Non-isomorphic two dimensional indecomposable modules over infinite dimensional hereditary path algebras are described. We infer that none of them can be determined by their dimension vectors.
Most of pointed Hopf algebras of dimension p^m with large coradtical are shown to be generalized path algebras. By the theory of generalized path algebras, the representations, homological dimensions and radicals of t...Most of pointed Hopf algebras of dimension p^m with large coradtical are shown to be generalized path algebras. By the theory of generalized path algebras, the representations, homological dimensions and radicals of these Hopf algebras are obtained. The relations between the radicals of path algebras and connectivity of directed graphs are given.展开更多
If K is a field with involution and E an arbitrary graph, the involution from K naturally induces an involution of the Leavitt path algebra LK（E）. We show that the involution on LK（E） is proper if the involution o...If K is a field with involution and E an arbitrary graph, the involution from K naturally induces an involution of the Leavitt path algebra LK（E）. We show that the involution on LK（E） is proper if the involution on K is positive-definite, even in the case when the graph E is not necessarily finite or row-finite. It has been shown that the Leavitt path algebra LK（E） is regular if and only if E is acyclic. We give necessary and sufficient conditions for LK（E） to be ＊-regular （i.e., regular with proper involution）. This characterization of ＊-regularity of a Leavitt path algebra is given in terms of an algebraic property of K, not just a graph-theoretic property of E. This differs from the known characterizations of various other algebraic properties of a Leavitt path algebra in terms of graph- theoretic properties of E alone. As a corollary, we show that Handelman＇s conjecture （stating that every ,-regular ring is unit-regular） holds for Leavitt path algebras. Moreover, its generalized version for rings with local units also continues to hold for Leavitt path algebras over arbitrary graphs.展开更多
We introduce two adjoint pairs （eλ^i（）i） and （（）i,ep^i） and give a new method to construct cotorsion pairs. As applications, we characterize all projective and injective representations of a generalized path ...We introduce two adjoint pairs （eλ^i（）i） and （（）i,ep^i） and give a new method to construct cotorsion pairs. As applications, we characterize all projective and injective representations of a generalized path algebra and exhibit projective and injective objects of the category Mp which is a generalization of monomorphisms category.展开更多
In this paper, by calculating the commutator subgroup of the unit group of finite path algebra k△ and the unit group abelianized, we explicitly characterize the K<sub>1</sub> group of finite dimensional p...In this paper, by calculating the commutator subgroup of the unit group of finite path algebra k△ and the unit group abelianized, we explicitly characterize the K<sub>1</sub> group of finite dimensional path algebra over an arbitrary field.展开更多
Let T be a tilting A-module over a path algebra of Dynkin type.We prove thatif the indecomposable direct summands of T are in the different T-orbits of the AR-quiverof A,then T is a complete slice module.
The goal of this paper is to investigate whether the Ext-groups of all pairs (M, N) of modules over Nakayama algebras of type (n,n,n) satisfy the condition ExtΛn(M,N)=0 for n >> 0 ? ExtΛn(M,N)=0 for n >>...The goal of this paper is to investigate whether the Ext-groups of all pairs (M, N) of modules over Nakayama algebras of type (n,n,n) satisfy the condition ExtΛn(M,N)=0 for n >> 0 ? ExtΛn(M,N)=0 for n >> 0. We achieve that by discussing the Ext-groups of Nakayama algebra with projectives of lengths 3n+1 and 3n+2 using combinations of modules of different lengths.展开更多
A path-integral representation of central spin system immersed in an antiferromagnetic environment was investigated. To carry out this study, we made use of the discrete-time propagator method associated with a basic ...A path-integral representation of central spin system immersed in an antiferromagnetic environment was investigated. To carry out this study, we made use of the discrete-time propagator method associated with a basic set involving coherent states of Grassmann variables which made it possible to obtain the analytical propagator which is the centerpiece of the study. In this study, we considered that the environment was in the low-temperature and low-excitation limit and was split into 2 subnets that do not interact with each other. The evaluation of our system was made by considering the first neighbor approximation. From the formalism of the path integrals, it is easy to evaluate the partition function and thermodynamic properties followed from an appropriate tracing over Grassmann variables in the imaginary time domain. We show that the energy of the system depends on the number of sites <em>n</em> when <em>β </em><em></em><span></span>→ 0.展开更多
基金The NSF(11371307)of ChinaResearch Culture Funds(2014xmpy11)of Anhui Normal University
文摘Non-isomorphic two dimensional indecomposable modules over infinite dimensional hereditary path algebras are described. We infer that none of them can be determined by their dimension vectors.
文摘Most of pointed Hopf algebras of dimension p^m with large coradtical are shown to be generalized path algebras. By the theory of generalized path algebras, the representations, homological dimensions and radicals of these Hopf algebras are obtained. The relations between the radicals of path algebras and connectivity of directed graphs are given.
基金supported by the Spanish MEC and Fondos FEDER through project MTM2007-60333the Junta de Andalucía and Fondos FEDER,jointly,through projects FQM-336 and FQM-2467
文摘If K is a field with involution and E an arbitrary graph, the involution from K naturally induces an involution of the Leavitt path algebra LK（E）. We show that the involution on LK（E） is proper if the involution on K is positive-definite, even in the case when the graph E is not necessarily finite or row-finite. It has been shown that the Leavitt path algebra LK（E） is regular if and only if E is acyclic. We give necessary and sufficient conditions for LK（E） to be ＊-regular （i.e., regular with proper involution）. This characterization of ＊-regularity of a Leavitt path algebra is given in terms of an algebraic property of K, not just a graph-theoretic property of E. This differs from the known characterizations of various other algebraic properties of a Leavitt path algebra in terms of graph- theoretic properties of E alone. As a corollary, we show that Handelman＇s conjecture （stating that every ,-regular ring is unit-regular） holds for Leavitt path algebras. Moreover, its generalized version for rings with local units also continues to hold for Leavitt path algebras over arbitrary graphs.
基金This work was carried out while the author was a visitor at University of California, Berkeley she thanks Prof. T. Y. Lam for the very helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China （Grant No. 11201424） and the Natural Science Foundation of Zhejiang Province （No. LY12A01026）.
文摘We introduce two adjoint pairs （eλ^i（）i） and （（）i,ep^i） and give a new method to construct cotorsion pairs. As applications, we characterize all projective and injective representations of a generalized path algebra and exhibit projective and injective objects of the category Mp which is a generalization of monomorphisms category.
基金Projoct supported by the National Natural Science Foundation of ChinaDoctoral Foundation of Education of China
文摘In this paper, by calculating the commutator subgroup of the unit group of finite path algebra k△ and the unit group abelianized, we explicitly characterize the K<sub>1</sub> group of finite dimensional path algebra over an arbitrary field.
基金supported by the National Natural Science Foundation of China(Grant No.10701062)the Huaqiao University Natural Science Foundation(Grant No.01HZR05).
文摘Let T be a tilting A-module over a path algebra of Dynkin type.We prove thatif the indecomposable direct summands of T are in the different T-orbits of the AR-quiverof A,then T is a complete slice module.
文摘The goal of this paper is to investigate whether the Ext-groups of all pairs (M, N) of modules over Nakayama algebras of type (n,n,n) satisfy the condition ExtΛn(M,N)=0 for n >> 0 ? ExtΛn(M,N)=0 for n >> 0. We achieve that by discussing the Ext-groups of Nakayama algebra with projectives of lengths 3n+1 and 3n+2 using combinations of modules of different lengths.
文摘A path-integral representation of central spin system immersed in an antiferromagnetic environment was investigated. To carry out this study, we made use of the discrete-time propagator method associated with a basic set involving coherent states of Grassmann variables which made it possible to obtain the analytical propagator which is the centerpiece of the study. In this study, we considered that the environment was in the low-temperature and low-excitation limit and was split into 2 subnets that do not interact with each other. The evaluation of our system was made by considering the first neighbor approximation. From the formalism of the path integrals, it is easy to evaluate the partition function and thermodynamic properties followed from an appropriate tracing over Grassmann variables in the imaginary time domain. We show that the energy of the system depends on the number of sites <em>n</em> when <em>β </em><em></em><span></span>→ 0.
