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1.
Lowell Abrams 《Israel Journal of Mathematics》2000,117(1):335-352
The Poincaré duality of classical cohomology and the extension of this duality to quantum cohomology endows these rings with
the structure of a Frobenius algebra. Any such algebra possesses a canonical “characteristic element;” in the classical case
this is the Euler class, and in the quantum case this is a deformation of the classical Euler class which we call the “quantum
Euler class.” We prove that the characteristic element of a Frobenius algebraA is a unit if and only ifA is semisimple, and then apply this result to the cases of the quantum cohomology of the finite complex Grassmannians, and
to the quantum cohomology of hypersurfaces. In addition we show that, in the case of the Grassmannians, the [quantum] Euler
class equals, as [quantum] cohomology element and up to sign, the determinant of the Hessian of the [quantum] Landau-Ginzbug
potential. 相似文献
2.
A key notion bridging the gap between quantum operator algebras [26] and vertex operator algebras [4, 9] is the definition of the commutativity of a pair of quantum operators (see Section 2). This is not commutativity in any ordinary sense, but it is clearly the correct generalization to the quantum context. In [26] we give a definition of a commutative quantum operator algebra. We show in [26] that a vertex operator algebra gives rise to a special case of a CQOA. The main purpose of the current paper is to further develop the foundations for a complete mathematical theory of CQOAs. We give proofs of most of the relevant results announced in [26], and we carry out some calculations with sufficient detail to enable the interested reader to become proficient with the algebra of commuting quantum operators. 相似文献
3.
A. Sevostyanov 《Selecta Mathematica, New Series》2002,8(4):637-703
We define deformations of W-algebras associated to comple semisimple Lie algebras by means of quantum Drinfeld-Sokolov reduction
procedure for affine quantum groups. We also introduce Wakimoto modules for arbitrary affine quantum groups and construct
free field resolutions and screening operators for the deformed W-algebras. We compare our results with earlier definitions
of q-W-algebras and of the deformed screening operators due to Awata, Kubo, Odake, Shiraishi [60],[6], [7], Feigin, E. Frenkel
[22] and E. Frenkel, Reshetikhin [34]. The screening operator and the free field resolution for the deformed W-algebra associated
to the simple Lie algebra sl2 coincide with those for the deformed Virasoro algebra introduced in [60].
The author is supported by the Swiss National Science Foundation. 相似文献
4.
量子群的基变换与范畴同构 总被引:5,自引:1,他引:5
令M是Z[v]的由v-1和奇素数p生成的理想,U是A=Z[v]M上相伴于对称Cartan矩阵的量子群, A-Γ是环同态, Uг=UAΓ[Uг]是Uг的量子坐标代数,本文建立了量子坐标代数的基变换:即在相关约束条件下有Г-Hopf同构 A[U]AГ≌Г[Uг].我们证明了有限秩 A自由 1型可积 U模范畴和有限秩 A自由 A[U]余模范畴是同构的.特别,当 Г是域时,局部有限 1型 Uг模范畴和Г[Uг]余模范畴是同构的.最后,我们还证明了在[1]中定义的诱导函子和B.Parshall与王建磐博士在[2]中研究的诱导函子的一致性. 相似文献
5.
This paper presents a connection between the defining basis presented by Beilinson-Lusztig-MacPherson [1] in their geometric setting for quantum GLn and the isomorphism classes of linear quiver representations. More precisely, the positive part of the basis in [1] identifies with the defining basis for the relevant Ringel-Hall algebra; hence, it is a PBW basis in the sense of quantum groups. This approach extends to q-Schur algebras, yielding a monomial basis property with respect to the Drinfeld-Jimbo type presentation for the positive (or negative) part of the q-Schur algebra. Finally, the paper establishes an explicit connection between the canonical basis for the positive part of quantum GLn and the Kazhdan-Lusztig basis for q-Schur algebras. 相似文献
6.
We introduce a class of quantum integrable systems generalizing the Gaudin model. The corresponding algebras of quantum Hamiltonians are obtained as quotients of the center of the enveloping algebra of an affine Kac-Moody algebra at the critical level, extending the construction of higher Gaudin Hamiltonians from B. Feigin et al. (1994) [17] to the case of non-highest weight representations of affine algebras. We show that these algebras are isomorphic to algebras of functions on the spaces of opers on P1 with regular as well as irregular singularities at finitely many points. We construct eigenvectors of these Hamiltonians, using Wakimoto modules of critical level, and show that their spectra on finite-dimensional representations are given by opers with trivial monodromy. We also comment on the connection between the generalized Gaudin models and the geometric Langlands correspondence with ramification. 相似文献
7.
A. K. Kwaśniewski 《Advances in Applied Clifford Algebras》1999,9(2):249-260
The non commuting matrix elements of matrices from quantum groupGL
q
(2;C) withq≡ω being then-th root of unity are given a representation as operators in Hilbert space with help ofC
4
(n)
generalized Clifford algebra generators appropriately tensored with unit 2×2 matrix infinitely many times. Specific properties
of such a representation are presented. Relevance of generalized Pauli algebra to azimuthal quantization of angular momentum
alà Lévy-Leblond [10] and to polar decomposition ofSU
q
(2;C) quantum algebra alà Chaichian and Ellinas [6] is also commented.
The case ofq∈C, |q|=1 may be treated parallely. 相似文献
8.
In [3] it was shown that a (real) signed measure on a cyclic coarse-grained quantum logic can be extended, as a signed measure,
over the entire power algebra. Later ([9]) this result was re-proved (and further improved on) and, moreover, the non-negative
measures were shown to allow for extensions as non-negative measures. In both cases the proof technique used was the technique
of linear algebra. In this paper we further generalize the results cited by extending group-valued measures on cyclic coarse-grained
quantum logics (or non-negative group-valued measures for lattice-ordered groups). Obviously, the proof technique is entirely
different from that of the preceding papers. In addition, we provide a new combinatorial argument for describing all atoms
of cyclic coarse-grained quantum logics. 相似文献
9.
Lieven Le Bruyn 《代数通讯》2013,41(3):865-876
In this note we describe two simple tricks derived from work of M. Artin , J. Tate and M. Van den Bergh [2] and apply them to the quantum enveloping algebra discovered by E. Witten [ll] encoding duality in three dimensional Chern-Simons gauge theory with structure group SU(2). In particular, we give a more conceptual interpretation of the dichotony in the representation theory and determine all finite dimensional simple representations. 相似文献
10.
A well known theorem of Duflo claims that the annihilator of a Verma module in the enveloping algebra of a complex semisimple
Lie algebra is generated by its intersection with the centre. For a Lie superalgebra this result fails to be true. For instance,
in the case of the orthosymplectic Lie superalgebra osp(1,2), Pinczon gave in [Pi] an example of a Verma module whose annihilator
is not generated by its intersection with the centre of universal enveloping algebra. More generally, Musson produced in [Mu1]
a family of such “singular” Verma modules for osp(1,2l) cases. In this article we give a necessary and sufficient condition
on the highest weight of a osp(1,2l)-Verma module for its annihilator to be generated by its intersection with the centre.
This answers a question of Musson. The classical proof of the Duflo theorem is based on a deep result of Kostant which uses
some delicate algebraic geometry reasonings. Unfortunately these arguments can not be reproduced in the quantum and super
cases. This obstruction forced Joseph and Letzter, in their work on the quantum case (see [JL]), to find an alternativeapproach
to the Duflo theorem. Following their ideas, we compute the factorization of the Parthasarathy–Ranga-Rao–Varadarajan (PRV)
determinants. Comparing it with the factorization of Shapovalov determinants we find, unlike to the classical and quantum
cases, that the PRV determinant contains some extrafactors. The set of zeroes of these extrafactors is precisely the set of
highest weights of Verma modules whose annihilators are not generated by their intersection with the centre. We also find
an analogue of Hesselink formula (see [He]) giving the multiplicity of every simple finite dimensional module in the graded
component of the harmonic space in the symmetric algebra.
Oblatum 1-IX-1998 & 4-XII-1998 / Published online: 10 May 1999 相似文献
11.
In the logico-algebraic foundation of quantum mechanics one often deals with the orthomodular lattices (OML) which enjoy state-separating properties of noncompatible pairs (see e.g. [18], [9] and [15]). These properties usually guarantee reasonable richness of the state space—an assumption needed in developing the theory of quantum logics. In this note we consider these classes of OMLs from the universal algebra standpoint, showing, as the main result, that these classes form quasivarieties. We also illustrate by examples that these classes may (and need not) be varieties. The results supplement the research carried on in [1], [3], [4], [5], [6], [11], [12], [13] and [16]. 相似文献
12.
《Comptes Rendus de l'Academie des Sciences Series IIA Earth and Planetary Science》1999,328(9):755-758
Using quantum methods, we introduce here the notion of “neo-algebra” which generalizes the notion of a commutative differential graded algebra. Under some mild finiteness conditions, we can associate functorially to a space a neo-algebra over the finite field Fp: its quasi-isomorphism's class determines the p-adic-homotopy type of X. As a matter of fact, from this data, we can describe in a simple way Steenrod operations in the cohomology of X, as well as the p primary part of its homotopy groups. This point of view extends to finite characteristics the well-known rational homotopy theory of D. Quillen [9] and D. Sullivan [11]. It is deeply related to previous works of P. May, I. Kriz [5] and M.A. Mandell [7], [8]. 相似文献
13.
A. Anikin 《Regular and Chaotic Dynamics》2008,13(5):377-402
According to classical result of Moser [1] a real-analytic Hamiltonian with one and a half degrees of freedom near a hyperbolic
fixed point can be reduced to the normal form by a real-analytic symplectic change of variables. In this paper the result
is extended to the case of the non-commutative algebra of quantum observables.We use an algebraic approach in quantum mechanics
presented in [2] and develop it to the non-autonomous case. We introduce the notion of quantum non-autonomous canonical transformations
and prove that they form a group and preserve the structure of the Heisenberg equation. We give the concept of a non-commutative
normal form and prove that a time-periodic quantum observable with one degree of freedom near a hyperbolic fixed point can
be reduced to a normal form by a canonical transformation. Unlike traditional results, where only formal theory of normal
forms is constructed, we prove a convergence of the normalizing procedure.
相似文献
14.
Zbigniew Lipecki 《manuscripta mathematica》2001,104(3):333-341
Cardinals that arise as the number of extreme quasi-measure extensions of a quasi-measure [resp., measure] μ defined on an
algebra [resp., σ-algebra] of sets to a larger algebra [resp., σ-algebra] of sets are characterized in the general case as
well as under some natural assumptions on μ.
Received: 19 September 2000 相似文献
15.
令M是Z[v]的由v-1和奇素数p生成的理想,U是A=Z[v]M上相伴于对称Cartan矩阵的量子代数.k是特征为零的代数闭域,A→k(v(?)ξ)是环同态.U_k=U(?)_Ak,u_k是U_k的无穷小量子代数.令ξ是1的p次本原根.本文证明了:若有限维可积U_k模M,V中至少有一个是内射模,或者M,V中有一个模作为u_k模是平凡的,则有U_k模同构M(?)V≌V(?)M.我们还证明了:若有限维可积U_k模V作为u_k模是不可分解的,有限维可积U_k模M是不可分解的,且M|_(uk)是平凡的,则V(?)M是不可分解U_k模.令V和M是有限维可积U_k模,作为u_k模是同构的且具有单基座,本文证明V和M作为U_k模也是同构的.由此得到:不可分解内射u_k模提升为U_k模是唯一的. 相似文献
16.
Global and local Weyl modules were introduced via generators and relations in the context of affine Lie algebras in [CP2] and were motivated by representations of quantum affine algebras. In [FL] a more general case was considered by replacing the polynomial ring with the coordinate ring of an algebraic variety and
partial results analogous to those in [CP2] were obtained. In this paper we show that there is a natural definition of the local and global Weyl modules via homological
properties. This characterization allows us to define the Weyl functor from the category of left modules of a commutative
algebra to the category of modules for a simple Lie algebra. As an application we are able to understand the relationships
of these functors to tensor products, generalizing results in [CP2] and [FL]. We also analyze the fundamental Weyl modules and show that, unlike the case of the affine Lie algebras, the Weyl functors
need not be left exact. 相似文献
17.
We construct bar-invariant Z[q ±1/2 ]-bases of the quantum cluster algebra of Kronecker quiver which are quantum analogues of the canonical basis, semicanonical basis and dual semicanonical basis of the corresponding cluster algebra. As a byproduct, we prove positivity of the elements in these bases. 相似文献
18.
Ya. P. Pugay 《Theoretical and Mathematical Physics》1994,100(1):900-911
We present Feigin's construction [Lectures given in Landau Institute] of latticeW algebras and give some simple results: lattice Virasoro andW
3 algebras. For the simplest caseg=sl(2), we introduce the wholeU
q(2)) quantum group on this lattice. We find the simplest two-dimensional module as well as the exchange relations and define the lattice Virasoro algebra as the algebra of invariants ofU
q(sl(2)). Another generalization is connected with the lattice integrals of motion as the invariants of the quantum affine groupU
q(ñ+). We show that Volkov's scheme leads to a system of difference equations for a function of non-commutative variables.Landau Institute for Theoretical Physics, 142432, Chernogolovka, Russia. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 1, pp. 132–147, July, 1994. 相似文献
19.
H. T. Koelink 《Acta Appl Math》1996,44(3):295-352
Generalised matrix elements of the irreducible representations of the quantum SU(2) group are defined using certain orthonormal bases of the representation space. The generalised matrix elements are relatively infinitesimal invariant with respect to Lie algebra like elements of the quantised universal enveloping algebra of sl(2). A full proof of the theorem announced by Noumi and Mimachi [Proc. Japan Acad. Sci. Ser. A
66 (1990), 146–149] describing the generalised matrix elements in terms of the full four-parameter family of Askey-Wilson polynomials is given. Various known and new applications of this interpretation are presented.Supported by a NATO-Science Fellowship of the Netherlands Organization for Scientific Research (NWO). 相似文献
20.
Suemi Rodríguez-Rorno 《代数通讯》2013,41(9):4307-4325
Following the method already developed for studying the actions of GLq (2,C) on the Clifford algebra C(l,3) and its quantum invariants [1], we study the action on C(l, 3) of the quantum GL 2 constructed by Dipper and Donkin [2]. We are able of proving that there exits only two non-equivalent cases of actions with nontrivial “perturbation” [1]. The spaces of invariants are trivial in both cases. We also prove that each irreducible finite dimensional algebra representation of the quantum GL 2 qm ≠1, is one dimensional. By studying the cases with zero “perturbation” we find that the cases with nonzero “perturbation” are the only ones with maximal possible dimension for the operator algebra ?. 相似文献