An algebraic group is called semi-reductive if it is a semi-direct product of a reductive subgroup and the unipotent radical.Such a semi-reductive algebraic group naturally arises and also plays a key role in the stud...An algebraic group is called semi-reductive if it is a semi-direct product of a reductive subgroup and the unipotent radical.Such a semi-reductive algebraic group naturally arises and also plays a key role in the study of modular representations of non-classical finite-dimensional simple Lie algebras in positive characteristic,and some other cases.Let G be a connected semi-reductive algebraic group over an algebraically closed field F and g=Lie(G).It turns out that G has many same properties as reductive groups,such as the Bruhat decomposition.In this note,we obtain an analogue of classical Chevalley restriction theorem for g,which says that the G-invariant ring F[g]~G is a polynomial ring if g satisfies a certain“positivity”condition suited for lots of cases we are interested in.As applications,we further investigate the nilpotent cones and resolutions of singularities for semi-reductive Lie algebras.展开更多
Advanced mathematical tools are used to conduct research on the kinematics analysis of hybrid mechanisms,and the generalized analysis method and concise kinematics transfer matrix are obtained.In this study,first,acco...Advanced mathematical tools are used to conduct research on the kinematics analysis of hybrid mechanisms,and the generalized analysis method and concise kinematics transfer matrix are obtained.In this study,first,according to the kinematics analysis of serial mechanisms,the basic principles of Lie groups and Lie algebras are briefly explained in dealing with the spatial switching and differential operations of screw vectors.Then,based on the standard ideas of Lie operations,the method for kinematics analysis of parallel mechanisms is derived,and Jacobian matrix and Hessian matrix are formulated recursively and in a closed form.Then,according to the mapping relationship between the parallel joints and corresponding equivalent series joints,a forward kinematics analysis method and two inverse kinematics analysis methods of hybrid mechanisms are examined.A case study is performed to verify the calculated matrices wherein a humanoid hybrid robotic arm with a parallel-series-parallel configuration is considered as an example.The results of a simulation experiment indicate that the obtained formulas are exact and the proposed method for kinematics analysis of hybrid mechanisms is practically feasible.展开更多
Let F be a field of characteristic not 2, and let A be a finite-dimensional semisimple F -algebra. All local automorphisms of A are characterized when all the degrees of A are larger than 1. If F is further assumed to...Let F be a field of characteristic not 2, and let A be a finite-dimensional semisimple F -algebra. All local automorphisms of A are characterized when all the degrees of A are larger than 1. If F is further assumed to be an algebraically closed field of characteristic zero, K a finite group, F K the group algebra of K over F , then all local automorphisms of F K are also characterized.展开更多
In this paper, we completely determine the structure of the unit group of the group algebra of some dihedral groups D2 n over the finite field Fpk, where p is a prime.
An embedding from a group algebra to a matrix algebra is given in this paper. By using it, a criterion for an invertible element in a group algebra is proven.
For a locally compact group G, L 1(G) is its group algebra and L ∞(G) is the dual of L 1(G). Lau has studied the bounded linear operators T : L ∞(G) → L ∞(G) which commute with convolutions and translations. For a...For a locally compact group G, L 1(G) is its group algebra and L ∞(G) is the dual of L 1(G). Lau has studied the bounded linear operators T : L ∞(G) → L ∞(G) which commute with convolutions and translations. For a subspace H of L ∞(G), we know that M(L ∞(G),H), the Banach algebra of all bounded linear operators on L ∞(G) into H which commute with convolutions, has been studied by Pym and Lau. In this paper, we generalize these problems to L(K)*, the dual of a hypergroup algebra L(K) in a very general setting, i. e. we do not assume that K admits a Haar measure. It should be noted that these algebras include not only the group algebra L 1(G) but also most of the semigroup algebras. Compact hypergroups have a Haar measure, however, in general it is not known that every hypergroup has a Haar measure. The lack of the Haar measure and involution presents many difficulties; however, we succeed in getting some interesting results.展开更多
We prove that the spectrum of a Cayley graph over a finite group with a normal generating set S containing with every its element s all generators of the cyclic group(s)is integral.In particular,a Cayley graph of a 2-...We prove that the spectrum of a Cayley graph over a finite group with a normal generating set S containing with every its element s all generators of the cyclic group(s)is integral.In particular,a Cayley graph of a 2-group generated by a normal set of involutions is integral.We prove that a Cayley graph over the symmetric group of degree n no less than 2 generated by all transpositions is integral.We find the spectrum of a Cayley graph over the alternating group of degree n no less than 4 with a generating set of 3-cycles of the form(k i j)with fixed k,a s{-n+1,1-n+1,2^2-n+1,...,(n-1)2-n+1}.展开更多
The torsion conjecture says: for any abelian variety A defined over a number field k, the order of the torsion subgroup of A(k) is bounded by a constant C(k,d) which depends only on the number field k and the dim...The torsion conjecture says: for any abelian variety A defined over a number field k, the order of the torsion subgroup of A(k) is bounded by a constant C(k,d) which depends only on the number field k and the dimension d of the abelian variety. The torsion conjecture remains open in general. However, in this paper, a short argument shows that the conjecture is true for more general fields if we consider linear groups instead of abelian varieties. If G is a connected linear algebraic group defined over a field k which is finitely generated over Q,Г is a torsion subgroup of G(k). Then the order of Г is bounded by a constant C'(k, d) which depends only on k and the dimension d of G.展开更多
Let A be a basic connected finite dimensional associative algebra over an algebraically closed field k and G be a cyclic group.There is a quiver QGwith relationsρG such that the skew group algebras A[G]is Morita equi...Let A be a basic connected finite dimensional associative algebra over an algebraically closed field k and G be a cyclic group.There is a quiver QGwith relationsρG such that the skew group algebras A[G]is Morita equivalent to the quotient algebra of path algebra kQGmodulo ideal(ρG).Generally,the quiver QGis not connected.In this paper we develop a method to determine the number of connect components of QG.Meanwhile,we introduce the notion of weight for underlying quiver of A such that A is G-graded and determine the connect components of smash product A#kG*.展开更多
In this paper, we study selfinjective Koszul algebras of finite complexity. We prove that the complexity is a nonnegative integer when it is finite; and that the category Yt of modules with complexity less or equal to...In this paper, we study selfinjective Koszul algebras of finite complexity. We prove that the complexity is a nonnegative integer when it is finite; and that the category Yt of modules with complexity less or equal to t, is resolving and coresolving. We show that for each 0 ≤ 1 ≤ m there exist a family of modules of complexity 1 parameterized by G(l, m), the Grassmannian of l-dimensional subspaces of an m-dimensional vector space V, for the exterior algebra of V. Using complexity, we also give a new approach to the representation theory of a tame symmetric algebra with vanishing radical cube over an algebraically closed field of characteristic 0, via skew group algebra of a finite subgroup of SL(2, C) over the exterior algebra of a 2-dimensional vector space.展开更多
We study the global dimensions of the coherent functors over two categories that are linked by a pair of adjoint functors. This idea is then exploited to compare the representation dimensions of two algebras. In parti...We study the global dimensions of the coherent functors over two categories that are linked by a pair of adjoint functors. This idea is then exploited to compare the representation dimensions of two algebras. In particular, we show that if an Artin algebra is switched from the other, then they have the same representation dimension.展开更多
Motivated by T-tilting theory developed by T. Adachi, O. Iyama, I. Reiten, for a finite-dimensional algebra A with action by a finite group G, we introduce the notion of G-stable support τ-tilting modules. Then we es...Motivated by T-tilting theory developed by T. Adachi, O. Iyama, I. Reiten, for a finite-dimensional algebra A with action by a finite group G, we introduce the notion of G-stable support τ-tilting modules. Then we establish bijections among G-stable support τ-tilting modules over ∧, G-stable two-term silting complexes in the homotopy category of bounded complexes of finitely generated projective ∧-modules, and G-stable functorially finite torsion classes in the category of finitely generated left ∧-modules. In the case when ∧ is the endomorphism of a G-stable cluster-tilting object T over a Horn-finite 2-Calabi- Yau triangulated category L with a G-action, these are also in bijection with G-stable cluster-tilting objects in L. Moreover, we investigate the relationship between stable support τ-tilitng modules over ∧ and the skew group algebra ∧G.展开更多
Let Fq be a finite field with order q and D2n be the dihedral group with 2n elements, and gcd(q, 2n) = 1. In this article, the authors give precise descriptions and enumerations of linear complementary dual(LCD) codes...Let Fq be a finite field with order q and D2n be the dihedral group with 2n elements, and gcd(q, 2n) = 1. In this article, the authors give precise descriptions and enumerations of linear complementary dual(LCD) codes and self-orthogonal codes in the finite dihedral group algebras Fq[D2n]. Some numerical examples are also presented to illustrate the main results.展开更多
Let K be a normal subgroup of the finite group H. To a point of a K-interior H-algebra we associate a group extension, and we prove that this extension is isomorphic to an extension associated to the point given by th...Let K be a normal subgroup of the finite group H. To a point of a K-interior H-algebra we associate a group extension, and we prove that this extension is isomorphic to an extension associated to the point given by the Brauer homomorphism. This may be regarded as a generalization and an alternative treatment of Dade's results [2, Section 12].展开更多
Let G be a finite p-solvable group and k be an algebraic closed field of characteristic p. It is proved that any projective indecomposable module of a G-graded k-algebra is an induced module of a module of the subalge...Let G be a finite p-solvable group and k be an algebraic closed field of characteristic p. It is proved that any projective indecomposable module of a G-graded k-algebra is an induced module of a module of the subalgebra graded by a Hall p′-subgroup. A necessary and sufficient condition for the indecomposability of an induced module from a Hall p′-subgroup is obtained.展开更多
We unify and generalize several approaches to constructing braid group representa-tions from finite groups,using iterated twisted tensor products.We provide some general characterizations and classification of these r...We unify and generalize several approaches to constructing braid group representa-tions from finite groups,using iterated twisted tensor products.We provide some general characterizations and classification of these representations,focusing on the size of their images,which are typically finite groups.The well-studied Gaussian representations associated with metaplectic modular categories can be understood in this framework,and we give some new examples to illustrate their ubiquity.Our results suggest a relationship between the braiding on the G-gaugings of a pointed modular category C(A,Q)and that of C(A,Q)itself.展开更多
基金Supported by NSFC(Grant Nos.12071136,11671138,11771279,12101544)Shanghai Key Laboratory of PMMP(Grant No.13dz2260400)the Fundamental Research Funds of Yunnan Province(Grant No.2020J0375)。
文摘An algebraic group is called semi-reductive if it is a semi-direct product of a reductive subgroup and the unipotent radical.Such a semi-reductive algebraic group naturally arises and also plays a key role in the study of modular representations of non-classical finite-dimensional simple Lie algebras in positive characteristic,and some other cases.Let G be a connected semi-reductive algebraic group over an algebraically closed field F and g=Lie(G).It turns out that G has many same properties as reductive groups,such as the Bruhat decomposition.In this note,we obtain an analogue of classical Chevalley restriction theorem for g,which says that the G-invariant ring F[g]~G is a polynomial ring if g satisfies a certain“positivity”condition suited for lots of cases we are interested in.As applications,we further investigate the nilpotent cones and resolutions of singularities for semi-reductive Lie algebras.
基金Supported by Zhejiang Province Foundation for Distinguished Young Scholars of China(Grant No.LR18E050003)National Natural Science Foundation of China(Grant Nos.51975523,51475424,51905481)Open Foundation of the State Key Laboratory of Fluid Power and Mechatronic Systems(Grant No.GZKF-201906).
文摘Advanced mathematical tools are used to conduct research on the kinematics analysis of hybrid mechanisms,and the generalized analysis method and concise kinematics transfer matrix are obtained.In this study,first,according to the kinematics analysis of serial mechanisms,the basic principles of Lie groups and Lie algebras are briefly explained in dealing with the spatial switching and differential operations of screw vectors.Then,based on the standard ideas of Lie operations,the method for kinematics analysis of parallel mechanisms is derived,and Jacobian matrix and Hessian matrix are formulated recursively and in a closed form.Then,according to the mapping relationship between the parallel joints and corresponding equivalent series joints,a forward kinematics analysis method and two inverse kinematics analysis methods of hybrid mechanisms are examined.A case study is performed to verify the calculated matrices wherein a humanoid hybrid robotic arm with a parallel-series-parallel configuration is considered as an example.The results of a simulation experiment indicate that the obtained formulas are exact and the proposed method for kinematics analysis of hybrid mechanisms is practically feasible.
基金Supported by the Fundamental Research Funds for the Central Universities
文摘Let F be a field of characteristic not 2, and let A be a finite-dimensional semisimple F -algebra. All local automorphisms of A are characterized when all the degrees of A are larger than 1. If F is further assumed to be an algebraically closed field of characteristic zero, K a finite group, F K the group algebra of K over F , then all local automorphisms of F K are also characterized.
文摘In this paper, we completely determine the structure of the unit group of the group algebra of some dihedral groups D2 n over the finite field Fpk, where p is a prime.
文摘An embedding from a group algebra to a matrix algebra is given in this paper. By using it, a criterion for an invertible element in a group algebra is proven.
文摘For a locally compact group G, L 1(G) is its group algebra and L ∞(G) is the dual of L 1(G). Lau has studied the bounded linear operators T : L ∞(G) → L ∞(G) which commute with convolutions and translations. For a subspace H of L ∞(G), we know that M(L ∞(G),H), the Banach algebra of all bounded linear operators on L ∞(G) into H which commute with convolutions, has been studied by Pym and Lau. In this paper, we generalize these problems to L(K)*, the dual of a hypergroup algebra L(K) in a very general setting, i. e. we do not assume that K admits a Haar measure. It should be noted that these algebras include not only the group algebra L 1(G) but also most of the semigroup algebras. Compact hypergroups have a Haar measure, however, in general it is not known that every hypergroup has a Haar measure. The lack of the Haar measure and involution presents many difficulties; however, we succeed in getting some interesting results.
基金The work was supported by the Program of Fundamental Scientific Research of the SB RAS N 1.5.1,project No.0314-2019-0016.
文摘We prove that the spectrum of a Cayley graph over a finite group with a normal generating set S containing with every its element s all generators of the cyclic group(s)is integral.In particular,a Cayley graph of a 2-group generated by a normal set of involutions is integral.We prove that a Cayley graph over the symmetric group of degree n no less than 2 generated by all transpositions is integral.We find the spectrum of a Cayley graph over the alternating group of degree n no less than 4 with a generating set of 3-cycles of the form(k i j)with fixed k,a s{-n+1,1-n+1,2^2-n+1,...,(n-1)2-n+1}.
基金Tianyuan Mathematics Foundation of NSFC (Grant No.10626050)the Knowledge Innovation Program of the Chinese Academy of Sciences
文摘The torsion conjecture says: for any abelian variety A defined over a number field k, the order of the torsion subgroup of A(k) is bounded by a constant C(k,d) which depends only on the number field k and the dimension d of the abelian variety. The torsion conjecture remains open in general. However, in this paper, a short argument shows that the conjecture is true for more general fields if we consider linear groups instead of abelian varieties. If G is a connected linear algebraic group defined over a field k which is finitely generated over Q,Г is a torsion subgroup of G(k). Then the order of Г is bounded by a constant C'(k, d) which depends only on k and the dimension d of G.
基金Supported by the National Natural Science Foundation of China(Grant Nos.11871404,11971398 and 12131018)。
文摘Let A be a basic connected finite dimensional associative algebra over an algebraically closed field k and G be a cyclic group.There is a quiver QGwith relationsρG such that the skew group algebras A[G]is Morita equivalent to the quotient algebra of path algebra kQGmodulo ideal(ρG).Generally,the quiver QGis not connected.In this paper we develop a method to determine the number of connect components of QG.Meanwhile,we introduce the notion of weight for underlying quiver of A such that A is G-graded and determine the connect components of smash product A#kG*.
基金Supported by NSFC #10671061SRFDP #200505042004the Cultivation Fund of the Key Scientific and Technical Innovation Project #21000115 of the Ministry of Education of China
文摘In this paper, we study selfinjective Koszul algebras of finite complexity. We prove that the complexity is a nonnegative integer when it is finite; and that the category Yt of modules with complexity less or equal to t, is resolving and coresolving. We show that for each 0 ≤ 1 ≤ m there exist a family of modules of complexity 1 parameterized by G(l, m), the Grassmannian of l-dimensional subspaces of an m-dimensional vector space V, for the exterior algebra of V. Using complexity, we also give a new approach to the representation theory of a tame symmetric algebra with vanishing radical cube over an algebraically closed field of characteristic 0, via skew group algebra of a finite subgroup of SL(2, C) over the exterior algebra of a 2-dimensional vector space.
基金supported by the Cultivation Fund of the Key Scientific and Technical Innovation Project(707004)the Doctorate Program FOUNDATION(20040027002)Ministry of Education of China,The partial support from NSF of China is also acknowledged
文摘We study the global dimensions of the coherent functors over two categories that are linked by a pair of adjoint functors. This idea is then exploited to compare the representation dimensions of two algebras. In particular, we show that if an Artin algebra is switched from the other, then they have the same representation dimension.
基金The authors would like to thank Dong Yang and Yuefei Zheng for their helpful discussion. This work was partially supported by the National Natural Science Foundation of China (Grant No. 11571164) and a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
文摘Motivated by T-tilting theory developed by T. Adachi, O. Iyama, I. Reiten, for a finite-dimensional algebra A with action by a finite group G, we introduce the notion of G-stable support τ-tilting modules. Then we establish bijections among G-stable support τ-tilting modules over ∧, G-stable two-term silting complexes in the homotopy category of bounded complexes of finitely generated projective ∧-modules, and G-stable functorially finite torsion classes in the category of finitely generated left ∧-modules. In the case when ∧ is the endomorphism of a G-stable cluster-tilting object T over a Horn-finite 2-Calabi- Yau triangulated category L with a G-action, these are also in bijection with G-stable cluster-tilting objects in L. Moreover, we investigate the relationship between stable support τ-tilitng modules over ∧ and the skew group algebra ∧G.
基金supported by the National Natural Science Foundation of China(Nos.61772015,11971321,12101326)Foundation of Nanjing Institute of Technology(No.CKJB202007)+4 种基金the NUPTSF(No.NY220137)the Guangxi Natural Science Foundation(No.2020GXNSFAA159053)the National Key Research and Development Program of China(No.2018YFA0704703)Foundation of Science and Technology on Information Assurance Laboratory(No.KJ-17-010)the Open Project of Shanghai Key Laboratory of Trustworthy Computing(No.OP202101)。
文摘Let Fq be a finite field with order q and D2n be the dihedral group with 2n elements, and gcd(q, 2n) = 1. In this article, the authors give precise descriptions and enumerations of linear complementary dual(LCD) codes and self-orthogonal codes in the finite dihedral group algebras Fq[D2n]. Some numerical examples are also presented to illustrate the main results.
基金supported by a grant of the Ministry of National Education,Romania,CNCS-UEFISCDI,project number PN-II-ID-PCE-2012-4-0100
文摘Let K be a normal subgroup of the finite group H. To a point of a K-interior H-algebra we associate a group extension, and we prove that this extension is isomorphic to an extension associated to the point given by the Brauer homomorphism. This may be regarded as a generalization and an alternative treatment of Dade's results [2, Section 12].
文摘Let G be a finite p-solvable group and k be an algebraic closed field of characteristic p. It is proved that any projective indecomposable module of a G-graded k-algebra is an induced module of a module of the subalgebra graded by a Hall p′-subgroup. A necessary and sufficient condition for the indecomposability of an induced module from a Hall p′-subgroup is obtained.
文摘We unify and generalize several approaches to constructing braid group representa-tions from finite groups,using iterated twisted tensor products.We provide some general characterizations and classification of these representations,focusing on the size of their images,which are typically finite groups.The well-studied Gaussian representations associated with metaplectic modular categories can be understood in this framework,and we give some new examples to illustrate their ubiquity.Our results suggest a relationship between the braiding on the G-gaugings of a pointed modular category C(A,Q)and that of C(A,Q)itself.