{"title":"Eigenfunction of the Laplacian as a Degenerate Case of a Function with its Fourier Transform Supported in an Annulus","authors":"Rudra P. Sarkar","doi":"10.4171/PRIMS/54-2-4","DOIUrl":"https://doi.org/10.4171/PRIMS/54-2-4","url":null,"abstract":"","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2018-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/PRIMS/54-2-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46045208","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Core of an Extended Affine Lie Superalgebra (A Characterization)","authors":"Rasul Aramian, M. Yousofzadeh","doi":"10.4171/PRIMS/54-2-1","DOIUrl":"https://doi.org/10.4171/PRIMS/54-2-1","url":null,"abstract":"","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2018-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/PRIMS/54-2-1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49389336","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We compute the Bass stable rank of the ring $Gamma(X,mathcal O_X)$ of global sections of the structure sheaf $mathcal O_X$ on a finite-dimensional Stein space $(X,mathcal O_X)$ and then apply this result to the problem of the factorization of invertible holomorphic matrices on $X$.
{"title":"On the Bass Stable Rank of Stein Algebras","authors":"A. Brudnyi","doi":"10.4171/PRIMS/*","DOIUrl":"https://doi.org/10.4171/PRIMS/*","url":null,"abstract":"We compute the Bass stable rank of the ring $Gamma(X,mathcal O_X)$ of global sections of the structure sheaf $mathcal O_X$ on a finite-dimensional Stein space $(X,mathcal O_X)$ and then apply this result to the problem of the factorization of invertible holomorphic matrices on $X$.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2018-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45138089","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $Ssubset mathbb{C}^n$ be a non-singular algebraic set and $f colon mathbb{C}^n to mathbb{C}$ be a polynomial function. It is well-known that the restriction $f|_S colon S to mathbb{C}$ of $f$ on $S$ is a locally trivial fibration outside a finite set $B(f|_S) subset mathbb{C}.$ In this paper, we give an explicit description of a finite set $T_infty(f|_S) subset mathbb{C}$ such that $B(f|_S) subset K_0(f|_S) cup T_infty(f|_S),$ where $K_0(f|_S)$ denotes the set of critical values of the $f|_S.$ Furthermore, $T_infty(f|_S)$ is contained in the set of critical values of certain polynomial functions provided that the $f|_S$ is Newton non-degenerate at infinity. Using these facts, we show that if ${f_t}_{t in [0, 1]}$ is a family of polynomials such that the Newton polyhedron at infinity of $f_t$ is independent of $t$ and the $f_t|_S$ is Newton non-degenerate at infinity, then the global monodromies of the $f_t|_S$ are all isomorphic.
{"title":"Bifurcation Sets and Global Monodromies of Newton Nondegenerate Polynomials on Algebraic Sets","authors":"T. Nguyen, P. Pham, T. Pham","doi":"10.4171/prims/55-4-6","DOIUrl":"https://doi.org/10.4171/prims/55-4-6","url":null,"abstract":"Let $Ssubset mathbb{C}^n$ be a non-singular algebraic set and $f colon mathbb{C}^n to mathbb{C}$ be a polynomial function. It is well-known that the restriction $f|_S colon S to mathbb{C}$ of $f$ on $S$ is a locally trivial fibration outside a finite set $B(f|_S) subset mathbb{C}.$ In this paper, we give an explicit description of a finite set $T_infty(f|_S) subset mathbb{C}$ such that $B(f|_S) subset K_0(f|_S) cup T_infty(f|_S),$ where $K_0(f|_S)$ denotes the set of critical values of the $f|_S.$ Furthermore, $T_infty(f|_S)$ is contained in the set of critical values of certain polynomial functions provided that the $f|_S$ is Newton non-degenerate at infinity. Using these facts, we show that if ${f_t}_{t in [0, 1]}$ is a family of polynomials such that the Newton polyhedron at infinity of $f_t$ is independent of $t$ and the $f_t|_S$ is Newton non-degenerate at infinity, then the global monodromies of the $f_t|_S$ are all isomorphic.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2018-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47553632","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a complex manifold endowed with a $mathbb{C}^times$-action and a DQ-algebra equipped with a compatible holomorphic Frobenius action (F-action), we prove that if the $mathbb{C}^times$-action is free and proper, then the category of F-equivariant DQ-modules is equivalent to the category of modules over the sheaf of invariant sections of the DQ-algebra. As an application, we deduce the codimension three conjecture for formal microdifferential modules from the one for DQ-modules on a symplectic manifold.
{"title":"Holomorphic Frobenius Actions for DQ-Modules","authors":"François Petit","doi":"10.4171/prims/58-1-5","DOIUrl":"https://doi.org/10.4171/prims/58-1-5","url":null,"abstract":"Given a complex manifold endowed with a $mathbb{C}^times$-action and a DQ-algebra equipped with a compatible holomorphic Frobenius action (F-action), we prove that if the $mathbb{C}^times$-action is free and proper, then the category of F-equivariant DQ-modules is equivalent to the category of modules over the sheaf of invariant sections of the DQ-algebra. As an application, we deduce the codimension three conjecture for formal microdifferential modules from the one for DQ-modules on a symplectic manifold.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2018-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41553973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Crystal basis theory for the queer Lie superalgebra was developed by Grantcharov et al. and it was shown that semistandard decomposition tableaux admit the structure of crystals for the queer Lie superalgebra or simply $mathfrak{q}$-crystal structure. In this paper, we explore the $mathfrak{q}$-crystal structure of primed tableaux (semistandard marked shifted tableaux) and that of signed unimodal factorizations of reduced words of type $B$. We give the explicit odd Kashiwara operators on primed tableaux and the forms of the highest and lowest weight vectors. We also give the explicit algorithms for odd Kashiwara operators on signed unimodal factorizations of reduced words of type $B$.
{"title":"$mathfrak{q}$-Crystal Structure on Primed Tableaux and on Signed Unimodal Factorizations of Reduced Words of Type $B$","authors":"Toya Hiroshima","doi":"10.4171/PRIMS/55-2-5","DOIUrl":"https://doi.org/10.4171/PRIMS/55-2-5","url":null,"abstract":"Crystal basis theory for the queer Lie superalgebra was developed by Grantcharov et al. and it was shown that semistandard decomposition tableaux admit the structure of crystals for the queer Lie superalgebra or simply $mathfrak{q}$-crystal structure. In this paper, we explore the $mathfrak{q}$-crystal structure of primed tableaux (semistandard marked shifted tableaux) and that of signed unimodal factorizations of reduced words of type $B$. We give the explicit odd Kashiwara operators on primed tableaux and the forms of the highest and lowest weight vectors. We also give the explicit algorithms for odd Kashiwara operators on signed unimodal factorizations of reduced words of type $B$.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2018-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/PRIMS/55-2-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46823675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce the notion of chiral symplectic cores in a vertex Poisson variety, which can be viewed as analogs of symplectic leaves in Poisson varieties. As an application we show that any quasi-lisse vertex algebra is a quantization of the arc space of its associated variety, in the sense that its reduced singular support coincides with the arc space of its associated variety. We also show that the coordinate ring of the arc space of Slodowy slices is free over its vertex Poisson center, and the latter coincides with the vertex Poisson center of the coordinate ring of the arc space of the dual of the corresponding simple Lie algebra.
{"title":"Arc Spaces and Chiral Symplectic Cores","authors":"T. Arakawa, Anne Moreau","doi":"10.4171/prims/57-3-3","DOIUrl":"https://doi.org/10.4171/prims/57-3-3","url":null,"abstract":"We introduce the notion of chiral symplectic cores in a vertex Poisson variety, which can be viewed as analogs of symplectic leaves in Poisson varieties. As an application we show that any quasi-lisse vertex algebra is a quantization of the arc space of its associated variety, in the sense that its reduced singular support coincides with the arc space of its associated variety. We also show that the coordinate ring of the arc space of Slodowy slices is free over its vertex Poisson center, and the latter coincides with the vertex Poisson center of the coordinate ring of the arc space of the dual of the corresponding simple Lie algebra.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2018-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42053784","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study Fourier integral operators with Shubin amplitudes and quadratic phase functions associated to twisted graph Lagrangians with respect to symplectic matrices. We factorize such an operator as a pseudodifferential operator and a metaplectic operator. Extending the conormal distributions adapted to the Shubin calculus, we define an adapted notion of Lagrangian tempered distribution. We show that the kernels of Fourier integral operators are identical to Lagrangian distributions with respect to twisted graph Lagrangians.
{"title":"Lagrangian Distributions and Fourier Integral Operators with Quadratic Phase Functions and Shubin Amplitudes","authors":"M. Cappiello, R. Schulz, P. Wahlberg","doi":"10.4171/PRIMS/56-3-5","DOIUrl":"https://doi.org/10.4171/PRIMS/56-3-5","url":null,"abstract":"We study Fourier integral operators with Shubin amplitudes and quadratic phase functions associated to twisted graph Lagrangians with respect to symplectic matrices. We factorize such an operator as a pseudodifferential operator and a metaplectic operator. Extending the conormal distributions adapted to the Shubin calculus, we define an adapted notion of Lagrangian tempered distribution. We show that the kernels of Fourier integral operators are identical to Lagrangian distributions with respect to twisted graph Lagrangians.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2018-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41804621","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study lattice operations on the set of idempotent states on a locally compact quantum group corresponding to the operations of intersection of compact subgroups and forming the subgroup generated by two compact subgroups. Normal ($sigma$-weakly continuous) idempotent states are investigated and a duality between normal idempotent states on a locally compact quantum group $mathbb{G}$ and on its dual $widehat{mathbb{G}}$ is established. Additionally we analyze the question when a left coideal corresponding canonically to an idempotent state is finite dimensional and give a characterization of normal idempotent states on compact quantum groups.
{"title":"Lattice of Idempotent States on a Locally Compact Quantum Group","authors":"P. Kasprzak, P. Sołtan","doi":"10.4171/prims/56-1-3","DOIUrl":"https://doi.org/10.4171/prims/56-1-3","url":null,"abstract":"We study lattice operations on the set of idempotent states on a locally compact quantum group corresponding to the operations of intersection of compact subgroups and forming the subgroup generated by two compact subgroups. Normal ($sigma$-weakly continuous) idempotent states are investigated and a duality between normal idempotent states on a locally compact quantum group $mathbb{G}$ and on its dual $widehat{mathbb{G}}$ is established. Additionally we analyze the question when a left coideal corresponding canonically to an idempotent state is finite dimensional and give a characterization of normal idempotent states on compact quantum groups.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2018-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/prims/56-1-3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41895623","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we explain the link between Pohlen's extended Hadamard product and the holomorphic cohomological convolution on $mathbb{C}^*$. For this purpose, we introduce a generalized Hadamard product, which is defined even if the holomorphic functions do not vanish at infinity, as well as a notion of strongly convolvable sets.
{"title":"Holomorphic Cohomological Convolution and Hadamard Product","authors":"Christophe Dubussy, Jean-Pierre Schneiders","doi":"10.4171/prims/58-1-2","DOIUrl":"https://doi.org/10.4171/prims/58-1-2","url":null,"abstract":"In this article, we explain the link between Pohlen's extended Hadamard product and the holomorphic cohomological convolution on $mathbb{C}^*$. For this purpose, we introduce a generalized Hadamard product, which is defined even if the holomorphic functions do not vanish at infinity, as well as a notion of strongly convolvable sets.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2018-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48888932","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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