Braid rigidity for path algebras

Path algebras are a convenient way of describing decompositions of tensor powers of an object in a tensor category. If the category is braided, one obtains representations of the braid groups $B_n$ for all $n\in \N$. We say that such representations are rigid if they are determined by the path algebra and the representations of $B_2$. We show that besides the known classical cases also the braid representations for the path algebra for the 7-dimensional representation of $G_2$ satisfies the rigidity condition, provided $B_3$ generates \End(V^{\otimes 3}). We obtain a complete classification of ribbon tensor categories with the fusion rules of \g(G_2) if this condition is satisfied

Thermodynamics of adsorption of organic compounds on C-120 silochrome surface with precise layers of nickel acidyl acetoneates, cobalt and copper

Studied the adsorption properties of the surface Silochrom C-120 and chemically modified on the basis of its sorption materials containing acetylacetonates of nickel, cobalt and copper. As test compounds were used n-alkanes (C[6]-C[9]) and the adsorbates whose molecules have different electron-withdrawing and electron donating properties. From the experimental data on the retention of adsorbates designed their differential molar heat of adsorption qdif, 1, change the standard differential molar entropy [delta]S[S1], C and for polar adsorbates contributions [delta] q[dif],1(spec) for energy dispersive and specific interactions

Quantum Gravity and the Algebra of Tangles

In Rovelli and Smolin's loop representation of nonperturbative quantum gravity in 4 dimensions, there is a space of solutions to the Hamiltonian constraint having as a basis isotopy classes of links in R^3. The physically correct inner product on this space of states is not yet known, or in other words, the *-algebra structure of the algebra of observables has not been determined. In order to approach this problem, we consider a larger space H of solutions of the Hamiltonian constraint, which has as a basis isotopy classes of tangles. A certain algebra T, the ``tangle algebra,'' acts as operators on H. The ``empty state'', corresponding to the class of the empty tangle, is conjectured to be a cyclic vector for T. We construct simpler representations of T as quotients of H by the skein relations for the HOMFLY polynomial, and calculate a *-algebra structure for T using these representations. We use this to determine the inner product of certain states of quantum gravity associated to the Jones polynomial (or more precisely, Kauffman bracket).Comment: 16 pages (with major corrections

Representation-theoretic derivation of the Temperley-Lieb-Martin algebras

Explicit expressions for the Temperley-Lieb-Martin algebras, i.e., the quotients of the Hecke algebra that admit only representations corresponding to Young diagrams with a given maximum number of columns (or rows), are obtained, making explicit use of the Hecke algebra representation theory. Similar techniques are used to construct the algebras whose representations do not contain rectangular subdiagrams of a given size.Comment: 12 pages, LaTeX, to appear in J. Phys.

A New Young Diagrammatic Method For Kronecker Products of O(n) and Sp(2m)

A new simple Young diagrammatic method for Kronecker products of O(n) and Sp(2m) is proposed based on representation theory of Brauer algebras. A general procedure for the decomposition of tensor products of representations for O(n) and Sp(2m) is outlined, which is similar to that for U(n) known as the Littlewood rules together with trace contractions from a Brauer algebra and some modification rules given by King.Comment: Latex, 11 pages, no figure

The induced representations of Brauer algebra and the Clebsch-Gordan coefficients of SO(n)

Induced representations of Brauer algebra $D_{f}(n)$ from $S_{f_{1}}\times S_{f_{2}}$ with $f_{1}+f_{2}=f$ are discussed. The induction coefficients (IDCs) or the outer-product reduction coefficients (ORCs) of $S_{f_{1}}\times S_{f_{2}}\uparrow D_{f}(n)$ with $f\leq 4$ up to a normalization factor are derived by using the linear equation method. Weyl tableaus for the corresponding Gel'fand basis of SO(n) are defined. The assimilation method for obtaining CG coefficients of SO(n) in the Gel'fand basis for no modification rule involved couplings from IDCs of Brauer algebra are proposed. Some isoscalar factors of $SO(n)\supset SO(n-1)$ for the resulting irrep $[\lambda_{1},~\lambda_{2},~ \lambda_{3},~\lambda_{4},\dot{0}]$ with $\sum\limits_{i=1}^{4}\lambda_{i}\leq .Comment: 48 pages latex, submitted to Journal of Phys.

Characteristic Relations for Quantum Matrices

General algebraic properties of the algebras of vector fields over quantum linear groups $GL_q(N)$ and $SL_q(N)$ are studied. These quantum algebras appears to be quite similar to the classical matrix algebra. In particular, quantum analogues of the characteristic polynomial and characteristic identity are obtained for them. The $q$-analogues of the Newton relations connecting two different generating sets of central elements of these algebras (the determinant-like and the trace-like ones) are derived. This allows one to express the $q$-determinant of quantized vector fields in terms of their $q$-traces.Comment: 11 pages, latex, an important reference [16] added

On Haagerup's list of potential principal graphs of subfactors

We show that any graph, in the sequence given by Haagerup in 1991 as that of candidates of principal graphs of subfactors, is not realized as a principal graph except for the smallest two. This settles the remaining case of a previous work of the first author.Comment: 19 page

Bin mapping of tomato diversity array (DArT) markers to genomic regions of Solanum lycopersicum × Solanum pennellii introgression lines

Marker-trait association studies in tomato have progressed rapidly due to the availability of several populations developed between wild species and domesticated tomato. However, in the absence of whole genome sequences for each wild species, molecular marker methods for whole genome comparisons and fine mapping are required. We describe the development and validation of a diversity arrays technology (DArT) platform for tomato using an introgression line (IL) population consisting of wild Solanumpennellii introgressed into Solanumlycopersicum (cv. M82). A tomato diversity array consisting of 6,912 clones from domesticated tomato and twelve wild tomato/Solanaceous species was constructed. We successfully bin-mapped 990 polymorphic DArT markers together with 108 RFLP markers across the IL population, increasing the number of markers available for each S.pennellii introgression by tenfold on average. A subset of DArT markers from ILs previously associated with increased levels of lycopene and carotene were sequenced, and 44% matched protein coding genes. The bin-map position and order of sequenced DArT markers correlated well with their physical position on scaffolds of the draft tomato genome sequence (SL2.40). The utility of sequenced DArT markers was illustrated by converting several markers in both the S.pennellii and S.lycopersicum phases to cleaved amplified polymorphic sequence (CAPS) markers. Genotype scores from the CAPS markers confirmed the genotype scores from the DArT hybridizations used to construct the bin map. The tomato diversity array provides additional “sequence-characterized” markers for fine mapping of QTLs in S.pennellii ILs and wild tomato species