The aim of this paper is to categorify the n-th tensor power of the vector representation of U（ ο（7,C））. The main tools are certain singular blocks and projective functors of the BGG category of the complex Lie a...The aim of this paper is to categorify the n-th tensor power of the vector representation of U（ ο（7,C））. The main tools are certain singular blocks and projective functors of the BGG category of the complex Lie algebra gln.展开更多
We categorify the notion of coalgebras by imposing a co-associative law up to some isomorphisms on the co-multiplication map and requiring that these isomorphisms satisfy certairl law of their own, which is called the...We categorify the notion of coalgebras by imposing a co-associative law up to some isomorphisms on the co-multiplication map and requiring that these isomorphisms satisfy certairl law of their own, which is called the copentagon identity. We also set up a 2-category of 2-coalgebras. The purpose of this study is from the idea of reconsidering the quasi-Hopf algebras by the categorification process, so that we can study the theory of quasi-Hopf algebras and their representations in some new framework of higher category theory in natural ways.展开更多
In this paper,we investigate the categorical description of the boson oscillator.Based on the categories constructed by Khovanov,we introduce a categorification of the Fock states and the corresponding inner product o...In this paper,we investigate the categorical description of the boson oscillator.Based on the categories constructed by Khovanov,we introduce a categorification of the Fock states and the corresponding inner product of these states.We find that there are two different categorical definitions of the inner product of the Fock states.These two definitions are consistent with each other,and the decategorification results also coincide with those in conventional quantum mechanics.We also find that there are some interesting properties of the 2-morphisms which relate to the inner product of the states.展开更多
In this paper, we study the diagrammatic categorification of q-boson algebra and also q-fermion algebra. We construct a graphical category corresponding to q-boson algebra, q-Fock states correspond to some kind of 1-m...In this paper, we study the diagrammatic categorification of q-boson algebra and also q-fermion algebra. We construct a graphical category corresponding to q-boson algebra, q-Fock states correspond to some kind of 1-morphisms, and the graded dimension of the graded vector space of 2-morphisms is exactly the inner product of the corresponding q-Fock states. We also find that this graphical category can be used to categorify q-fermion algebra.展开更多
In this paper, we study the diagrammatic categorification of the fermion algebra. We construct a graphical category corresponding to the one-dimensional （1D） fermion algebra, and we investigate the properties of thi...In this paper, we study the diagrammatic categorification of the fermion algebra. We construct a graphical category corresponding to the one-dimensional （1D） fermion algebra, and we investigate the properties of this category. The categorical analogues of the Fock states are some kind of 1-morphisms in our category, and the dimension of the vector space of 2-morphisms is exactly the inner product of the corresponding Fock states. All the results in our categorical framework coincide exnetlv with those in normal quantum mechanics.展开更多
In this paper, we categorify the algebra Uq（sl2） with the same approach as in [A. Lauda, Adv. Math. （2010）, arXiv：math.QA/0803.3662; M. Khovanov, Comm. Algebra 11 （2001） 5033]. The algebra U =Uq（sl2） is obtai...In this paper, we categorify the algebra Uq（sl2） with the same approach as in [A. Lauda, Adv. Math. （2010）, arXiv：math.QA/0803.3662; M. Khovanov, Comm. Algebra 11 （2001） 5033]. The algebra U =Uq（sl2） is obtained from Uq（sl2） by adjoining a collection of orthogonal idempotents 1λ,λ ∈ P, in which P is the weight lattice of Uq（sl2）. Under such construction the algebra U is decomposed into a direct sum λ∈p 1λ,U1λ. We set the collection of λ∈ P as the objects of the category U, 1-morphisms from λ to λ′ are given by 1λ,U1λ, and 2-morphisms are constructed by some semilinear form defined on U. Hence we get a 2-category u from the algebra Uq（sl2）.展开更多
In this paper we give a categorification of the n-th tensor products of the spin modules of enveloping algebra of lie algebra of type D4 via some subcategories and projective functors of the BGG category of the genera...In this paper we give a categorification of the n-th tensor products of the spin modules of enveloping algebra of lie algebra of type D4 via some subcategories and projective functors of the BGG category of the general linear Lie algebra gln over the complex field C.展开更多
We categorify the highest weight integrable representations and their tensor products of a symmetric quantum Kac–Moody algebra. As a byproduct, we obtain a geometric realization of Lusztig＇s canonical bases of these...We categorify the highest weight integrable representations and their tensor products of a symmetric quantum Kac–Moody algebra. As a byproduct, we obtain a geometric realization of Lusztig＇s canonical bases of these representations as well as a new positivity result. The main ingredient in the underlying geometric construction is a class of micro-local perverse sheaves on quiver varieties.展开更多
In this paper, we prove one case of conjecture given by Hemandez and Leclerc. We give a cluster algebra structuure on the Grothendieck ring of a full subcategory of the finite dimensional representations of affine qua...In this paper, we prove one case of conjecture given by Hemandez and Leclerc. We give a cluster algebra structuure on the Grothendieck ring of a full subcategory of the finite dimensional representations of affine quantum group Uq（A3）. As a conclusion, for every exchange relation of cluster algebra, there exists an exact sequence of the full subcategory corresponding to it.展开更多
In this paper,we introduce Frobenius von Neumann algebras and study quantum convolution inequalities.In this framework,we unify quantum Young's inequality on quantum symmetries such as subfactors,and fusion bi-alg...In this paper,we introduce Frobenius von Neumann algebras and study quantum convolution inequalities.In this framework,we unify quantum Young's inequality on quantum symmetries such as subfactors,and fusion bi-algebras studied in quantum Fourier analysis.Moreover,we prove quantum entropic convolution inequalities and characterize the extremizers in the subfactor case.We also prove quantum smooth entropic convolution inequalities.We obtain the positivity of comultiplications of subfactor planar algebras,which is stronger than the quantum Schur product theorem.All these inequalities provide analytic obstructions of unitary categorification of fusion rings stronger than the Schur product criterion.展开更多
We introduce a class of categories, called clustered hyperbolic categories, which are constructed from a categorized version of preseeds called categorical preseeds using categorical mutations that are a ＂functorial...We introduce a class of categories, called clustered hyperbolic categories, which are constructed from a categorized version of preseeds called categorical preseeds using categorical mutations that are a ＂functorial＂ edition of preseed mutations. Every Weyl preseed p gives rise to a categorical preseed P which generates a clustered hyperbolic category; this is formed by copies of categories each one of which is equivalent to the category of representations of the Weyl cluster algebra H（p）. A ＂categorical realization＂ of Weyl cluster algebra is provided in the sense of defining a map Fp from any clustered hyperbolic category induced from p to the Weyl cluster algebra H（p）, where the image of Fp generates H（p）.展开更多
基金Supported by the Natural Science Foundation of Beijing(Grant No.1122006)the Specialized Research Fund for the Doctoral Program of Higher Education of China(Grant No.201111103110011)Science and Technology Foundation of BJUT(Grant No.ykj-4787)
文摘The aim of this paper is to categorify the n-th tensor power of the vector representation of U（ ο（7,C））. The main tools are certain singular blocks and projective functors of the BGG category of the complex Lie algebra gln.
基金Supported by National Natural Science Foundation of China under Grant Nos. 10975102, 11031005 10871135, 10871227, and PHR201007107
文摘We categorify the notion of coalgebras by imposing a co-associative law up to some isomorphisms on the co-multiplication map and requiring that these isomorphisms satisfy certairl law of their own, which is called the copentagon identity. We also set up a 2-category of 2-coalgebras. The purpose of this study is from the idea of reconsidering the quasi-Hopf algebras by the categorification process, so that we can study the theory of quasi-Hopf algebras and their representations in some new framework of higher category theory in natural ways.
基金Supported by National Natural Science Foundation of China under Grant Nos. 10975102,10871135,11031005,11075014
文摘In this paper,we investigate the categorical description of the boson oscillator.Based on the categories constructed by Khovanov,we introduce a categorification of the Fock states and the corresponding inner product of these states.We find that there are two different categorical definitions of the inner product of the Fock states.These two definitions are consistent with each other,and the decategorification results also coincide with those in conventional quantum mechanics.We also find that there are some interesting properties of the 2-morphisms which relate to the inner product of the states.
基金supported by the National Natural Science Foundation of China (Grant Nos.10975102,10871135,11031005,and 11075014)
文摘In this paper, we study the diagrammatic categorification of q-boson algebra and also q-fermion algebra. We construct a graphical category corresponding to q-boson algebra, q-Fock states correspond to some kind of 1-morphisms, and the graded dimension of the graded vector space of 2-morphisms is exactly the inner product of the corresponding q-Fock states. We also find that this graphical category can be used to categorify q-fermion algebra.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.10975102,10871135,11031005,and 11075014)
文摘In this paper, we study the diagrammatic categorification of the fermion algebra. We construct a graphical category corresponding to the one-dimensional （1D） fermion algebra, and we investigate the properties of this category. The categorical analogues of the Fock states are some kind of 1-morphisms in our category, and the dimension of the vector space of 2-morphisms is exactly the inner product of the corresponding Fock states. All the results in our categorical framework coincide exnetlv with those in normal quantum mechanics.
基金Supported by National Natural Science Foundation of China under Grant Nos. 10975102, 10871135, 11031005, and 10871227
文摘In this paper, we categorify the algebra Uq（sl2） with the same approach as in [A. Lauda, Adv. Math. （2010）, arXiv：math.QA/0803.3662; M. Khovanov, Comm. Algebra 11 （2001） 5033]. The algebra U =Uq（sl2） is obtained from Uq（sl2） by adjoining a collection of orthogonal idempotents 1λ,λ ∈ P, in which P is the weight lattice of Uq（sl2）. Under such construction the algebra U is decomposed into a direct sum λ∈p 1λ,U1λ. We set the collection of λ∈ P as the objects of the category U, 1-morphisms from λ to λ′ are given by 1λ,U1λ, and 2-morphisms are constructed by some semilinear form defined on U. Hence we get a 2-category u from the algebra Uq（sl2）.
基金Supported by National Natural Science Foundation of China(Grant No.11271043)Natural Science Foundation of Beijing(Grant No.1122006)Specialized Research Fund for the Doctoral Program of Higher Education of China(Grant No.201111103110011)
文摘In this paper we give a categorification of the n-th tensor products of the spin modules of enveloping algebra of lie algebra of type D4 via some subcategories and projective functors of the BGG category of the general linear Lie algebra gln over the complex field C.
基金Supported by National Natural Science Foundation of China(Grant Nos.10631060 and 11131008)
文摘We categorify the highest weight integrable representations and their tensor products of a symmetric quantum Kac–Moody algebra. As a byproduct, we obtain a geometric realization of Lusztig＇s canonical bases of these representations as well as a new positivity result. The main ingredient in the underlying geometric construction is a class of micro-local perverse sheaves on quiver varieties.
基金Project supported by the National Natural Science Foundation of China(Grant No.11475178)
文摘In this paper, we prove one case of conjecture given by Hemandez and Leclerc. We give a cluster algebra structuure on the Grothendieck ring of a full subcategory of the finite dimensional representations of affine quantum group Uq（A3）. As a conclusion, for every exchange relation of cluster algebra, there exists an exact sequence of the full subcategory corresponding to it.
基金supported by Beijing Municipal Science&Technology Commission,Administrative Commission of Zhongguancun Science Park(Grant No.Z221100002722017)supported by grants from Beijing Institute of Mathematical Sciences and Applications+4 种基金supported by China’s National Key R&D Programmes(Grant No.2020YFA0713000)Tsinghua University(Grant No.04200100122)Templeton Religion Trust(TRT 159)supported by Beijing Natural Science Foundation Key Program(Grant No.Z220002)supported by National Natural Science Foundation of China(Grant Nos.12031004 and 12371124)。
文摘In this paper,we introduce Frobenius von Neumann algebras and study quantum convolution inequalities.In this framework,we unify quantum Young's inequality on quantum symmetries such as subfactors,and fusion bi-algebras studied in quantum Fourier analysis.Moreover,we prove quantum entropic convolution inequalities and characterize the extremizers in the subfactor case.We also prove quantum smooth entropic convolution inequalities.We obtain the positivity of comultiplications of subfactor planar algebras,which is stronger than the quantum Schur product theorem.All these inequalities provide analytic obstructions of unitary categorification of fusion rings stronger than the Schur product criterion.
文摘We introduce a class of categories, called clustered hyperbolic categories, which are constructed from a categorized version of preseeds called categorical preseeds using categorical mutations that are a ＂functorial＂ edition of preseed mutations. Every Weyl preseed p gives rise to a categorical preseed P which generates a clustered hyperbolic category; this is formed by copies of categories each one of which is equivalent to the category of representations of the Weyl cluster algebra H（p）. A ＂categorical realization＂ of Weyl cluster algebra is provided in the sense of defining a map Fp from any clustered hyperbolic category induced from p to the Weyl cluster algebra H（p）, where the image of Fp generates H（p）.
