Three dimensional Sklyanin algebras and Gröbner bases

We consider Sklyanin algebras $S$ with 3 generators, which are quadratic algebras over a field $\K$ with $3$ generators $x,y,z$ given by $3$ relations $pxy+qyx+rzz=0$, $pyz+qzy+rxx=0$ and $pzx+qxz+ryy=0$, where $p,q,r\in\K$. this class of algebras has enjoyed much attention. In particular, using tools from algebraic geometry, Feigin, Odesskii \cite{odf}, and Artin, Tate and Van Den Bergh, showed that if at least two of the parameters $p$, $q$ and $r$ are non-zero and at least two of three numbers $p^3$, $q^3$ and $r^3$ are distinct, then $S$ is Artin--Schelter regular. More specifically, $S$ is Koszul and has the same Hilbert series as the algebra of commutative polynomials in 3 indeterminates (PHS). It has became commonly accepted that it is impossible to achieve the same objective by purely algebraic and combinatorial means like the Groebner basis technique. The main purpose of this paper is to trace the combinatorial meaning of the properties of Sklyanin algebras, such as Koszulity, PBW, PHS, Calabi-Yau, and to give a new constructive proof of the above facts due to Artin, Tate and Van Den Bergh. Further, we study a wider class of Sklyanin algebras, namely
the situation when all parameters of relations could be different. We call them generalized Sklyanin algebras. We classify up to isomorphism all generalized Sklyanin algebras with the same Hilbert series as commutative polynomials on
3 variables. We show that generalized Sklyanin algebras in general position have a Golod–Shafarevich Hilbert series (with exception of the case of field with two elements).

Reconstruction of graded groupoids from graded Steinberg algebras

We show how to reconstruct a graded ample Hausdorff groupoid with topologically principal neutrally-graded component from the ring structure of its graded Steinberg algebra over any commutative integral domain with 1, together with the embedding of the canonical abelian subring of functions supported on the unit space. We deduce that diagonal-preserving ring isomorphism of Leavitt path algebras implies $C^*$-isomorphism of $C^*$-algebras for graphs $E$ and $F$ in which every cycle has an exit. This is a joint work with Joan Bosa, Roozbeh Hazrat and Aidan Sims.

Directed Abelian algebras and their application to stochastic models

With each directed acyclic graph (this includes some D-dimensional lattices) one can associate some Abelian algebras that we call directed Abelian algebras (DAAs). On each site of the graph one attaches a generator of the algebra. These algebras depend on several parameters and are semisimple. Using any DAA, one can define a family of Hamiltonians which give the continuous time evolution of a stochastic process. The calculation of the spectra and ground-state wave functions (stationary state probability distributions) is an easy algebraic exercise. If one considers D-dimensional lattices and chooses Hamiltonians linear in the generators, in finite-size scaling the Hamiltonian spectrum is gapless with a critical dynamic exponent z=D. One possible application of the DAA is to sandpile models. In the paper we present this application, considering one- and two-dimensional lattices. In the one-dimensional case, when the DAA conserves the number of particles, the avalanches belong to the random walker universality class (critical exponent sigma(tau)=3/2). We study the local density of particles inside large avalanches, showing a depletion of particles at the source of the avalanche and an enrichment at its end. In two dimensions we did extensive Monte-Carlo simulations and found sigma(tau)=1.780 +/- 0.005.

DJKM ALGEBRAS I: THEIR UNIVERSAL CENTRAL EXTENSION

The purpose of this paper is to explicitly describe in terms of generators and relations the universal central extension of the infinite dimensional Lie algebra, g circle times C[t, t(-1), u vertical bar u(2) = (t(2) - b(2))(t(2) - c(2))], appearing in the work of Date, Jimbo, Kashiwara and Miwa in their study of integrable systems arising from the Landau-Lifshitz differential equation.

Induced Modules for Affine Lie Algebras

We study induced modules of nonzero central charge with arbitrary multiplicities over affine Lie algebras. For a given pseudo parabolic subalgebra P of an affine Lie algebra G, our main result establishes the equivalence between a certain category of P-induced G-modules and the category of weight P-modules with injective action of the central element of G. In particular, the induction functor preserves irreducible modules. If P is a parabolic subalgebra with a finite-dimensional Levi factor then it defines a unique pseudo parabolic subalgebra P(ps), P subset of P(ps). The structure of P-induced modules in this case is fully determined by the structure of P(ps)-induced modules. These results generalize similar reductions in particular cases previously considered by V. Futorny, S. Konig, V. Mazorchuk [Forum Math. 13 (2001), 641-661], B. Cox [Pacific J. Math. 165 (1994), 269-294] and I. Dimitrov, V. Futorny, I. Penkov [Comm. Math. Phys. 250 (2004), 47-63].

AUSLANDER GENERATORS OF ITERATED TILTED ALGEBRAS

Let A be an iterated tilted algebra. We will construct an Auslander generator M in order to show that the representation dimension of A is three in case A is representation infinite.

Industrial apiculture in the Jordan valley during Biblical times with Anatolian honeybees

Although texts and wall paintings suggest that bees were kept in the Ancient Near East for the production of precious wax and honey, archaeological evidence for beekeeping has never been found. The Biblical term ""honey"" commonly was interpreted as the sweet product of fruits, such as dates and figs. The recent discovery of unfired clay cylinders similar to traditional hives still used in the Near East at the site of Tel Rehov in the Jordan valley in northern Israel suggests that a large-scale apiary was located inside the town, dating to the 10th-early 9th centuries B.C.E. This paper reports the discovery of remains of honeybee workers, drones, pupae, and larvae inside these hives. The exceptional preservation of these remains provides unequivocal identification of the clay cylinders as the most ancient beehives yet found. Morphometric analyses indicate that these bees differ from the local subspecies Apis mellifera syriaca and from all subspecies other than A. m. anatoliaca, which presently resides in parts of Turkey. This finding suggests either that the Western honeybee subspecies distribution has undergone rapid change during the last 3,000 years or that the ancient inhabitants of Tel Rehov imported bees superior to the local bees in terms of their milder temper and improved honey yield.

Flat tori, lattices and bounds for commutative group codes

We show that commutative group spherical codes in R(n), as introduced by D. Slepian, are directly related to flat tori and quotients of lattices. As consequence of this view, we derive new results on the geometry of these codes and an upper bound for their cardinality in terms of minimum distance and the maximum center density of lattices and general spherical packings in the half dimension of the code. This bound is tight in the sense it can be arbitrarily approached in any dimension. Examples of this approach and a comparison of this bound with Union and Rankin bounds for general spherical codes is also presented.

Quantum mechanics as an approximation to classical mechanics in Hilbert space

Classical mechanics is formulated in complex Hilbert space with the introduction of a commutative product of operators, an antisymmetric bracket and a quasidensity operator that is not positive definite. These are analogues of the star product, the Moyal bracket, and the Wigner function in the phase space formulation of quantum mechanics. Quantum mechanics is then viewed as a limiting form of classical mechanics, as Planck's constant approaches zero, rather than the other way around. The forms of semiquantum approximations to classical mechanics, analogous to semiclassical approximations to quantum mechanics, are indicated.

The structure of quantum Lie algebras for the classical series, B-l, C-l and D-l

The structure constants of quantum Lie algebras depend on a quantum deformation parameter q and they reduce to the classical structure constants of a Lie algebra at q = 1. We explain the relationship between the structure constants of quantum Lie algebras and quantum Clebsch-Gordan coefficients for adjoint x adjoint --> adjoint We present a practical method for the determination of these quantum Clebsch-Gordan coefficients and are thus able to give explicit expressions for the structure constants of the quantum Lie algebras associated to the classical Lie algebras B-l, C-l and D-l. In the quantum case the structure constants of the Cartan subalgebra are non-zero and we observe that they are determined in terms of the simple quantum roots. We introduce an invariant Killing form on the quantum Lie algebras and find that it takes values which are simple q-deformations of the classical ones.

Exactly solvable quantum spin ladders associated with the orthogonal and symplectic Lie algebras

We extend the results of spin ladder models associated with the Lie algebras su(2(n)) to the case of the orthogonal and symplectic algebras o(2(n)), sp(2(n)) where n is the number of legs for the system. Two classes of models are found whose symmetry, either orthogonal or symplectic, has an explicit n dependence. Integrability of these models is shown for an arbitrary coupling of XX-type rung interactions and applied magnetic field term.

Jordan-Wigner fermionization of the one-dimensional Bariev model of three coupled XY chains

The Jordan-Wigner fermionization for the one-dimensional Bariev model of three coupled XY chains is formulated. The L-matrix in terms of fermion operators and the R-matrix are presented explicitly. Furthermore, the graded reflection equations and their solutions are discussed.

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 L-h(g) of the quantized enveloping algebras U-h(g). On them the quantum Lie product is given by the quantum adjoint action. Here we define for any finite-dimensional simple complex Lie algebra g an abstract quantum Lie algebra g(h) 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 L-h(g) are isomorphic to an abstract quantum Lie algebra g(h). In this way we prove two important properties of quantum Lie algebras: 1) all quantum Lie algebras L-h(g) associated to the same g are isomorphic, 2) the quantum Lie product of any Ch(B) is q-antisymmetric. We also describe a construction of L-h(g) which establishes their existence.

Strong Grobner bases for polynomials over a principal ideal ring

Grobner bases have been generalised to polynomials over a commutative ring A in several ways. Here we focus on strong Grobner bases, also known as D-bases. Several authors have shown that strong Grobner bases can be effectively constructed over a principal ideal domain. We show that this extends to any principal ideal ring. We characterise Grobner bases and strong Grobner bases when A is a principal ideal ring. We also give algorithms for computing Grobner bases and strong Grobner bases which generalise known algorithms to principal ideal rings. In particular, we give an algorithm for computing a strong Grobner basis over a finite-chain ring, for example a Galois ring.