When the base connected cochain DG algebra is cohomologically bounded, it is proved that the difference between the amplitude of a compact DG module and that of the DG algebra is just the projective dimension of that ...When the base connected cochain DG algebra is cohomologically bounded, it is proved that the difference between the amplitude of a compact DG module and that of the DG algebra is just the projective dimension of that module. This yields the unboundedness of the cohomology of non-trivial regular DG algebras.When A is a regular DG algebra such that H(A) is a Koszul graded algebra, H(A) is proved to have the finite global dimension. And we give an example to illustrate that the global dimension of H(A) may be infinite, if the condition that H(A) is Koszul is weakened to the condition that A is a Koszul DG algebra. For a general regular DG algebra A, we give some equivalent conditions for the Gorensteiness.For a finite connected DG algebra A, we prove that Dc(A) and Dc(A op) admit Auslander-Reiten triangles if and only if A and A op are Gorenstein DG algebras. When A is a non-trivial regular DG algebra such that H(A) is locally finite, Dc(A) does not admit Auslander-Reiten triangles. We turn to study the existence of Auslander-Reiten triangles in D lf b (A) and D lf b (A op) instead, when A is a regular DG algebra.展开更多
Poisson algebras are fundamental algebraic structures in physics and symplectic geometry. However, the structure theory of Poisson algebras has not been well developed. In this paper, we determine the structure of the...Poisson algebras are fundamental algebraic structures in physics and symplectic geometry. However, the structure theory of Poisson algebras has not been well developed. In this paper, we determine the structure of the central simple Poisson algebras related to locally finite derivations, over an algebraically closed field of characteristic zero.The Lie algebra structures of these Poisson algebras are in general not finitely-graded.展开更多
The classification of extended affine Lie algebras of type A 1 depends on the Tits-Kantor-Koecher (TKK) algebras constructed from semilattices of Euclidean spaces. One can define a unitary Jordan algebra J(S) from a s...The classification of extended affine Lie algebras of type A 1 depends on the Tits-Kantor-Koecher (TKK) algebras constructed from semilattices of Euclidean spaces. One can define a unitary Jordan algebra J(S) from a semilattice S of ?v (v ≥ 1), and then construct an extended affine Lie algebra of type A 1 from the TKK algebra T(J(S)) which is obtained from the Jordan algebra J(S) by the so-called Tits-Kantor-Koecher construction. In ?2 there are only two non-similar semilattices S and S’, where S is a lattice and S’ is a non-lattice semilattice. In this paper we study the ?2-graded automorphisms of the TKK algebra T(J(S)).展开更多
Let A be a connected cochain DG algebra, whose underlying graded algebra is an Artin-Schelter regular algebra of global dimension 2 generated in degree 1. We give a description of all possible differential of A and co...Let A be a connected cochain DG algebra, whose underlying graded algebra is an Artin-Schelter regular algebra of global dimension 2 generated in degree 1. We give a description of all possible differential of A and compute H（A）. Such kind of DG algebras are proved to be strongly Gorenstein. Some of them serve as examples to indicate that a connected DG algebra with Koszul underlying graded algebra may not be a Koszul DG algebra defined in He and We （J Algebra, 2008, 320： 2934-2962）. Unlike positively graded chain DG algebras, we give counterexamples to show that a bounded below DC A-module with a free underlying graded A^#-module may not be semi-projective.展开更多
We introduce the category of t-fold modules which is a full subcategory of graded modules over a graded algebra. We show that this subcategory and hence the subcategory of t-Koszul modules are both closed under extens...We introduce the category of t-fold modules which is a full subcategory of graded modules over a graded algebra. We show that this subcategory and hence the subcategory of t-Koszul modules are both closed under extensions and cokernels of monomorphisms. We study the one-point extension algebras, and a necessary and sufficient condition for such an algebra to be t-Koszul is given. We also consider the conditions such that the category of t-Koszul modules and the category of quadratic modules coincide.展开更多
In 2014, Vargas first defined a super-shuffle product and a cut-box coproduct on permutations. In 2020, Aval, Bergeron and Machacek introduced the super-shuffle product and the cut-box coproduct on labeled simple grap...In 2014, Vargas first defined a super-shuffle product and a cut-box coproduct on permutations. In 2020, Aval, Bergeron and Machacek introduced the super-shuffle product and the cut-box coproduct on labeled simple graphs. In this paper, we generalize the super-shuffle product and the cut-box coproduct from labeled simple graphs to (0,1)-matrices. Then we prove that the vector space spanned by (0,1)-matrices with the super-shuffle product is a graded algebra and with the cut-box coproduct is a graded coalgebra.展开更多
We construct the maximal graded left quotient algebra of every graded algebra A without homogeneous total right zero divisors as the direct limit of graded homomorphisms （of left A-modules） from graded dense left id...We construct the maximal graded left quotient algebra of every graded algebra A without homogeneous total right zero divisors as the direct limit of graded homomorphisms （of left A-modules） from graded dense left ideals of A into a graded left quotient algebra of A. In the case of a superalgebra, and with some extra hypothesis, we prove that the component in the neutral element of the group of the maximal graded left quotient algebra coincides with the maximal left quotient algebra of the component in the neutral element of the group of the superalgebra.展开更多
In the paper we will give a complete classification of finite-dimemsional simple Novikov algebras over an algebraically closed field with prime characteristic p>2.
Let K〈X〉 = K（X1,..., Xn） be the free K-algebra on X = {X1,..., Xn} over a field K, which is equipped with a weight N-gradation （i.e., each Xi is assigned a positive degree）, and let G be a finite homogeneous GrS...Let K〈X〉 = K（X1,..., Xn） be the free K-algebra on X = {X1,..., Xn} over a field K, which is equipped with a weight N-gradation （i.e., each Xi is assigned a positive degree）, and let G be a finite homogeneous GrSbner basis for the ideal I = （G） of K（X） with respect to some monomial ordering 〈 on K（X）. It is shown that if the monomial algebra K（X）/（LM（6）） is semiprime, where LM（6） is the set of leading monomials of 6 with respect to 〈, then the N-graded algebra A ： K（X）/I is semiprimitive in the sense of Jacobson. In the case that G is a finite nonhomogeneous Gr6bner basis with respect to a graded monomial ordering 〈gr, and the N-filtration FA of the algebra A = K（X）/I induced by the N-grading filtration FK（X） of K（X） is considered, if the monomial algebra K（X）/（LM（6）） is semiprime, then it is shown that the associated N-graded algebra G（A） and the Rees algebra A of A determined by FA are all semiprimitive.展开更多
We study diagonal invariant ideals of topologically graded C*-algebras over discrete groups. Since all Toeplitz algebras defined on discrete groups are topologically graded, the results in this paper have improved the...We study diagonal invariant ideals of topologically graded C*-algebras over discrete groups. Since all Toeplitz algebras defined on discrete groups are topologically graded, the results in this paper have improved the first author's previous works on this topic.展开更多
In Shen Guangyu constructed two classes of simple Lie algebras ∑, , and proved that ∑ is new. This paper gives the invariant filtrations and the associated graded algebras of ∑, . Itfollows that is new too. Moreov...In Shen Guangyu constructed two classes of simple Lie algebras ∑, , and proved that ∑ is new. This paper gives the invariant filtrations and the associated graded algebras of ∑, . Itfollows that is new too. Moreover, the author gives some new invariants of ∑, and refines some results of Shen Guangyu [1, 4].展开更多
For any K-algebra A,based on Hochschild complex and Hochschild coho-mology of A,we construct a new Gerstenhaber algebra,and give Gerstenhaber algebra epimorphism from the new Gerstenhaber algebra to the Gerstenhaber a...For any K-algebra A,based on Hochschild complex and Hochschild coho-mology of A,we construct a new Gerstenhaber algebra,and give Gerstenhaber algebra epimorphism from the new Gerstenhaber algebra to the Gerstenhaber algebra of the Hochschild cohomology of A.展开更多
Let G and Gt be two finite groups, and p be a prime number, k is an algebraically closed field of characteristic p. We denote by b and b~ the block idempotents of G and Gt over k, respectively. We assume that the bloc...Let G and Gt be two finite groups, and p be a prime number, k is an algebraically closed field of characteristic p. We denote by b and b~ the block idempotents of G and Gt over k, respectively. We assume that the block algebras kGb and kG＇b＇ are basically Morita equivalent. Puig and Zhou （2007） proved that the corresponding block algebras of some special subgroups of G and G＇ are also basically Morita equivalent. We investigate the relationships between the basic Morita equivalences of two kinds of subgroups of G and G＇： We find a module such that its induced module and its restricted module induce the basic Morita equivalences respectively, hence give a precise description of these basic Morita equivalences.展开更多
基金supported by the National Natural Science Foundation of China (Grant No. 10731070)the Doctorate Foundation of Ministry of Education of China (Grant No. 20060246003)
文摘When the base connected cochain DG algebra is cohomologically bounded, it is proved that the difference between the amplitude of a compact DG module and that of the DG algebra is just the projective dimension of that module. This yields the unboundedness of the cohomology of non-trivial regular DG algebras.When A is a regular DG algebra such that H(A) is a Koszul graded algebra, H(A) is proved to have the finite global dimension. And we give an example to illustrate that the global dimension of H(A) may be infinite, if the condition that H(A) is Koszul is weakened to the condition that A is a Koszul DG algebra. For a general regular DG algebra A, we give some equivalent conditions for the Gorensteiness.For a finite connected DG algebra A, we prove that Dc(A) and Dc(A op) admit Auslander-Reiten triangles if and only if A and A op are Gorenstein DG algebras. When A is a non-trivial regular DG algebra such that H(A) is locally finite, Dc(A) does not admit Auslander-Reiten triangles. We turn to study the existence of Auslander-Reiten triangles in D lf b (A) and D lf b (A op) instead, when A is a regular DG algebra.
基金This work is supported by the National Natural Science Foundation of China (Grant No.10171064)two grants 'Excellent Young Teacher Program' and 'Trans-Century Training Programme Foundation for the Talents' from Ministry of Education of China.
文摘Poisson algebras are fundamental algebraic structures in physics and symplectic geometry. However, the structure theory of Poisson algebras has not been well developed. In this paper, we determine the structure of the central simple Poisson algebras related to locally finite derivations, over an algebraically closed field of characteristic zero.The Lie algebra structures of these Poisson algebras are in general not finitely-graded.
基金the National Natural Science Foundation of China (Grant No.10671160)
文摘The classification of extended affine Lie algebras of type A 1 depends on the Tits-Kantor-Koecher (TKK) algebras constructed from semilattices of Euclidean spaces. One can define a unitary Jordan algebra J(S) from a semilattice S of ?v (v ≥ 1), and then construct an extended affine Lie algebra of type A 1 from the TKK algebra T(J(S)) which is obtained from the Jordan algebra J(S) by the so-called Tits-Kantor-Koecher construction. In ?2 there are only two non-similar semilattices S and S’, where S is a lattice and S’ is a non-lattice semilattice. In this paper we study the ?2-graded automorphisms of the TKK algebra T(J(S)).
基金supported by National Natural Science Foundation of China (Grant No. 11001056)by the China Postdoctoral Science Foundation (Grant No. 20090450066),by the China Postdoctoral Science Foundation (Grant No. 201003244)by Key Disciplines of Shanghai Municipality (Grant No. S30104)
文摘Let A be a connected cochain DG algebra, whose underlying graded algebra is an Artin-Schelter regular algebra of global dimension 2 generated in degree 1. We give a description of all possible differential of A and compute H（A）. Such kind of DG algebras are proved to be strongly Gorenstein. Some of them serve as examples to indicate that a connected DG algebra with Koszul underlying graded algebra may not be a Koszul DG algebra defined in He and We （J Algebra, 2008, 320： 2934-2962）. Unlike positively graded chain DG algebras, we give counterexamples to show that a bounded below DC A-module with a free underlying graded A^#-module may not be semi-projective.
基金the National Natural Science Foundation of China(Grants Nos.10301033,10501041)
文摘We introduce the category of t-fold modules which is a full subcategory of graded modules over a graded algebra. We show that this subcategory and hence the subcategory of t-Koszul modules are both closed under extensions and cokernels of monomorphisms. We study the one-point extension algebras, and a necessary and sufficient condition for such an algebra to be t-Koszul is given. We also consider the conditions such that the category of t-Koszul modules and the category of quadratic modules coincide.
文摘In 2014, Vargas first defined a super-shuffle product and a cut-box coproduct on permutations. In 2020, Aval, Bergeron and Machacek introduced the super-shuffle product and the cut-box coproduct on labeled simple graphs. In this paper, we generalize the super-shuffle product and the cut-box coproduct from labeled simple graphs to (0,1)-matrices. Then we prove that the vector space spanned by (0,1)-matrices with the super-shuffle product is a graded algebra and with the cut-box coproduct is a graded coalgebra.
文摘We construct the maximal graded left quotient algebra of every graded algebra A without homogeneous total right zero divisors as the direct limit of graded homomorphisms （of left A-modules） from graded dense left ideals of A into a graded left quotient algebra of A. In the case of a superalgebra, and with some extra hypothesis, we prove that the component in the neutral element of the group of the maximal graded left quotient algebra coincides with the maximal left quotient algebra of the component in the neutral element of the group of the superalgebra.
文摘In the paper we will give a complete classification of finite-dimemsional simple Novikov algebras over an algebraically closed field with prime characteristic p>2.
基金Project supported by the National Natural Science Foundation of China （10971044）.
文摘Let K〈X〉 = K（X1,..., Xn） be the free K-algebra on X = {X1,..., Xn} over a field K, which is equipped with a weight N-gradation （i.e., each Xi is assigned a positive degree）, and let G be a finite homogeneous GrSbner basis for the ideal I = （G） of K（X） with respect to some monomial ordering 〈 on K（X）. It is shown that if the monomial algebra K（X）/（LM（6）） is semiprime, where LM（6） is the set of leading monomials of 6 with respect to 〈, then the N-graded algebra A ： K（X）/I is semiprimitive in the sense of Jacobson. In the case that G is a finite nonhomogeneous Gr6bner basis with respect to a graded monomial ordering 〈gr, and the N-filtration FA of the algebra A = K（X）/I induced by the N-grading filtration FK（X） of K（X） is considered, if the monomial algebra K（X）/（LM（6）） is semiprime, then it is shown that the associated N-graded algebra G（A） and the Rees algebra A of A determined by FA are all semiprimitive.
基金Supported by the National Natural Science Foundation of China(10371051)
文摘We study diagonal invariant ideals of topologically graded C*-algebras over discrete groups. Since all Toeplitz algebras defined on discrete groups are topologically graded, the results in this paper have improved the first author's previous works on this topic.
文摘In Shen Guangyu constructed two classes of simple Lie algebras ∑, , and proved that ∑ is new. This paper gives the invariant filtrations and the associated graded algebras of ∑, . Itfollows that is new too. Moreover, the author gives some new invariants of ∑, and refines some results of Shen Guangyu [1, 4].
基金Supported by the National Natural Science Foundation of China(Grant No.12201182).
文摘For any K-algebra A,based on Hochschild complex and Hochschild coho-mology of A,we construct a new Gerstenhaber algebra,and give Gerstenhaber algebra epimorphism from the new Gerstenhaber algebra to the Gerstenhaber algebra of the Hochschild cohomology of A.
文摘Let G and Gt be two finite groups, and p be a prime number, k is an algebraically closed field of characteristic p. We denote by b and b~ the block idempotents of G and Gt over k, respectively. We assume that the block algebras kGb and kG＇b＇ are basically Morita equivalent. Puig and Zhou （2007） proved that the corresponding block algebras of some special subgroups of G and G＇ are also basically Morita equivalent. We investigate the relationships between the basic Morita equivalences of two kinds of subgroups of G and G＇： We find a module such that its induced module and its restricted module induce the basic Morita equivalences respectively, hence give a precise description of these basic Morita equivalences.
