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.      

Tensor Products of Hecke Algebras

Tensor product of irreducible representations of Hecke algebras are discussed. It is found that the tensor product of irreps of Hecke algebras generates representations of Birman-Wenzl algebra C_ƒ(γ, q) with γ = q³ or-q^-3. A procedure for the evaluation of tensor product coefficients (TPC's) of H_ƒ (q)oH_ƒ(q) ↓ C_ƒ(γ,q) is established when the representations of Cƒ(γ, q) remain irreducible. An example of deriving TPC's of H_ƒ (q)oH_ƒ(q) ↓ C_ƒ(γ, q) is given. It is also found that indecomposable representation of C₄(γ q) occurs in the tensor product [211]o[31].     

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.     

Universal Differential Calculus on Ternary Algebras

General concept of ternary algebras is introduced in this article, along with several examples of its realization. Universal envelope of such algebras is defined, as well as the concept of tri-modules over ternary algebras. The universal differential calculus on these structures is then defined and its basic properties investigated.     

Reduction of Quantum Systems with Arbitrary First Class Constraints and Hecke Algebras

We propose a method for reduction of quantum systems with arbitrary first-class constraints. An appropriate mathematical setting for the problem is the homology of associative algebras. For every such algebra A and subalgebra B with augmentation ɛ there exists a cohomological complex which is a generalization of the BRST one. Its cohomology is an associative graded algebra Hk ^*(A,B) which we call the Hecke algebra of the triple (A,B,ɛ). It acts in the cohomology space H ^*(B,V) for every left A module V. In particular the zeroth graded component $Hk^{0}(A,B)$ acts in the space of B invariants of $V$ and provides the reduction of the quantum system. Received: 15 June 1998 / Accepted: 25 January 1999     

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.     

On Gibbs Measures of Models with Competing Ternary and Binary Interactions and Corresponding von Neumann Algebras

In the present paper a model with competing ternary (J ₂) and binary (J ₁) interactions with spin values ±1, on a Cayley tree is considered. One studies the structure of Gibbs measures for the model considered. It is known, that under some conditions on parameters J ₁,J ₂ (resp. in the opposite case) there are three (resp. a unique) translation-invariant Gibbs measures. We prove, that two of them (minimal and maximal) are extreme in the set of all Gibbs measures and also construct two periodic (with period 2) and uncountable number of distinct non-translation-invariant Gibbs measures. One shows that they are extreme. Besides, types of von Neumann algebras, generated by GNS-representation associated with diagonal states corresponding to extreme periodic Gibbs measures, are determined. Namely, it is shown that an algebra associated with the unordered phase is a factor of type III _λ , where λ=exp{?2βJ ₂}, β>0 is the inverse temperature. We find conditions, which ensure that von Neumann algebras, associated with the periodic Gibbs measures, are factors of type III _δ , otherwise they have type III₁.     

RSOS Models and Jantzen–Seitz Representations of Hecke Algebras at Roots of Unity

A special family of partitions occurs in two apparently unrelated contexts: the evaluation of one-dimensional configuration sums of certain RSOS models, and the modular representation theory of symmetric groups or their Hecke algebras H_m. We provide an explanation of this coincidence by showing how the irreducible H_m-modules which remain irreducible under restriction to H_{m_1} (Jantzen–Seitz modules) can be determined from the decomposition of a tensor product of representations sl_n.     

On Gibbs Measures of Models with Competing Ternary and Binary Interactions and Corresponding Von Neumann Algebras II

In the present paper the Ising model with competing binary (J) and binary (J₁) interactions with spin values ±1, on a Cayley tree of order 2 is considered. The structure of Gibbs measures for the model is studied. We completely describe the set of all periodic Gibbs easures for the model with respect to any normal subgroup of finite index of a group representation of the Cayley tree. Types of von Neumann algebras, generated by GNS-representation associated with diagonal states corresponding to the translation invariant Gibbs measures, are determined. It is proved that the factors associated with minimal and maximal Gibbs states are isomorphic, and if they are of type III then the factor associated with the unordered phase of the model can be considered as a subfactors of these factors respectively. Some concrete examples of factors are given too.     

On the Li Coefficients for the Hecke L-functions

In this paper, we compute and verify the positivity of the Li coefficients for the Hecke L-functions using an arithmetic formula established in Omar and Mazhouda, J. Number Theory 125(1), 50–58 (2007) and J. Number Theory 130(4), 1098–1108 (2010) and the Serre trace formula. Additional results are presented, including new formulas for the Li coefficients and a formulation of a criterion for the partial Riemann hypothesis. Basing on the numerical computations made below, we conjecture that these coefficients are increasing in n.     

On Fusion Algebras and Modular Matrices

We consider the fusion algebras arising in e.g. Wess–Zumino–Witten conformal field theories, affine Kac–Moody algebras at positive integer level, and quantum groups at roots of unity. Using properties of the modular matrix S, we find small sets of primary fields (equivalently, sets of highest weights) which can be identified with the variables of a polynomial realization of the A _r fusion algebra at level k. We prove that for many choices of rank r and level k, the number of these variables is the minimum possible, and we conjecture that it is in fact minimal for most r and k. We also find new, systematic sources of zeros in the modular matrix S. In addition, we obtain a formula relating the entries of S at fixed points, to entries of S at smaller ranks and levels. Finally, we identify the number fields generated over the rationals by the entries of S, and by the fusion (Verlinde) eigenvalues. Received: 7 October 1997 / Accepted: 7 March 1999     

On the Leibniz Homology of Poisson Algebras

We study the Leibniz homology of the Poisson algebra of polynomial functions over (²ⁿ ,) where is the standard symplectic structure. We identify it with certain highest-weight vectors of some _2n ( )-modules and obtain some explicit result in low degree.     

On Centralizer Algebras for Spin Representations

We give a presentation of the centralizer algebras for tensor products of spinor representations of quantum groups via generators and relations. In the even-dimensional case, this can be described in terms of non-standard q-deformations of orthogonal Lie algebras; in the odd-dimensional case only a certain subalgebra will appear. In the classical case q?= 1 the relations boil down to Lie algebra relations.     

On Ternary Fission Induced by Neutrons

Experiments on measuring the rotational effect of the ²³⁴U fissile nucleus at the scission point showed that the fissile nucleus rotates as a right screw with respect to the longitudinally polarized neutron beam direction in the ternary fission of the ²³³U target nucleus induced by polarized s-neutrons; in the binary fission of the same nuclei it rotates in the opposite direction. Moreover, it was found that ternary fission “prefers” the spin state of J = I +1/2. This phenomenon cannot be explained within the existing concepts of ternary fission as one of the two “final” states after neck rupture. The same “parent” ²³⁴U nucleus cannot rotate in opposite directions in the two different final states. It should be assumed that ternary fission is a special branch of descent from the saddle point to the point of neck rupture. It can also be assumed that this branch is formed at the saddle point in a configuration favorable for cluster formation. Why does it prefer the spin state of J = I + 1/2? This is an interesting question for further studies.

     

On the Classification of Automorphic Lie Algebras

The problem of reduction of integrable equations can be formulated in a uniform way using the theory of invariants. This provides a powerful tool of analysis and it opens the road to new applications of Automorphic Lie Algebras, beyond the context of integrable systems. In this paper it is shown that \mathfraksl₂(\mathbbC){\mathfrak{sl}_{2}(\mathbb{C})}–based Automorphic Lie Algebras associated to the icosahedral group \mathbb I{{\mathbb I}}, the octahedral group \mathbb O{{\mathbb O}}, the tetrahedral group \mathbb T{{\mathbb T}}, and the dihedral group \mathbb D_n{{\mathbb D}_n} are isomorphic. The proof is based on techniques from classical invariant theory and makes use of Clebsch-Gordan decomposition and transvectants, Molien functions and the trace-form. This result provides a complete classification of \mathfraksl₂(\mathbbC){\mathfrak{sl}_{2}(\mathbb{C})}–based Automorphic Lie Algebras associated to finite groups when the group representations are chosen to be the same and it is a crucial step towards the complete classification of Automorphic Lie Algebras.