The 2-cocycles on the algebra of differential operators was studied in Fef. [1]. In this note the Lie algebras of derivations of the algebra and the Lie algebra of differential operators are determined. 1 The Lie Alge...The 2-cocycles on the algebra of differential operators was studied in Fef. [1]. In this note the Lie algebras of derivations of the algebra and the Lie algebra of differential operators are determined. 1 The Lie Algebra of Derivations of the Algebra of Differential Operators Let =C[t, t<sup>-1</sup>] be the Laurent polynomial algebra, d/dt be the differential展开更多
In this paper, automorphisms of the algebra of q-difference operators, as an associative algebra for arbitrary q and as a Lie algebra for q being not a root of unity, are determined.
Creative telescoping is the method of choice for obtaining information about definite sums or integrals. It has been intensively studied since the early 1990s, and can now be considered as a classical technique in com...Creative telescoping is the method of choice for obtaining information about definite sums or integrals. It has been intensively studied since the early 1990s, and can now be considered as a classical technique in computer algebra. At the same time, it is still a subject of ongoing research.This paper presents a selection of open problems in this context. The authors would be curious to hear about any substantial progress on any of these problems.展开更多
Let K be a field of characteristic p>0 . We prove that the derivative algebra of K[x 1,…,x n] is a proer subring of the ring of differential operators of K[x 1,…,x n] . A concrete example is given to show that th...Let K be a field of characteristic p>0 . We prove that the derivative algebra of K[x 1,…,x n] is a proer subring of the ring of differential operators of K[x 1,…,x n] . A concrete example is given to show that there is a differential operator of order p that does not belong to the derivative algebra. By these results, is follows that the derivative algebra is Morita equivalent to K[x p 1,…,x p n] , and hence its global homological dimension, Krull dimension, K 0 group and some other properties are got.展开更多
We study some properties of first order differential operators from an algebraic viewpoint. We show this last can be decomposed in sum of an element of a module and a derivation. From a geometric viewpoint, we give so...We study some properties of first order differential operators from an algebraic viewpoint. We show this last can be decomposed in sum of an element of a module and a derivation. From a geometric viewpoint, we give some properties on the algebra of smooth functions. The Dirac mass at a point is the best example of first order differential operators at this point. This allows to construct a basis of this set and its dual basis.展开更多
The aim of this paper is to investigate the first Hochschild cohomology of ad- missible algebras which can be regarded as a generalization of basic algebras. For this purpose, the authors study differential operators ...The aim of this paper is to investigate the first Hochschild cohomology of ad- missible algebras which can be regarded as a generalization of basic algebras. For this purpose, the authors study differential operators on an admissible algebra. Firstly, dif- ferential operators from a path algebra to its quotient algebra as an admissible algebra ave discussed. Based on this discussion, the first cohomology with admissible algebras as coefficient modules is characterized, including their dimension formula. Besides, for planar quivers, the k-linear bases of the first cohomology of acyclic complete monomial algebras and acyclic truncated quiver algebras are constructed over the field k of characteristic 0.展开更多
In the paper, we further realize the higher rank quantized universal enveloping algebra Uq（sln＋1） as certain quantum differential operators in the quantum Weyl algebra Wq （2n） defined over the quantum divided pow...In the paper, we further realize the higher rank quantized universal enveloping algebra Uq（sln＋1） as certain quantum differential operators in the quantum Weyl algebra Wq （2n） defined over the quantum divided power algebra Sq（n） of rank n. We give the quantum differential operators realization for both the simple root vectors and the non-simple root vectors of Uq（sln＋1）. The nice behavior of the quantum root vectors formulas under the action of the Lusztig symmetries once again indicates that our realization model is naturally matched.展开更多
基金Project supported by the National Natural Science Foundation of China.
文摘The 2-cocycles on the algebra of differential operators was studied in Fef. [1]. In this note the Lie algebras of derivations of the algebra and the Lie algebra of differential operators are determined. 1 The Lie Algebra of Derivations of the Algebra of Differential Operators Let =C[t, t<sup>-1</sup>] be the Laurent polynomial algebra, d/dt be the differential
文摘In this paper, automorphisms of the algebra of q-difference operators, as an associative algebra for arbitrary q and as a Lie algebra for q being not a root of unity, are determined.
基金supported by the National Natural Science Foundation of China under Grant No.11501552the President Fund of the Academy of Mathematics and Systems Science,CAS(2014-cjrwlzx-chshsh)+1 种基金a Starting Grant from the Ministry of Education of Chinasupported by the Austrian FWF under Grant Nos.F5004,Y464-N18,and W1214
文摘Creative telescoping is the method of choice for obtaining information about definite sums or integrals. It has been intensively studied since the early 1990s, and can now be considered as a classical technique in computer algebra. At the same time, it is still a subject of ongoing research.This paper presents a selection of open problems in this context. The authors would be curious to hear about any substantial progress on any of these problems.
文摘Let K be a field of characteristic p>0 . We prove that the derivative algebra of K[x 1,…,x n] is a proer subring of the ring of differential operators of K[x 1,…,x n] . A concrete example is given to show that there is a differential operator of order p that does not belong to the derivative algebra. By these results, is follows that the derivative algebra is Morita equivalent to K[x p 1,…,x p n] , and hence its global homological dimension, Krull dimension, K 0 group and some other properties are got.
文摘We study some properties of first order differential operators from an algebraic viewpoint. We show this last can be decomposed in sum of an element of a module and a derivation. From a geometric viewpoint, we give some properties on the algebra of smooth functions. The Dirac mass at a point is the best example of first order differential operators at this point. This allows to construct a basis of this set and its dual basis.
基金supported by the National Natural Science Foundation of China(Nos.11271318,11171296,11401522,J1210038)the Specialized Research Fund for the Doctoral Program of Higher Education of China(No.20110101110010)the Zhejiang Provincial Natural Science Foundation of China(No.LZ13A010001)
文摘The aim of this paper is to investigate the first Hochschild cohomology of ad- missible algebras which can be regarded as a generalization of basic algebras. For this purpose, the authors study differential operators on an admissible algebra. Firstly, dif- ferential operators from a path algebra to its quotient algebra as an admissible algebra ave discussed. Based on this discussion, the first cohomology with admissible algebras as coefficient modules is characterized, including their dimension formula. Besides, for planar quivers, the k-linear bases of the first cohomology of acyclic complete monomial algebras and acyclic truncated quiver algebras are constructed over the field k of characteristic 0.
基金Supported by National Natural Science Foundation of China(Grant No.11271131)
文摘In the paper, we further realize the higher rank quantized universal enveloping algebra Uq（sln＋1） as certain quantum differential operators in the quantum Weyl algebra Wq （2n） defined over the quantum divided power algebra Sq（n） of rank n. We give the quantum differential operators realization for both the simple root vectors and the non-simple root vectors of Uq（sln＋1）. The nice behavior of the quantum root vectors formulas under the action of the Lusztig symmetries once again indicates that our realization model is naturally matched.
