Hecke Symmetries and Characteristic Relations on Reflection Equation Algebras

We analyze the relation between the properties of Hecke symmetry (i.e., Hecke type R-matrix) and the algebraic structure of the corresponding reflection equation (RE) algebra. Analogues of the Newton relations and Cayley–Hamilton theorem for the matrix of generators of the RE algebra associated with a finite rank even Hecke symmetry are derived.     

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。     

Baxterization of Solutions to Reflection Equation with Hecke R-matrix

Let R be a Hecke solution to the Yang–Baxter equation and K be a reflection equation matrix with coefficients in an associative algebra . Let R(x) be the baxterization of R and suppose that K satisfies a polynomial equation with coefficients in the center of . We construct solutions to the reflection equation with spectral parameter relative to R(x), in the form of polynomials in K.     

Indecomposable Representations of the Square—Root Lie Algebras of Vector Type and Their Boson Realizations

The explicit expressions for indecomposable representations of nine square-root Lie algebras of vector type, R_νλ (ν, λ=0, ±1), are obtained on the space of universal enveloping algebra of two-state Heisenberg-Weyl algebra, the invariant subspaces and the quotient spaces. From Fock representations corresponding to these indecomposable representations, the inhomogeneous boson realizations of R_νλ are given. The expectation values of R_νλ in the angular momentum coherent states are calculated as well as the corresponding classical limits.     

Tensor Product of Distributive Sequential Effect Algebras and Product Effect Algebras

A distributive sequential effect algebra (DSEA) is an effect algebra on which a distributive sequential product with natural properties is defined. We define the tensor product of two arbitrary DSEA’s and we give a necessary and sufficient condition for it to exist. As a corollary we obtain the result (see Gudder, S. in Math. Slovaca 54:1–11, 2004, to appear) that the tensor product of a pair of commutative sequential effect algebras exists if and only if they admit a bimorphism. We further obtain a similar result for the tensor product of a pair of product effect algebras.     

Tensor Products of Quantum Structures and Their Applications in Quantum Measurements

Tensor products of quantum logics and effect algebras with some known problems are reviewed. It is noticed that although tensor products of effect algebras having at least one state exist, in the category of complex Hilbert space effect algebras similar problems as with tensor products of projection lattices occur. Nevertheless, if one of the two coupled physical systems is classical, tensor product exists and can be considered as a Boolean power. Some applications of tensor products (in the form of Boolean powers) to quantum measurements are reviewed.     

The Structure of Affine Hecke Algebra HN for the XXZ Model with Boundary

A new affine Hecke algebra is found in studying the XXZ model with a diagonal boundary condition. In the special case of T = 0, we give the double structure of this new affine Hecke algebra, which is similar to the B_n type affine Hecke algebra.     

On representations of Hecke algebras

In this report we review some facts about representation theory of Hecke algebras. For Hecke algebras we adapt the approach of A. Okounkov and A. Vershik [Selecta Math., New Ser., 2 (1996) 581], which was developed for the representation theory of symmetric groups. We justify explicit construction of idempotents for Hecke algebras in terms of Jucys-Murphy elements. Ocneanu's traces for these idempotents (which can be interpreted as q-dimensions of corresponding irreducible representations of quantum linear groups) are presented. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005. This work was supported in part by the grants INTAS 03-51-3350 and RFBR 05-01-01086-a.     

Commutativity of Quantum Family Algebras

Recently, A. A. Kirillov introduced an important notion of classical and quantum family algebras. Here the criterion of commutativity is given. The quantum eigenvalues of are computed.     

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(英文)

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。     

Tensor products of Hilbert space effect algebras

A definition of a tensor product in the category of Hilbert space effect algebras is introduced such that the tensor product reflects as much as possible of the physically important properties of the components. It is shown that in the complex case, there are two candidates to the tensor product, which are not equivalent. The situation is similar to the tensor product in the category of projection lattices, but also the two dimensional case is included. The case of tensor product of two classical systems is also investigated.     

Some Tensor Product Representations of Uq(s $${\hat l}{\kern 1pt} $$ 2) at Roots of Unity

When q is a root of unity, a triangular decomposition of U _q(s ₂) is given and irreducibility conditions concerning some tensor product representations of U _q(s ₂) are presented. Their connection with physics is also pointed out.     

On the Idempotents of Hecke Algebras

We give a new construction of primitive idempotents of the Hecke algebras associated with the symmetric groups. The idempotents are found as evaluated products of certain rational functions thus providing a new version of the fusion procedure for the Hecke algebras. We show that the normalization factors which occur in the procedure are related to the Ocneanu–Markov trace of the idempotents.      

P-Twisted Affine Lie Algebras and Their Associated Vertex Algebras

For any finite-dimensional semisimple Lie algebra g, a Z＋-graded vertex algebra is construsted on the vacuum representation Vk（g[θ]of g[θ]）,which is a one-dimentionM central extension of 8-invariant subspace on the loop algebra Lg=g C（（t^1/p））.     

On Ternary Quotients of Cubic Hecke Algebras

We prove that the quotient of the group algebra of the braid group introduced by Funar (Commun Math Phys 173:513–558, 1995) collapses in characteristic distinct from 2. In characteristic 2 we define several quotients of it, which are connected to the classical Hecke and Birman-Wenzl-Murakami quotients, but which admit in addition a symmetry of order 3. We also establish conditions on the possible Markov traces factorizing through it.     

P-Twisted Affine Lie Algebras and Their Associated Vertex Algebras

For any finite-dimensional semisimple Lie algebra g, a Z₊-graded vertex algebra is construsted on the vacuum representation V_k(\hat{g}[ θ]) of \hat{g}[θ], which is a one-dimentional central extension of θ-invariant subspace on the loop algebra Lg=g\otimes C((t^1/p)).     

Product Effect Algebras

We introduce a product on an effect algebra. We prove that every product effect algebra with the Riesz decomposition property (RDP), is an interval in an Abelian unital interpolation po-ring, and we show that the category of product effect algebras with the RDP is categorically equivalent with the category of unital Abelian interpolation po-rings. In addition, we show that every product effect algebra with the RDP and with 1 as a product unity is a subdirect product of antilattice product effect algebras with the RDP.     

Entropy Treatment of Evolution Algebras

In this paper, by introducing an entropy of Markov evolution algebras, we treat the isomorphism of S-evolution algebras. A family of Markov evolution algebras is defined through the Hadamard product of structural matrices of non-negative real S-evolution algebras, and their isomorphism is studied by means of their entropy. Furthermore, the isomorphism of S-evolution algebras is treated using the concept of relative entropy.     

States on Operator Algebras and Axiomatic System of Quantum Theory

Recent development brings new results on the interplay of states on operator algebras and axiomatics of quantum mechanics. Neither hidden space in the sense of Kochen and Specker nor approximate hidden variables exist on von Neumann algebras. Tracial properties of states are connected with dispersions. The axioms on composite systems simplify to state extension properties.