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1.
For an algebraically closed field , we investigate a class of noncommutative -algebras called connected quantized Weyl algebras. Such an algebra has a PBW basis for a set of generators such that each pair satisfies a relation of the form , where and , with, in some sense, sufficiently many pairs for which . For such an algebra it turns out that there is a single parameter q such that each . Assuming that , we classify connected quantized Weyl algebras, showing that there are two types linear and cyclic. When q is not a root of unity we determine the prime spectra for each type. The linear case is the easier, although the result depends on the parity of n, and all prime ideals are completely prime. In the cyclic case, which can only occur if n is odd, there are prime ideals for which the factors have arbitrarily large Goldie rank.We apply connected quantized Weyl algebras to obtain presentations of two classes of quantum cluster algebras. Let be an odd integer. We present the quantum cluster algebra of a Dynkin quiver of type as a factor of a linear connected quantized Weyl algebra by an ideal generated by a central element. We also consider the quiver identified by Fordy and Marsh in their analysis of periodic quiver mutation. When n is odd, we show that the quantum cluster algebra of this quiver is generated by a cyclic connected quantized Weyl algebra in n variables and one further generator. We also present it as the factor of an iterated skew polynomial algebra in variables by an ideal generated by a central element. For both classes, the quantum cluster algebras are simple noetherian.We establish Poisson analogues of the results on prime ideals and quantum cluster algebras. We determine the Poisson prime spectra for the semiclassical limits of the linear and cyclic connected quantized Weyl algebras and show that, when n is odd, the cluster algebras of and are simple Poisson algebras that can each be presented as a Poisson factor of a polynomial algebra, with an appropriate Poisson bracket, by a principal ideal generated by a Poisson central element. 相似文献
2.
Nai Hong HU Shen You WANG 《数学学报(英文版)》2014,30(10):1674-1688
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. 相似文献
3.
Deformation theory can be used to compute the cohomology of a deformed algebra with coefficients in itself from that of the original. Using the invariance of the Euler–Poincaré characteristic under deformation, it is applied here to compute the cohomology of the Weyl algebra, the algebra of the quantum plane, and the q-Weyl algebra. The behavior of the cohomology when q is a root of unity may encode some number theoretic information. 相似文献
4.
Jason Gaddis 《Journal of Pure and Applied Algebra》2017,221(10):2511-2524
Bell and Zhang have shown that if A and B are two connected graded algebras finitely generated in degree one that are isomorphic as ungraded algebras, then they are isomorphic as graded algebras. We exploit this result to solve the isomorphism problem in the cases of quantum affine spaces, quantum matrix algebras, and homogenized multiparameter quantized Weyl algebras. Our result involves determining the degree one normal elements, factoring out, and then repeating. This creates an iterative process that allows one to determine relationships between relative parameters. 相似文献
5.
E. Horozov 《Bulletin des Sciences Mathématiques》2002,126(7):605-614
In this paper we give another characterization of the strictly nilpotent elements in the Weyl algebra, which (apart from the polynomials) turn out to be all bispectral operators with polynomial coefficients. This also allows to reformulate in terms of bispectral operators the famous conjecture, that all the endomorphisms of the Weyl algebra are automorphisms (Dixmier, Kirillov, etc). 相似文献
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This paper discusses the properties of Quantum bit (Qubit) and Quantum logic gates (Quantum not-gate, Hadamard gate and Quantum
controlled not-gate etc.) by the generating element of Pauli algebra (Clifford algebra Cl3). 相似文献
8.
We give explicit constructions of quantum symplectic affine algebras at level one using vertex operators. 相似文献
9.
We study the connections between one-sided Hopf algebras and one-sided quantum quasigroups, tracking the four possible invertibility conditions for the left and right composite morphisms that combine comultiplications and multiplications in these structures. The genuinely one-sided structures exhibit precisely two of the invertibilities, while it emerges that imposing one more condition often entails the validity of all four. A main result shows that under appropriate conditions, just one of the invertibility conditions is su?cient for the existence of a one-sided antipode. In the left Hopf algebra which is a variant of the quantum special linear group of two-dimensional matrices, it is shown explicitly that the right composite is not injective, and the left composite is not surjective. 相似文献
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A. Kundu 《Theoretical and Mathematical Physics》2007,151(3):831-842
We discover an operator-deformed quantum algebra using the quantum Yang-Baxter equation with the trigonometric R-matrix. This
novel Hopf algebra together with its q→1 limit seems the most general Yang-Baxter algebra underlying quantum integrable systems.
We identify three different directions for applying this algebra in integrable systems depending on different sets of values
of the deforming operators. Fixed values on the whole lattice yield subalgebras linked to standard quantum integrable models,
and the associated Lax operators generate and classify them in a unified way. Variable values yield a new series of quantum
integrable inhomogeneous models. Fixed but different values at different lattice sites can produce a novel class of integrable
hybrid models including integrable matter-radiation models and quantum field models with defects, in particular, a new quantum
integrable sine-Gordon model with defect.
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 3, pp. 470–485, June, 2007. 相似文献
13.
Let (Γ,I) be the bound quiver of a cyclic quiver whose vertices correspond to the Abelian group Zd. In this paper, we list all indecomposable representations of (Γ,I) and give the conditions that those representations of them can be extended to representations of deformed preprojective algebra Πλ(Γ,I). It is shown that those representations given by extending indecomposable representations of (Γ,I) are all simple representations of Πλ(Γ,I). Therefore, it is concluded that all simple representa-tions of rest... 相似文献
14.
F. Plastria 《Journal of Optimization Theory and Applications》1988,57(3):463-484
Nonlinear, possibly nonsmooth, minimization problems are considered with boundedly lower subdifferentiable objective and constraints.
An algorithm of the cutting plane type is developed, which has the property that the objective needs to be considered at feasible
points only. It generates automatically a nondecreasing sequence of lower bounds converging to the optimal function value,
thus admitting a rational rule for stopping the calculations when sufficient precision in the objective value has been obtained.
Details are given concerning the efficient implementation of the algorithm. Computational results are reported concerning
the algorithm as applied to continuous location problems with distance constraints.
The author thanks the referees for several constructive remarks and for pointing out an error in an earlier version of the
proof of Lemma 2.1. 相似文献
15.
V.V. Bavula 《Journal of Pure and Applied Algebra》2007,210(1):147-159
Let An be the nth Weyl algebra and Pm be a polynomial algebra in m variables over a field K of characteristic zero. The following characterization of the algebras {An⊗Pm} is proved: an algebraAadmits a finite setδ1,…,δsof commuting locally nilpotent derivations with generic kernels andiffA?An⊗Pmfor somenandmwith2n+m=s, and vice versa. The inversion formula for automorphisms of the algebra An⊗Pm (and for ) has been found (giving a new inversion formula even for polynomials). Recall that (see [H. Bass, E.H. Connell, D. Wright, The Jacobian Conjecture: Reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. (New Series) 7 (1982) 287-330]) given, then (the proof is algebro-geometric). We extend this result (using [non-holonomic] D-modules): given, then. Any automorphism is determined by its face polynomials [J.H. McKay, S.S.-S. Wang, On the inversion formula for two polynomials in two variables, J. Pure Appl. Algebra 52 (1988) 102-119], a similar result is proved for .One can amalgamate two old open problems (the Jacobian Conjecture and the Dixmier Problem, see [J. Dixmier, Sur les algèbres de Weyl, Bull. Soc. Math. France 96 (1968) 209-242. [6]] problem 1) into a single question, (JD): is aK-algebra endomorphismσ:An⊗Pm→An⊗Pman algebra automorphism providedσ(Pm)⊆Pmand? (Pm=K[x1,…,xm]). It follows immediately from the inversion formula that this question has an affirmative answer iff both conjectures have (see below) [iff one of the conjectures has a positive answer (as follows from the recent papers [Y. Tsuchimoto, Endomorphisms of Weyl algebra and p-curvatures, Osaka J. Math. 42(2) (2005) 435-452. [10]] and [A. Belov-Kanel, M. Kontsevich, The Jacobian conjecture is stably equivalent to the Dixmier Conjecture. ArXiv:math.RA/0512171. [5]])]. 相似文献
16.
将经典“试探函数组”1,x,x^2应用于扩展乘数法,建立了一个判别线性正算子能否改造为逼近任何无界连续函数的充要条件。利用该条件给出了一类变形的插值多项式算子的收敛性定理,得到了具有一般性的结论。 相似文献
17.
WU JingYan WEI JunChao & LI LiBin School of Mathematics Yangzhou University Yangzhou China Shijiazhuang Experimental Middle School Shijiazhuang 《中国科学 数学(英文版)》2011,(1)
Suppose that q is not a root of unity, it is proved in this paper that the center of the quantum group Uq(sl4) is isomorphic to a quotient algebra of polynomial algebra with four variables and one relation. 相似文献
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Arnold Knopfmacher Augustine Munagi 《Journal of Difference Equations and Applications》2013,19(1):115-127
The aim of this paper is to study analytical and combinatorial methods to solve a special type of recurrence relation with two indices. It is shown that the recurrence relation enumerates a natural combinatorial object called a plane composition. In addition, further interesting recurrence relations arise in the study of statistics for these plane compositions. 相似文献
20.
We calculate the projection of the product of the Drinfeld currents on the intersection of the different Borel subalgebras
in the current realization of the quantum affine algebra
. This projection yields a universal weight function and has the structure of nested Bethe vectors.
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 2, pp. 286–303, February, 2007. 相似文献