X-2009-001","DOIUrl":"https://doi.org/10.1215/0023608X-2009-001","url":null,"abstract":"Toeplitz CAR flows are a class of E_0-semigroups including the first type III example constructed by R. T. Powers. We show that the Toeplitz CAR flows contain uncountably many mutually non cocycle conjugate E_0-semigroups of type III. We also generalize the type III criterion for Toeplitz CAR flows employed by Powers (and later refined by W. Arveson), and show that Toeplitz CAR flows are always either of type I or type III.","PeriodicalId":50142,"journal":{"name":"Journal of Mathematics of Kyoto University","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2009-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1215/0023608X-2009-001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66040714","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2009-04-10DOI: 10.1215/21562261-1424875
B. Feigin, A. Tsymbaliuk
In this paper we construct the action of Ding-Iohara and shuffle algebras in the sum of localized equivariant K-groups of Hilbert schemes of points on C^2. We show that commutative elements K_i of shuffle algebra act through vertex operators over positive part {h_i}_{i>0} of the Heisenberg algebra in these K-groups. Hence we get the action of Heisenberg algebra itself. Finally, we normalize the basis of the structure sheaves of fixed points in such a way that it corresponds to the basis of Macdonald polynomials in the Fock space k[h_1,h_2,...].
{"title":"Equivariant K-theory of Hilbert schemes via shuffle algebra","authors":"B. Feigin, A. Tsymbaliuk","doi":"10.1215/21562261-1424875","DOIUrl":"https://doi.org/10.1215/21562261-1424875","url":null,"abstract":"In this paper we construct the action of Ding-Iohara and shuffle algebras in the sum of localized equivariant K-groups of Hilbert schemes of points on C^2. We show that commutative elements K_i of shuffle algebra act through vertex operators over positive part {h_i}_{i>0} of the Heisenberg algebra in these K-groups. Hence we get the action of Heisenberg algebra itself. Finally, we normalize the basis of the structure sheaves of fixed points in such a way that it corresponds to the basis of Macdonald polynomials in the Fock space k[h_1,h_2,...].","PeriodicalId":50142,"journal":{"name":"Journal of Mathematics of Kyoto University","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2009-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1215/21562261-1424875","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66024579","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}