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941.
Michael E. Hoffman 《Journal of Algebraic Combinatorics》2000,11(1):49-68
Given a locally finite graded set A and a commutative, associative operation on A that adds degrees, we construct a commutative multiplication * on the set of noncommutative polynomials in A which we call a quasi-shuffle product; it can be viewed as a generalization of the shuffle product III. We extend this commutative algebra structure to a Hopf algebra (U, *, ); in the case where A is the set of positive integers and the operation on A is addition, this gives the Hopf algebra of quasi-symmetric functions. If rational coefficients are allowed, the quasi-shuffle product is in fact no more general than the shuffle product; we give an isomorphism exp of the shuffle Hopf algebra (U, III, ) onto (U, *, ) the set L of Lyndon words on A and their images { exp(w) w L} freely generate the algebra (U, *). We also consider the graded dual of (U, *, ). We define a deformation *q of * that coincides with * when q = 1 and is isomorphic to the concatenation product when q is not a root of unity. Finally, we discuss various examples, particularly the algebra of quasi-symmetric functions (dual to the noncommutative symmetric functions) and the algebra of Euler sums. 相似文献
942.
Philippe Caldero 《Proceedings of the American Mathematical Society》2000,128(4):943-951
Let be a complex not a root of unity and be a semi-simple Lie -algebra. Let be the quantized enveloping algebra of Drinfeld and Jimbo, be its triangular decomposition, and the associated quantum group. We describe explicitly and as a quantum Weyl field. We use for this a quantum analogue of the Taylor lemma.
943.
Both building upon and revising previous literature, this paper formulates the general notion of a Borel subalgebra B of a quasi-hereditary algebra A. We present various general constructions of Borel subalgebras, establish a triangular factorization of A, and relate the concept to graded Kazhdan–Lusztig theories in the sense of Cline et al. (Tôhoku Math. J.
45 (1993), 511–534). Various interesting types of Borel subalgebras arise naturally in different contexts. For example, `excellent" Borel subalgebras come about by abstracting the theory of Schubert varieties. Numerous examples from algebraic groups, q-Schur algebras, and quantum groups are considered in detail. 相似文献
944.
Based on the 1/n-expansion derived in a previous paper, the displacement fluctuations are analyzed in a quantum n-vector model of anharmonic crystal in the large n regime. It is shown that in the ferroelectric phase the n limit of the local fluctuation field has faster large-distance correlation decay than its Hartree–Fock approximation. Also, the critical exponent of the global displacement fluctuation is strictly smaller there than the Hartree–Fock exponent. In particular, the displacement fluctuations may be normal in the ferroelectric phase in spite of the Hartree–Fock prediction. 相似文献
945.
The problem is the classification of the ideals of free differential algebras, or the associated quotient algebras, the q-algebras; being finitely generated, unital C-algebras with homogeneous relations and a q-differential structure. This family of algebras includes the quantum groups, or at least those that are based on simple (super) Lie or Kac–Moody algebras. Their classification would encompass the so far incompleted classification of quantized (super) Kac–Moody algebras and of the (super) Kac–Moody algebras themselves. These can be defined as singular limits of q-algebras, and it is evident that to deal with the q-algebras in their full generality is more rational than the examination of each singular limit separately. This is not just because quantization unifies algebras and superalgebras, but also because the points q=1 and q=–1 are the most singular points in parameter space. In this Letter, one of two major hurdles in this classification program has been overcome. Fix a set of integers n
1,...,n
k, and consider the space
of homogeneous polynomials of degree n
1 in the generator e
1, and so on. Assume that there are no constants among the polynomials of lower degree, in any one of the generators; in this case all constants in the space
have been classified. The task that remains, the more formidable one, is to remove the stipulation that there are no constants of lower degree. 相似文献
946.
It is known that the trigonometric Calogero–Sutherland model is obtained by the trigonometric limit (–1) of the elliptic Calogero–Moser model, where (1, ) is a basic period of the elliptic function. We show that for all square-integrable eigenstates and eigenvalues of the Hamiltonian of the Calogero–Sutherland model, if exp(2–1) is small enough then there exist square-integrable eigenstates and eigenvalues of the Hamiltonian of the elliptic Calogero–Moser model which converge to the ones of the Calogero–Sutherland model for the 2-particle and the coupling constant l is positive integer cases and the 3-particle and l=1 case. In other words, we justify the regular perturbation with respect to the parameter exp(2–1). With some assumptions, we show analogous results for N-particle and l is positive integer cases. 相似文献
947.
R. Plaga 《Foundations of Physics Letters》2000,13(5):461-476
Karl Popper proposed a way to test whether a proposed relation of a quantum-mechanical state to perceived reality in the Copenhagen interpretation (CI) of quantum mechanics—namely that the state of a particle is merely an expression of what is known about the system—is in agreement with all experimental facts. A conceptual flaw in Popper's proposal is identified and an improved version of his experiment (called Extension step 1)—which fully serves its original purpose—is suggested. The main purpose of this paper is to suggest to perform this experiment. The results of this experiment predicted under the alternative assumptions that the CI or the many-worlds interpretation (MWI) is correct are shown to be identical. Only after a further modification (called Extension step 2) (the use of an ion isolated from the macroscopic environment as particle detector) the predictions using the respective interpretations become qualitatively different. This is because what is known by a human observer H can fail as a basis for the prediction of the statistical distribution of measurement results within the MWI in special cases: The temporal evolution of a system un-entangled with H (like the isolated ion) can depend on another system's state components that are entangled with states ortogonal to H. Thus—within the CI—for H they are known not to exist. Yet H can infer their existence by studying the evolution of the ion. 相似文献
948.
A. I. Belousova Yu. E. Lozovikb 《The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics》2000,8(2):251-264
A two dimensional (2D) classical system of dipole particles confined by a quadratic potential is studied. This system can
be used as a model for rare electrons in semiconductor structures near a metal electrode, indirect excitons in coupled quantum
dots etc. For clusters of N ≤ 80 particles ground state configurations and appropriate eigenfrequencies and eigenvectors for the normal modes are found.
Monte-Carlo and molecular dynamic methods are used to study the order-disorder transition (the “melting” of clusters). In
mesoscopic clusters (N < 37) there is a hierarchy of transitions: at lower temperatures an intershell orientational disordering of pairs of shells
takes place; at higher temperatures the intershell diffusion sets in and the shell structure disappears. In “macroscopic”
clusters (N > 37) an orientational “melting” of only the outer shell is possible. The most stable clusters (having both maximal lowest
nonzero eigenfrequencies and maximal temperatures of total melting) are those of completed crystal shells which are concentric
groups of nodes of 2D hexagonal lattice with a number of nodes placed in the center of them. The picture of disordering in
clusters is compared with that in an infinite 2D dipole system. The study of the radial diffusion constant, the structure
factor, the local minima distribution and other quantities shows that the melting temperature is a nonmonotonic function of
the number of particles in the system. The dynamical equilibrium between “solid-like” and “orientationally disordered” forms
of clusters is considered. 相似文献
949.
R. Unanyan S. Guérin B. W. Shore K. Bergmann 《The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics》2000,8(3):443-449
In traditional schemes of multilevel multilaser excitation, each laser pulse interacts with only one pair of states, and the
rotating wave approximation (RWA) is applicable. Here we study the population transfer process in a three-state system when
each of the two lasers interacts with each of the pair of states and when the Rabi frequencies characterizing the interaction
strengths of the system are comparable to or larger than the difference of the transition frequencies. We show that complete
and robust population transfer is possible under conditions more general than those hitherto considered necessary for stimulated
Raman adiabatic passage (STIRAP) or for successive π pulses. Using adiabatic Floquet theory we show that successful population transfer can be interpreted as adiabatic passage
by means of a transfer state which connects the initial and final states. The Floquet picture offers a convenient interpretation
of the population transfer as accompanied by multiple absorption of photons from or emission into the laser fields. 相似文献
950.