Schlesinger Transformations for Linearisable Equations

A simple axiomatic characterization of the noncommutative Itô algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. It is proved that every Itô algebra with a quotient identity has a faithful representation in a Minkowski space and is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Lévy (Poisson) algebra. In particular, every quantum thermal noise of a finite number of degrees of freedom is the orthogonal sum of a quantum Wiener noise and a quantum Poisson noise as it is stated by the Lévy–Khinchin Theorem in the classical case. Two basic examples of noncommutative Itô finite group algebras are considered.     

Quantum Double for Quasi-Hopf Algebras

We introduce a quantum double quasi-triangular quasi-Hopf algebra D(H) associated to any quasi-Hopf algebra H. The algebra structure is a cocycle double cross-product. We use categorical reconstruction methods. As an example, we recover the quasi-Hopf algebra of Dijkgraaf, Pasquier and Roche as the quantum double D(G) associated to a finite group G and group 3-cocycle .     

差错基、量子码与群代数

本文找到了一种研究优质差错基和量子纠错码的新方法,即群代数方法, 它为差错基和量子码提供了一种代数表示. 利用这种代数表示, 建立了一系列关于最一般量子纠错码的线性规划限. 关键词： 群代数 差错基 量子纠错码 量子信息     

Center for Elliptic Quantum Group Eτ,η(sln)

We give the center of the elliptic quantum group in general cases. Based on the dynamical Yang-Baxter relation and the fusion method, we prove that the center commutes with all generators of the elliptic quantum group. Then for a kind of assumed form of these generators, we find that the coefficients of these generators form a new type of closed algebra. We also give the center for the algebra.     

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.     

A van Est Isomorphism for Bicrossed Product Hopf Algebras

To any locally finite representation of a given double crossed sum (product) Lie algebra (group), we associate a stable anti Yetter-Drinfeld (SAYD) module over the bicrossed product Hopf algebra which arises from the semidualization procedure. We prove a van Est isomorphism between the relative Lie algebra cohomology of the total Lie algebra and the Hopf cyclic cohomology of the corresponding Hopf algebra with coefficients in the associated SAYD module.     

Universal Framework for Quantum Error-Correcting Codes

We present a universal framework for quantum error-correcting codes, i.e., a framework that applies to the most general quantum error-correcting codes. This framework is based on the group algebra, an algebraic notation associated with nice error bases of quantum systems. The nicest thing about this framework is that we can characterize the properties of quantum codes by the properties of the group algebra. We show how it characterizes the properties of quantum codes as well as generates some new results about quantum codes.     

Hochschild cohomology of the Weyl algebra and Vasiliev’s equations

We propose a simple injective resolution for the Hochschild complex of the Weyl algebra. By making use of this resolution, we derive explicit expressions for nontrivial cocycles of the Weyl algebra with coefficients in twisted bimodules as well as for the smash products of the Weyl algebra and a finite group of linear symplectic transformations. A relationship with the higher-spin field theory is briefly discussed.     

On a Trialgebraic Deformation of the Manin Plane

Quantum groups (or more precisely, function algebras on quantum groups), i.e. bialgebras with certain additional properties, can be gained by deforming an appropriate function algebra on a group. In a similar way, we show that a polynomial-like algebra on the (function algebra of the) Manin plane leads to a so-called trialgebra (as suggested by Crane and Frenkel), i.e. an algebraic structure possessing a coproduct and two products in a compatible way. We show how to deform this trialgebra to a noncocommutative and totally noncommutative (i.e. in both products) one. Trialgebras are of interest for various reasons, e.g. the search for topological invariants for four manifolds or the duality operation for non-Abelian lattice gauge theories, recently suggested by the authors.     

Norms in group algebras of finite groups under inducing

We compare norms of an element of a group algebra of a normal subgroup of a finite group in a representation of the normal subgroup and the corresponding induced representation (under the natural embedding of the group algebra of the normal subgroup in the group algebra of the entire group).     

Quantized superfields

We construct quantized free superfields and represent them as operator‐valued distributions in Fock space starting with the Majorana field. We then analyse the algebras generated by free component quantum fields together with the Susy generators Q, . This enables us to obtain the quantized chiral superfield by finite Susy transformation from its scalar component. To get hermitian superfields we study by the same method a second scalar field algebra from which various scalar superfields can be obtained by exponentiation. Next we investigate the vector algebra and use it to construct the massive vector superfield. Surprisingly enough, the result is totally different from the vector multiplet in the literature. It contains two hermitian four‐vector components instead of one and a spin‐3/2 field similar to the gravitino in supergravity.     

Entropy of Quantum States

Given the algebra of observables of a quantum system subject to selection rules, a state can be represented by different density matrices. As a result, different von Neumann entropies can be associated with the same state. Motivated by a minimality property of the von Neumann entropy of a density matrix with respect to its possible decompositions into pure states, we give a purely algebraic definition of entropy for states of an algebra of observables, thus solving the above ambiguity. The entropy so-defined satisfies all the desirable thermodynamic properties and reduces to the von Neumann entropy in the quantum mechanical case. Moreover, it can be shown to be equal to the von Neumann entropy of the unique representative density matrix belonging to the operator algebra of a multiplicity-free Hilbert-space representation.     

An Algebra of Effects in the Formalism of Quantum Mechanics on Phase Space

Defining an addition of the effects in the formalism of quantum mechanics on phase space, we obtain a new effect algebra that is strictly contained in the effect algebra of all effects. A new property of the phase space formalism comes to light, namely that the new effect algebra does not contain any pair of noncommuting projections. In fact, in this formalism, there are no nontrivial projections at all. We illustrate this with the spin-1/2 algebra and the momentum/position algebra. Next, we equip this algebra of effects with the sequential product and get an interpretation of why certain properties fail to hold. PACS: 02.10.Gd, 03.65.Bz. This paper was a submission to the Fifth International Quantum Structure Association Conference (QS5), which took place in Cesena, Italy, March 31–April 5, 2001.     

De Morgan Property for Effect Algebras of von Neumann Algebras

It is shown that the unit interval of a von Neumann algebra is a Sum Brouwer–Zadeh algebra when equipped with another unary operation sending each element to the complement of its range projection. The main result of this Letter says that a von Neumann algebra is finite if and only if the corresponding Brouwer–Zadeh structure is de Morgan or, equivalently, if the range projection map preserves infima in the unit interval. This provides a new characterization of finiteness in the Murray–von Neumann structure theory of von Neumann algebras in terms of Brouwer–Zadeh structures.     

Hidden quantum groups inside Kac-Moody algebra

A lattice analogue of the Kac-Moody algebra is constructed. It is shown that the generators of the quantum algebra with the deformation parameterq=exp(iπ/k+h) can be constructed in terms of generators of the lattice Kac-Moody algebra (LKM) with the central chargek. It appears that there exists a natural correspondence between representations of the LKM algebra and the finite dimensional quantum group. The tensor product for representations of the LKM algebra and the finite dimensional quantum algebra is suggested.     

Noncommutative Unification of General Relativity and Quantum Mechanics. A Finite Model

We construct a model unifying general relativity and quantum mechanics in a broader structure of noncommutative geometry. The geometry in question is that of a transformation groupoid given by the action of a finite group on a space E. We define the algebra of smooth complex valued functions on , with convolution as multiplication, in terms of which the groupoid geometry is developed. Owing to the fact that the group G is finite the model can be computed in full details. We show that by suitable averaging of noncommutative geometric quantities one recovers the standard space-time geometry. The quantum sector of the model is explored in terms of the regular representation of the algebra , and its correspondence with the standard quantum mechanics is established.     

On the Finite-Dimensional Quantum Group M3⊕(M2|1(Λ2))0

We describe a few properties of the nonsemisimple associative algebra =M₃ (M_2|1 (²))₀, where ² is the Grassmann algebra with two generators. We show that is not only a finite-dimensional algebra but also a (noncommutative) Hopf algebra, hence a finite-dimensional quantum group. By selecting a system of explicit generators, we show how it is related with the quantum enveloping of SL_q(2) when the parameter q is a cubic root of unity. We describe its indecomposable projective representations as well as the irreducible ones. We also comment about the relation between this object and the theory of modular representation of the group SL(2, F₃), i.e. the binary tetrahedral group. Finally, we briefly discuss its relation with the Lorentz group and, as already suggested by A.~Connes, make a few comments about the possible use of this algebra in a modification of the Standard Model of particle physics (the unitary group of the semisimple algebra associated with is U(3) × U(2) × U(1)).     

Field Theory on Kappa-spacetime

A general formalism is developed, that allows the construction of field theory on quantum spaces which are deformations of ordinary spacetime. The symmetry group of spacetime is replaced by a quantum group. This formalism is demonstrated for the -deformed Poincaré algebra and its quantum space. The algebraic setting is mapped to the algebra of functions of commuting variables with a suitable -product. Fields are elements of this function algebra. As an example, the Klein-Gordon equation is defined and derived from an action.     

Quasi-idempotent Rota–Baxter operators arising from quasi-idempotent elements

In this short note, we construct quasi-idempotent Rota–Baxter operators by quasi-idempotent elements and show that every finite dimensional Hopf algebra admits nontrivial Rota–Baxter algebra structures and tridendriform algebra structures. Several concrete examples are provided, including finite quantum groups and Iwahori–Hecke algebras.