Generalization of Drinfeld Quantum Affine Algebras

Drinfeld gave a current realization of the quantum affine algebras as a Hopf algebra with a simple comultiplication for the quantum current operators. In this Letter, we will present a generalization of such a realization of quantum Hopf algebras. As a special case, we will choose the structure functions for this algebra to be elliptic functions to derive certain elliptic quantum groups as a Hopf algebra, which degenerates into quantum affine algebras if we take certain degeneration of the structure functions.     

Quasi-Hopf Deformations of Quantum Groups

The search for elliptic quantum groups leads to a modified quantum Yang–Baxter relation and to a special class of quasi-triangular quasi-Hopf algebras. This Letter calculates deformations of standard quantum groups (with or without spectral parameter) in the category of quasi-Hopf algebras. An earlier investigation of the deformations of quantum groups, in the category of Hopf algebras, showed that quantum groups are generically rigid: Hopf algebra deformations exist only under some restrictions on the parameters. In particular, affine Kac–Moody algebras are more rigid than their loop algebra quotients and only the latter (in the case of sl(n)) can be deformed to elliptic Hopf algebras. The generalization to quasi-Hopf deformations lifts this restriction. The full elliptic quantum groups (with central extension) associated with sl(n) are thus quasi-Hopf algebras. The universal R-matrices satisfy a modified Yang–Baxter relation and are calculated more or less explicitly. The modified classical Yang–Baxter relation is obtained and the elliptic solutions are worked out explicitly.The same method is used to construct the Universal R-matrices associated with Felder's quantization of the Knizhnik–Zamolodchikov–Bernard equation, to throw some light on the quasi-Hopf structure of conformal field theory and (perhaps) the Calogero–Moser models.     

Quantum Lie Algebras, Their Existence, Uniqueness and q-Antisymmetry

Quantum Lie algebras are generalizations of Lie algebras which have the quantum parameter h built into their structure. They have been defined concretely as certain submodules of the quantized enveloping algebras . On them the quantum Lie product is given by the quantum adjoint action. Here we define for any finite-dimensional simple complex Lie algebra an abstract quantum Lie algebra independent of any concrete realization. Its h-dependent structure constants are given in terms of inverse quantum Clebsch-Gordan coefficients. We then show that all concrete quantum Lie algebras are isomorphic to an abstract quantum Lie algebra . In this way we prove two important properties of quantum Lie algebras: 1) all quantum Lie algebras associated to the same are isomorphic, 2) the quantum Lie product of any is q-antisymmetric. We also describe a construction of which establishes their existence. Received: 23 May 1996 / Accepted: 17 October 1996     

A remark on quasi-triangular Lie algebras

Quasi-triangular quantum Lie algebras are algebras of quantum Lie derivatives. We show how to associate a quasi-triangular quantum Lie algebra to any quantum group defined by exchange relations. This provides a systematical way for constructing bi-covariant differential calculus on exchange quantum groups.     

Jimbo's q-analogues and current algebras

We give a realization of the quantum groups of Jimbo and Drinfeld in terms of current algebras.     

Central Charge Extended Supersymmetric Structures for Fundamental Fermions Around Non-Abelian Vortices

Fermionic zero modes around non-abelian vortices are shown that they constitute two N = 2, d = 1 supersymmetric quantum mechanics algebras. These two algebras can be combined under certain circumstances to form a central charge extended N = 4 supersymmetric quantum algebra. We thoroughly discuss the implications of the existence of supersymmetric quantum mechanics algebras, in the quantum Hilbert space of the fermionic zero modes.     

Yangians,integrable quantum systems and Dorey's rule

It was pointed out by P. Dorey that the three-point couplings between the quantum particles in affine Toda field theories have a remarkable Lie-theoretic interpretation. It is also well known that such theories admit quantum affine algebras as quantum symmetry groups, and widely believed that the quantum particles correspond to the so-called fundamental representations of these algebras. This led to the conjecture that Dorey's rule should describe when a fundamental representation occurs with non-zero multiplicity in a tensor product of two other fundamental representations. The purpose of this paper is to prove this conjecture, both for quantum affine algebras and for Yangians. The result reveals a hitherto unsuspected role played by Coxeter elements (and their twisted analogues) in the representation theory of these algebras.     

A quiver quantum group

We construct quantum groups at a root of unity and we describe their monoidal module category using techniques from the representation theory of finite dimensional associative algebras.Supported by the FNSRS (Swiss National Foundation)     

Three lectures on quantum groups: representations, duality, real forms

In these lectures three topics are discussed: the representations of quantum groups, duality between quantum algebras and matrix quantum groups and q-deformations of real forms of quantum groups.     

Moment maps at the quantum level

We introduce the notion of moment maps for quantum groups acting on their module algebras. When the module algebras are quantizations of Poisson manifolds, we prove that the construction at the quantum level is a quantization of that at the semi-classical level. We also prove that the corresponding smashed product algebras are quantizations of the semi-direct product Poisson structures.Research partially supported by NSF grant DMS-89-07710     

Type-Decomposition of an Effect Algebra

Effect algebras (EAs), play a significant role in quantum logic, are featured in the theory of partially ordered Abelian groups, and generalize orthoalgebras, MV-algebras, orthomodular posets, orthomodular lattices, modular ortholattices, and boolean algebras. We study centrally orthocomplete effect algebras (COEAs), i.e., EAs satisfying the condition that every family of elements that is dominated by an orthogonal family of central elements has a supremum. For COEAs, we introduce a general notion of decomposition into types; prove that a COEA factors uniquely as a direct sum of types I, II, and III; and obtain a generalization for COEAs of Ramsay’s fourfold decomposition of a complete orthomodular lattice.     

黑克、布劳、及伯曼-温采尔代数的诱导及分导系数和量子经典李代数的耦合与重新耦合系数

本文总结了计算黑克、布劳、及伯曼 温采尔代数在各种工数链下诱导及分导系数的线性方程方法(LEM)。特别强调了关于A,B,C,D型李代数及其量子情形与其中心代数之间的舒尔 魏尔 布劳双关性关系。这一关系使我们能够利用相应中心代数的诱导及分导系数计算出经典李代数及其量子情形的耦合与重新耦合系数。讨论了从该方法得到B,C,D型李代数不可约表示克罗内克积分解的应用。基于LEM还得到了处理对应于置换群CG系列问题的黑克代数张量积的方法。     

Central extensions of classical and quantum q-Virasoro algebras

We investigate the central extensions of the q-deformed (classical and quantum) Virasoro algebras constructed from the elliptic quantum algebra A_q,p(sl(N))_c. After establishing the expressions of the cocycle conditions, we solve them, both in the classical and in the quantum case (for sl(2)). We find that the consistent central extensions are much more general that those found previously in the literature.     

Generalized Ehrenfest Relations,Deformation Quantization,and the Geometry of Inter-model Reduction

This study attempts to spell out more explicitly than has been done previously the connection between two types of formal correspondence that arise in the study of quantum–classical relations: one the one hand, deformation quantization and the associated continuity between quantum and classical algebras of observables in the limit \(\hbar \rightarrow 0\), and, on the other, a certain generalization of Ehrenfest’s Theorem and the result that expectation values of position and momentum evolve approximately classically for narrow wave packet states. While deformation quantization establishes a direct continuity between the abstract algebras of quantum and classical observables, the latter result makes in-eliminable reference to the quantum and classical state spaces on which these structures act—specifically, via restriction to narrow wave packet states. Here, we describe a certain geometrical re-formulation and extension of the result that expectation values evolve approximately classically for narrow wave packet states, which relies essentially on the postulates of deformation quantization, but describes a relationship between the actions of quantum and classical algebras and groups over their respective state spaces that is non-trivially distinct from deformation quantization. The goals of the discussion are partly pedagogical in that it aims to provide a clear, explicit synthesis of known results; however, the particular synthesis offered aspires to some novelty in its emphasis on a certain general type of mathematical and physical relationship between the state spaces of different models that represent the same physical system, and in the explicitness with which it details the above-mentioned connection between quantum and classical models.     

A Class of Baker–Akhiezer Arrangements

We study a class of arrangements of lines with multiplicities on the plane which admit the Chalykh–Veselov Baker–Akhiezer function. These arrangements are obtained by adding multiplicity one lines in an invariant way to any dihedral arrangement with invariant multiplicities. We describe all the Baker–Akhiezer arrangements when at most one line has multiplicity higher than 1. We study associated algebras of quasi-invariants which are isomorphic to the commutative algebras of quantum integrals for the generalized Calogero–Moser operators. We compute the Hilbert series of these algebras and we conclude that the algebras are Gorenstein. We also show that there are no other arrangements with Gorenstein algebras of quasi-invariants when at most one line has multiplicity bigger than 1.     

Quantum group symmetries and non-local currents in 2D QFT

We construct and study the implications of some new non-local conserved currents that exist is a wide variety of massive integrable quantum field theories in 2 dimensions, including the sine-Gordon theory and its generalization to affine Toda theory. These non-local currents provide a non-perturbative formulation of the theories. The symmetry algebras correspond to the quantum affine Kac-Moody algebras. TheS-matrices are completely characterized by these symmetries. FormalS-matrices for the imaginary-coupling affine Toda theories are thereby derived. The application of theseS-matrices to perturbed coset conformal field theory is studied. Non-local charges generating the finite dimensional Quantum Group in the Liouville theory are briefly presented. The formalism based on non-local charges we describe provides an algernative to the quantum inverse scattering method for solving integrable quantum field theories in 2d.     

Algebras with operator and Campbell-Hausdorff formula

We introduce some new classes of algebras and establish within these algebras Campbell-Hausdorff-like formulae. We describe the application of these constructions to the problem of the connectivity of the Feynman graphs corresponding to Green functions in quantum fields theory.     

Quantum group symmetry of partition functions of IRF models and its application to Jones' index theory

For each Boltzmann weight of a face model, we associate two quantum groups (face algebras) which describe the dependence of the partition function on boundary value condition. Using these, we give a proof of (non-)flatness of A-D-E connections of A. Ocneanu, which is a crucial algebraic part of the classification of subfactors with Jones' index less than 4.     

Deformed traces and covariant quantum algebras for quantum groups GLqp(2) and GLqp (1¦1)

The q-deformed traces and orbits for the two parametric quantum groups GL_qp(2) and GL_qp(1¦1) are defined, they are subsequently used in the construction of q-orbit invariants for these groups. General qp-(super)oscillator commutation relations are obtained, which remain invariant under the coactions of groups GL_qp(2) and GL_qp(1¦1). The GL_qp(2)-covariant deformed algebra is deduced in terms of the bilinears of bosonic qp-oscillators, which turns out to be a central extension of the Witten-type deformation of sl(2) algebra. In the case of the supergroup GL_qp(1¦1), the corresponding covariant algebras contain the N = 2 supersymmetric quantum mechanical subalgebras.