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
We study the cluster automorphism group of a skew-symmetric cluster algebra with geometric coefficients. We introduce the notion of gluing free cluster algebra, and show that under a weak condition the cluster automor...We study the cluster automorphism group of a skew-symmetric cluster algebra with geometric coefficients. We introduce the notion of gluing free cluster algebra, and show that under a weak condition the cluster automorphism group of a gluing free cluster algebra is a subgroup of the cluster automorphism group of its principal part cluster algebra(i.e., the corresponding cluster algebra without coefficients). We show that several classes of cluster algebras with coefficients are gluing free, for example, cluster algebras with principal coefficients,cluster algebras with universal geometric coefficients, and cluster algebras from surfaces(except a 4-gon) with coefficients from boundaries. Moreover, except four kinds of surfaces, the cluster automorphism group of a cluster algebra from a surface with coefficients from boundaries is isomorphic to the cluster automorphism group of its principal part cluster algebra; for a cluster algebra with principal coefficients, its cluster automorphism group is isomorphic to the automorphism group of its initial quiver.展开更多
We use the quantum version of Chebyshev polynomials to explicitly construct the recursive formulas for the Kronecker quantum cluster algebra with principal coefficients.As a byproduct,we obtain two barinvariant positi...We use the quantum version of Chebyshev polynomials to explicitly construct the recursive formulas for the Kronecker quantum cluster algebra with principal coefficients.As a byproduct,we obtain two barinvariant positive ZP-bases with one being the atomic basis.展开更多
We construct bar-invariant Z[q ±1/2]-bases of the quantum cluster algebra of Kronecker quiver which are quantum analogues of the canonical basis, semicanonical basis and dual semicanonical basis of the correspond...We construct bar-invariant Z[q ±1/2]-bases of the quantum cluster algebra of Kronecker quiver which are quantum analogues of the canonical basis, semicanonical basis and dual semicanonical basis of the corresponding cluster algebra. As a byproduct, we prove positivity of the elements in these bases.展开更多
Our motivation is to build a systematic method in order to investigate the structure of cluster algebras of geometric type. The method is given through the notion of mixing-type sub-seeds, the theory of seed homomorph...Our motivation is to build a systematic method in order to investigate the structure of cluster algebras of geometric type. The method is given through the notion of mixing-type sub-seeds, the theory of seed homomorphisms and the view-point of gluing of seeds. As an application, for(rooted) cluster algebras, we completely classify rooted cluster subalgebras and characterize rooted cluster quotient algebras in detail. Also,we build the relationship between the categorification of a rooted cluster algebra and that of its rooted cluster subalgebras. Note that cluster algebras of geometric type studied here are of the sign-skew-symmetric case.展开更多
We study the relations between two groups related to cluster automorphism groups which are defined by Assem,Schiffler and Shamchenko.We establish the relation-ships among(strict)direct cluster automorphism groups and ...We study the relations between two groups related to cluster automorphism groups which are defined by Assem,Schiffler and Shamchenko.We establish the relation-ships among(strict)direct cluster automorphism groups and those groups consisting of periodicities of labeled seeds and exchange matrices,respectively,in the language of short exact sequences.As an application,we characterize automorphism-finite cluster algebras in the cases of bipartite seeds or finite mutation type.Finally,we study the relation between the group Aut(A)for a cluster algebra A and the group AutMn(S)for a mutation group Mn and a labeled mutation class S,and we give a negative answer via counter-examples to King and Pressland's problem.展开更多
We construct quantized versions of generic bases in quantum cluster algebras of finite and affine types.Under the specialization of q and coefficients to 1,these bases are generic bases of finite and affine cluster al...We construct quantized versions of generic bases in quantum cluster algebras of finite and affine types.Under the specialization of q and coefficients to 1,these bases are generic bases of finite and affine cluster algebras.展开更多
Given a symmetrizable generalized Cartan matrix A, for any index k, one can define an automorphism associated with A, of the field Q(u1,…, un) of rational functions of n independent indeterminates u1,…,un.It is an i...Given a symmetrizable generalized Cartan matrix A, for any index k, one can define an automorphism associated with A, of the field Q(u1,…, un) of rational functions of n independent indeterminates u1,…,un.It is an isomorphism between two cluster algebras associated to the matrix A (see sec. 4 for the precise meaning). When A is of finite type, these isomorphisms behave nicely; they are compatible with the BGP-reflection functors of cluster categories defined in a previous work if we identify the indecomposable objects in the categories with cluster variables of the corresponding cluster algebras, and they are also compatible with the 'truncated simple reflections' defined by Fomin-Zelevinsky. Using the construction of preprojective or preinjective modules of hereditary algebras by DIab-Ringel and the Coxeter automorphisms (i.e. a product of these isomorphisms), we construct infinitely many cluster variables for cluster algebras of infinite type and all cluster variables for finite types.展开更多
We initiate a study of the dependence of the choice of ground ring on the problem on whether a cluster algebra is equal to its upper cluster algebra. A condition for when there is equality of the cluster algebra and u...We initiate a study of the dependence of the choice of ground ring on the problem on whether a cluster algebra is equal to its upper cluster algebra. A condition for when there is equality of the cluster algebra and upper cluster algebra is given by using a variation of Muller's theory of cluster localization. An explicit example exhibiting dependence on the ground ring is provided. We also present a maximal green sequence for this example.展开更多
Symplectic symmetry approach to clustering(SSAC)in atomic nuclei,recently proposed,is modified and further developed in more detail.It is firstly applied to the light two-cluster^(20)Ne+αsystem of^(24)Mg,the latter e...Symplectic symmetry approach to clustering(SSAC)in atomic nuclei,recently proposed,is modified and further developed in more detail.It is firstly applied to the light two-cluster^(20)Ne+αsystem of^(24)Mg,the latter exhibiting well developed low-energy K^(π)=0_(1)^(+),k^(π)=2_(1)^(+) and π^(π)=0_(1)^(-) rotational bands in its spectrum.A simple algebraic Hamiltonian,consisting of dynamical symmetry,residual and vertical mixing parts is used to describe these three lowest rotational bands of positive and negative parity in^(24)Mg.A good description of the excitation energies is obtained by considering only the SU(3)cluster states restricted to the stretched many-particle Hilbert subspace,built on the leading Pauli allowed SU(3)multiplet for the positive-and negative-parity states,respectively.The coupling to the higher cluster-model configurations allows us to describe the known low-lying experimentally observed B(E2)transition probabilities within and between the cluster states of the three bands under consideration without the use of an effective charge.展开更多
We study an anisotropic spin cluster of 3 spin S=1/2 particles with antiferromagnetic exchange interactionwith non-uniform coupling constants.A time-dependent magnetic field is applied to control the time evolution of...We study an anisotropic spin cluster of 3 spin S=1/2 particles with antiferromagnetic exchange interactionwith non-uniform coupling constants.A time-dependent magnetic field is applied to control the time evolution of thecluster.It is well known that for an odd number of sites a spin cluster qubit can be defined in terms of the ground statedoublet.The universal one-qubit logic gate can be constructed from the time evolution operator of the non-autonomousmany-body system,and the six basic one-qubit gates can be realized by adjusting the applied time-dependent magneticfield.展开更多
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 National Natural Science Foundation of China(Grant No.11131001)
文摘We study the cluster automorphism group of a skew-symmetric cluster algebra with geometric coefficients. We introduce the notion of gluing free cluster algebra, and show that under a weak condition the cluster automorphism group of a gluing free cluster algebra is a subgroup of the cluster automorphism group of its principal part cluster algebra(i.e., the corresponding cluster algebra without coefficients). We show that several classes of cluster algebras with coefficients are gluing free, for example, cluster algebras with principal coefficients,cluster algebras with universal geometric coefficients, and cluster algebras from surfaces(except a 4-gon) with coefficients from boundaries. Moreover, except four kinds of surfaces, the cluster automorphism group of a cluster algebra from a surface with coefficients from boundaries is isomorphic to the cluster automorphism group of its principal part cluster algebra; for a cluster algebra with principal coefficients, its cluster automorphism group is isomorphic to the automorphism group of its initial quiver.
基金supported by National Natural Science Foundation of China(Grant No.11771217)supported by National Natural Science Foundation of China(Grant No.12031007)。
文摘We use the quantum version of Chebyshev polynomials to explicitly construct the recursive formulas for the Kronecker quantum cluster algebra with principal coefficients.As a byproduct,we obtain two barinvariant positive ZP-bases with one being the atomic basis.
基金supported by the Fundamental Research Funds for the Central Universitiespartially supported by the Ph.D. Programs Foundation of Ministry of Education of China (Grant No.200800030058)
文摘We construct bar-invariant Z[q ±1/2]-bases of the quantum cluster algebra of Kronecker quiver which are quantum analogues of the canonical basis, semicanonical basis and dual semicanonical basis of the corresponding cluster algebra. As a byproduct, we prove positivity of the elements in these bases.
基金supported by National Natural Science Foundation of China (Grant Nos. 11671350 and 11571173)
文摘Our motivation is to build a systematic method in order to investigate the structure of cluster algebras of geometric type. The method is given through the notion of mixing-type sub-seeds, the theory of seed homomorphisms and the view-point of gluing of seeds. As an application, for(rooted) cluster algebras, we completely classify rooted cluster subalgebras and characterize rooted cluster quotient algebras in detail. Also,we build the relationship between the categorification of a rooted cluster algebra and that of its rooted cluster subalgebras. Note that cluster algebras of geometric type studied here are of the sign-skew-symmetric case.
文摘We study the relations between two groups related to cluster automorphism groups which are defined by Assem,Schiffler and Shamchenko.We establish the relation-ships among(strict)direct cluster automorphism groups and those groups consisting of periodicities of labeled seeds and exchange matrices,respectively,in the language of short exact sequences.As an application,we characterize automorphism-finite cluster algebras in the cases of bipartite seeds or finite mutation type.Finally,we study the relation between the group Aut(A)for a cluster algebra A and the group AutMn(S)for a mutation group Mn and a labeled mutation class S,and we give a negative answer via counter-examples to King and Pressland's problem.
基金supported by the Fundamental Research Funds for the Central UniversitiesNational Natural Science Foudation of China(Grant No.11071133)
文摘We construct quantized versions of generic bases in quantum cluster algebras of finite and affine types.Under the specialization of q and coefficients to 1,these bases are generic bases of finite and affine cluster algebras.
基金This work was supported by the National Natural Science Foundation of China(Grant No.10471071)partially by the Cultivation Fund of the Key Scientific and Technical Innovation Project,Ministry of Education of China 2005.
文摘Given a symmetrizable generalized Cartan matrix A, for any index k, one can define an automorphism associated with A, of the field Q(u1,…, un) of rational functions of n independent indeterminates u1,…,un.It is an isomorphism between two cluster algebras associated to the matrix A (see sec. 4 for the precise meaning). When A is of finite type, these isomorphisms behave nicely; they are compatible with the BGP-reflection functors of cluster categories defined in a previous work if we identify the indecomposable objects in the categories with cluster variables of the corresponding cluster algebras, and they are also compatible with the 'truncated simple reflections' defined by Fomin-Zelevinsky. Using the construction of preprojective or preinjective modules of hereditary algebras by DIab-Ringel and the Coxeter automorphisms (i.e. a product of these isomorphisms), we construct infinitely many cluster variables for cluster algebras of infinite type and all cluster variables for finite types.
基金supported by National Science Foundation of USA (Grant No. DMS1702115)
文摘We initiate a study of the dependence of the choice of ground ring on the problem on whether a cluster algebra is equal to its upper cluster algebra. A condition for when there is equality of the cluster algebra and upper cluster algebra is given by using a variation of Muller's theory of cluster localization. An explicit example exhibiting dependence on the ground ring is provided. We also present a maximal green sequence for this example.
文摘Symplectic symmetry approach to clustering(SSAC)in atomic nuclei,recently proposed,is modified and further developed in more detail.It is firstly applied to the light two-cluster^(20)Ne+αsystem of^(24)Mg,the latter exhibiting well developed low-energy K^(π)=0_(1)^(+),k^(π)=2_(1)^(+) and π^(π)=0_(1)^(-) rotational bands in its spectrum.A simple algebraic Hamiltonian,consisting of dynamical symmetry,residual and vertical mixing parts is used to describe these three lowest rotational bands of positive and negative parity in^(24)Mg.A good description of the excitation energies is obtained by considering only the SU(3)cluster states restricted to the stretched many-particle Hilbert subspace,built on the leading Pauli allowed SU(3)multiplet for the positive-and negative-parity states,respectively.The coupling to the higher cluster-model configurations allows us to describe the known low-lying experimentally observed B(E2)transition probabilities within and between the cluster states of the three bands under consideration without the use of an effective charge.
文摘We study an anisotropic spin cluster of 3 spin S=1/2 particles with antiferromagnetic exchange interactionwith non-uniform coupling constants.A time-dependent magnetic field is applied to control the time evolution of thecluster.It is well known that for an odd number of sites a spin cluster qubit can be defined in terms of the ground statedoublet.The universal one-qubit logic gate can be constructed from the time evolution operator of the non-autonomousmany-body system,and the six basic one-qubit gates can be realized by adjusting the applied time-dependent magneticfield.
