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We study the manifold of clusters of nonintersecting congruent solid bodies, all touching the central ball $B\subset\mathbb{R}^{3}$ of radius one. Two main examples are clusters of balls and clusters of infinite cylinders. We introduce the notion of \textit{critical cluster} and we study several critical clusters of balls and of cylinders. For the case of cylinders some of our critical clusters are new. We also establish the criticality properties of clusters, introduced earlier by W. Kuperberg., Comment: arXiv admin note: text overlap with arXiv:1812.09543

We study the path $$\Gamma =\{ C_{6,x}\mid x\in [0,1]\}$$ in the moduli space of configurations of six equal cylinders touching the unit sphere. Among the configurations $$C_{6,x}$$ is the record configuration $$C_{\mathfrak {m}}$$ of Ogievetsky and Shlosman (Discrete Comput Geom 2019, https://doi.org/10.1007/s00454-019-00064-3 ). We show that $$C_{\mathfrak {m}}$$ is a local sharp maximum of the distance function, so in particular the configuration $$C_{\mathfrak {m}}$$ is not only unlockable but rigid. We show that if $${(1 + x) (1 + 3 x)}/{3}$$ is a rational number but not a square of a rational number, the configuration $$C_{6,x}$$ has some hidden symmetries, part of which we explain.

Motivated by a question of W. Kuperberg, we study the 18-dimensional manifold of configurations of six non-intersecting infinite cylinders of radius r, all touching the unit ball in $$\mathbb {R}^{3}$$ . We find a configuration with $$\begin{aligned} r=\frac{1}{8}\Big ( 3+\sqrt{33}\Big ) \approx 1.093070331. \end{aligned}$$ We believe that this value is the maximum possible.

We establish the analogue of the Cayley--Hamilton theorem for the quantum matrix algebras of the symplectic type., Comment: arXiv admin note: text overlap with arXiv:math/0511618

We establish two versions of the fusion procedure for the walled Brauer algebras. In each of them, a complete system of primitive pairwise orthogonal idempotents for the walled Brauer algebra is constructed by consecutive evaluations of a rational function in several variables on contents of standard walled tableaux.

We define contravariant forms on diagonal reduction algebras, algebras of h -deformed differential operators and on standard modules over these algebras. We study properties of these forms and their specializations. We show that the specializations of the forms on the spaces of h -commuting variables present zero singular vectors iff they are in the kernel of the specialized form. As an application we compute norms of highest weight vectors in the tensor product of an irreducible finite dimensional representation of the Lie algebra gl n with a symmetric or wedge tensor power of its fundamental representation.

Properties of relative traces and symmetrizing forms on chains of cyclotomic and affine Hecke algebras are studied. The study relies on the use of bases of these algebras which generalize a normal form for elements of the complex reflection groups G ( m , 1 , n ) , m = 1 , 2 , … , ∞ , constructed by a recursive use of the Coxeter–Todd algorithm. Formulas for inducing, from representations of an algebra in the chain, representations of the next member of the chain are presented.

L'anneau $\Diff(n)$ des operateurs differentiels $\h$-deformes apparait dans la theorie des algebres de reduction.Dans cette these, nous construisons les anneaux des operateurs differentiels generalises sur les espaces vectoriels $\h$-deformes de type $\gl$. Contrairement aux espaces vectoriels $q$-deformes pour lequel l'anneau des operateurs differentiels est unique \`a isomorphisme pr\`es, l'anneau generalise des operateurs differentiels $\h$-deformes $\Diffs(n)$ est indexee par une fonction rationnelle $\sigma$ en $n$ variables, solution d'un syst\`eme d\'eg\'en\'er\'e d'\'equations aux diff\'erences finies. Nous obtenons la solution g\'en\'erale de ce syst\`eme. Nous montrons que le centre de $\Diffs(n)$ est un anneau des polynomes en $n$ variables. Nous construisons un isomorphisme entre des localisations de l'anneau $\Diffs(n)$ et de l’algebre de Weyl $\text{W}_n$ l’etendue par $n$ indetermines. Nous presentons des conditions irreductibilite des modules de dimension fini de $\Diffs(n)$. Finalement, nous discutons des difficultes a trouver les constructions analogues pour l'anneau $\Diff(n,N)$ correspondant \`a $N$ copies de $\Diff(n)$.

A complete system of primitive pairwise orthogonal idempotents for the Coxeter groups of type B B and, more generally, for the complex reflection groups G ( m , 1 , n ) G(m,1,n) is constructed by a sequence of evaluations of a rational function in several variables with values in the group ring. The evaluations correspond to the eigenvalues of the two arrays of Jucys–Murphy elements.

For a Coxeter system (G, S) the multi-parametric alternating subalgebra H +(G) of the Hecke algebra and the alternating subgroup ℬ+(G) of the braid group are defined. Two presentations for H +(G) and ℬ+(G) are given; one generalizes the Bourbaki presentation for the alternating subgroups of Coxeter groups, another one uses generators related to edges of the Coxeter graph.

A complete system of pairwise orthogonal minimal idempotents for Birman–Murakami–Wenzl algebras is obtained by a consecutive evaluation of a rational function in several variables on sequences of quantum contents of up-down tableaux. A byproduct of the construction is a one-parameter family of fusion procedures for Hecke algebras. Classical limits to two different fusion procedures for Brauer algebras are described.

We establish the absence of zero divisors in the reduction algebra of a Lie algebra ${\mathfrak{g}}$ with respect to its reductive Lie subalgebra ${\mathfrak{k}}$ . We identify the field of fractions of the diagonal reduction algebra of ${\mathfrak{sl}}_2$ with the standard skew field; as a by-product we obtain a two-parametric family of realizations of this diagonal reduction algebra by differential operators. We also present a new proof of the Poincare–Birkhoff–Witt theorem for reduction algebras.

We propose an inductive approach to the representation theory of the chain of complex reflection groups G(m, 1, n). We obtain the Jucys-Murphy elements of G(m, 1, n) from the Jucys-Murphy elements of the cyclotomic Hecke algebra and study their common spectrum using representations of a degenerate cyclotomic affine Hecke algebra. We construct representations of G(m, 1, n) using a new associative algebra whose underlying vector space is the tensor product of the group ring ℂG(m, 1, n) with a free associative algebra generated by the standard m-tableaux.

The Burman-Wenzl-Murakami algebra, considered as the quotient of the braid group algebra, possesses the commutative set of Jucys-Murphy elements. We show that the set of Jucys-Murphy elements is maximal commutative for the generic Birman-Wenzl-Murakami algebra and reconstruct the representation theory of the tower of Birman-Wenzl-Murakami algebras.

International audience; An approach, based on Jucys-Murphy elements, to the representation theory of cyclotomic Hecke algebras is developed. The maximality (in the cyclotomic Hecke algebra) of the set of the Jucys-Murphy elements is established. A basis of the cyclotomic Hecke algebra is suggested; this basis is used to establish the flatness of the deformation without using the representation theory.

We describe the diagonal reduction algebra D(gl n ) of the Lie algebra gl n in the R-matrix formalism. As a byproduct we present two families of central elements and the braided bialgebra structure of D(gl n ).

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.

We continue our study of quantum Lie algebras, an important class of quadratic algebras arising in the Woronowicz calculus on a quantum group. Quantum Lie algebras are generalizations of Lie (super)algebras. Many notions from the theory of Lie (super)algebras admit "quantum" analogues. In particular, there is a BRST operator Q(Q2=0) which generates the differential in the Woronowicz theory and gives information about (co)homologies of quantum Lie algebras. In our previous papers a recurrence relation for the operator Q for quantum Lie algebras was given. Here we solve this recurrence relation and obtain an explicit formula for the BRST operator.

We give the inflation rules for the decorated Mosseri—Sadoc tiles in the projection class of tilings T ( MS ) . Dehn invariants related to the stone inflation of the Mosseri–Sadoc tiles provide eigenvectors of the inflation matrix with eigenvalues equal to τ =(1+√5)/2 and (− τ −1 ).

The quantum dynamical Yang-Baxter (or Gervais-Neveu-Felder) equation defines an R-matrix R(p), where $p$ stands for a set of mutually commuting variables. A family of SL(n)-type solutions of this equation provides a new realization of the Hecke algebra. We define quantum antisymmetrizers, introduce the notion of quantum determinant and compute the inverse quantum matrix for matrix algebras of the type R(p) a_1 a_2 = a_1 a_2 R. It is pointed out that such a quantum matrix algebra arises in the operator realization of the chiral zero modes of the WZNW model., 28 pages, LaTeX

The Cayley-Hamilton-Newton identities which generalize both the characteristic identity and the Newton relations have been recently obtained for the algebras of the RTT-type. We extend this result to a wider class of algebras M(R,F) defined by a pair of compatible solutions of the Yang-Baxter equation. This class includes the RTT-algebras as well as the Reflection equation algebras.

The q-generalizations of the two fundamental statements of matrix algebra -- the Cayley-Hamilton theorem and the Newton relations -- to the cases of quantum matrix algebras of an "RTT-" and of a "Reflection equation" types have been obtained in [2]-[6]. We construct a family of matrix identities which we call Cayley-Hamilton-Newton identities and which underlie the characteristic identity as well as the Newton relations for the RTT- and Reflection equation algebras, in the sence that both the characteristic identity and the Newton relations are direct consequences of the Cayley-Hamilton-Newton identities.

A complete system of primitive pairwise orthogonal idempotents for cyclotomic Hecke algebras is constructed by consecutive evaluations of a rational function in several variables on quantum contents of multi-tableaux. This function is a product of two terms, one of which depends only on the shape of the multi-tableau and is proportional to the inverse of the corresponding Schur element.

International audience; Given a partially ordered set S, we define, for a linear extension P of S, a multivariate polynomial, counting certain reverse partitions on S called P-pedestals. We establish a remarkable property of this polynomial: it does not depend on the choice of P. For S a Young diagram, we show that this polynomial generalizes the hook polynomial.

Quantum Lie algebras related to multi-parametric Drinfeld-Jimbo $R$-matrices of type $GL(m|n)$ are classified., Comment: 9 pages

Let G be a discrete Coxeter group, G + its alternating subgroup and G + the spinor cover of G + . A presentation of the groups G + and G + is proved for an arbitrary Coxeter system ( G , S ) ; the generators are related to edges of the Coxeter graph. Results of the Coxeter–Todd algorithm–with this presentation–for the chains of alternating groups of types A, B and D are given.

We give a new fusion procedure for the Brauer algebra by showing that all primitive idempotents can be found by evaluating a rational function in several variables which has the form of a product of R-matrix type factors. In particular, this provides a new fusion procedure for the symmetric group involving an arbitrary parameter. The R-matrices are solutions of the Yang--Baxter equation associated with the classical Lie algebras g_N of types B, C and D. Moreover, we construct an evaluation homomorphism from a reflection equation algebra B(g_N) to U(g_N) and show that the fusion procedure provides an equivalence between natural tensor representations of B(g_N) with the corresponding evaluation modules., Comment: 31 pages

Let $V$ be a braided vector space, that is, a vector space together with a solution $\hat{R}\in {\text{End}}(V\otimes V)$ of the Yang--Baxter equation. Denote $T(V):=\bigoplus_k V^{\otimes k}$. We associate to $\hat{R}$ a solution $T(\hat{R})\in {\text{End}}(T(V)\otimes T(V))$ of the Yang--Baxter equation on the tensor space $T(V)$. The correspondence $\hat{R}\rightsquigarrow T(\hat{R})$ is functorial with respect to $V$., Comment: 10 pages, no figures

Theq-differential calculus for theq-Minkowski space is developed. The algebra of theq-derivatives with theq-Lorentz generators is found giving theq-deformation of the Poincare algebra. The reality structure of theq-Poincare algebra is given. The reality structure of theq-differentials is also found. The real Laplacian is constructed. Finally the comultiplication, counit and antipode for theq-Poincare algebra are obtained making it a Hopf algebra.

We construct for all $N$ a solution of the Frenkel--Moore $N$--simplex equation which generalizes the $R$--matrix for the Jordanian quantum group., Comment: 6 pages

We prove that the rings of q-differential operators on quantum planes of the GL q (n) and SO q (n) types are isomorphic to the rings of classical differential operators. Also, we construct decompositions of the rings of q-differential operators into tensor products of the rings of q-differential operators with less variables.

An algebra given by quadratic relations in a polynomial algebra on infinite set of generators is introduced. Using it, we prove some explicit formulas for the coefficients of the Sturm sequence of a polynomial. In the second part we discuss a numerical example of polynomials studied by Euler. There, the Hilbert matrices and the Cauchy determinants appear in the asymptotics of the Sturm sequence.

Multiplicative analogues of the shuffle elements of the braid group rings are introduced; in local representations they give rise to certain graded associative algebras (b-shuffle algebras). For the Hecke and BMW algebras, the (anti)-symmetrizers have simple expressions in terms of the multiplicative shuffles. The (anti)-symmetrizers can be expressed in terms of the highest multiplicative 1-shuffles (for the Hecke and BMW algebras) and in terms of the highest additive 1-shuffles (for the Hecke algebras). The spectra and multiplicities of eigenvalues of the operators of the multiplication by the multiplicative and additive 1-shuffles are examined., 18 pages

Quantum matrices in two dimensions, admitting left and right quantum spaces, are classified: they fall into two families, the 2-parametric family GLp,q(2) and a 1-parametric family GL infα sup J(2). Phenomena previously found for GLp,q(2) hold in this general situation: (a) powers of quantum matrices are again quantum and (b) entries of the logarithm of a two-dimensional quantum matrix form a Lie algebra.

Quantum groups have some peculiar properties is two dimensions. We formulate conditions sufficient for the product of two quantum matrices (with not necessarily the same values of deformation parameters) to be a quantum matrix again. This is then used to study the powers and exponential form of matrices fromGLp,q(2), generalising this way properties ofGLq(2)-matrices.

Some ingredients of the BRST construction for quantum Lie algebras are applied to a wider class of quadratic algebras of constraints. We build the BRST charge for a quantum Lie algebra with three generators and ghost-anti-ghosts commuting with constraints. We consider a one-parametric family of quadratic algebras with three generators and show that the BRST charge acquires the conventional form after a redefinition of ghosts. The modified ghosts form a quadratic algebra. The family possesses a non-linear involution, which implies the existence of two independent BRST charges for each algebra in the family. These BRST charges anticommute and form a double BRST complex., 10 pages

We construct a family of automorphisms of Mickelsson algebra, satisfying braid group relations. The construction uses ‘Zhelobenko cocycle’ and includes the dynamical Weyl group action as a particular case.

Non-polynomial Baxterized solutions of reflection equations associated with affine Hecke and affine Birman-Murakami-Wenzl algebras are found. Relations to integrable spin chain models with nontrivial boundary conditions are discussed., Comment: LaTeX, 18 pages, typos fixed, journal-ref added

The zero modes of the chiral SU(n) WZNW model give rise to an intertwining quantum matrix algebra A generated by an n x n matrix a=(a^i_\alpha) (with noncommuting entries) and by rational functions of n commuting elements q^{p_i}. We study a generalization of the Fock space (F) representation of A for generic q (q not a root of unity) and demonstrate that it gives rise to a model of the quantum universal enveloping algebra U_q(sl_n), each irreducible representation entering F with multiplicity 1. For an integer level k the complex parameter q is an even root of unity, q^h=-1 (h=k+n) and the algebra A has an ideal I_h such that the factor algebra A_h = A/I_h is finite dimensional., Comment: 48 pages, LaTeX, uses amsfonts; final version to appear in J. Phys. A

We introduce Dehn invariants as a useful tool in the study of the inflation of quasiperiodic space tilings. The tilings by ``golden tetrahedra'' are considered. We discuss how the Dehn invariants can be applied to the study of inflation properties of the six golden tetrahedra. We also use geometry of the faces of the golden tetrahedra to analyze their inflation properties. We give the inflation rules for decorated Mosseri-Sadoc tiles in the projection class of tilings ${\cal T}^{(MS)}$. The Dehn invariants of the Mosseri-Sadoc tiles provide two eigenvectors of the inflation matrix with eigenvalues equal to $\tau = \frac{1+\sqrt{5}}{2}$ and $-\frac{1}{\tau}$, and allow to reconstruct the inflation matrix uniquely., Comment: LaTeX file, 25 pages + 9 figures (Fig1.gif, Fig2.gif... Fig9.gif); hard copies with all figures are available from the authors

A quantum version of the Cayley-Hamilton theorem is found for the matrix T of the generators of the RTT-algebra. In the quasitriangular case, a connection between the characteristic identities in the RTT and RE-algebras is established.

The concept of symmetry groups has a mathematically well defined generalization in the framework of Hopf algebras. Such generalizations have become known as quantum groups- these are Hopf algebras with an algebraic structure which depends on one or more parameters q (q ∊ C,q ≠ 0), such that for a particular value of these parameters, say q = 1, the quantum group coincides with the group. In this sense a quantum group is a deformation of a group, q being a deformation parameter. With the concept of a group goes the concept of representations and representation spaces. These representation spaces find a natural generalization as well, called quantum spaces. These are algebraic structures that depend on the deformation parameter q and for q = 1 coincide with the linear space in which the corresponding group is represented. For q ≠ 1, the quantum group acts as a linear morphism of the algebraic structure of the quantum space. The algebraic structure of the group and the quantum space are closely related.