Nil graded self-similar algebras

In [19], [24] we introduced a family of self-similar nil Lie algebras L over fields of prime characteristic p > 0 whose properties resemble those of Grigorchuk and Gupta-Sidki groups. The Lie algebra L is generated by two derivations v(1) = partial derivative(1) + t(0)(p-1) (partial derivative(2) + t(1)(p-1) (partial derivative(3) + t(2)(p-1) (partial derivative(4) + t(3)(p-1) (partial derivative(5) + t(4)(p-1) (partial derivative(6) + ...))))), v(2) = partial derivative(2) + t(1)(p-1) (partial derivative(3) + t(2)(p-1) (partial derivative(4) + t(3)(p-1) (partial derivative(5) + t(4)(p-1) (partial derivative(6) + ...)))) of the truncated polynomial ring K[t(i), i is an element of N vertical bar t(j)(p) =0, i is an element of N] in countably many variables. The associative algebra A generated by v(1), v(2) is equipped with a natural Z circle plus Z-gradation. In this paper we show that for p, which is not representable as p = m(2) + m + 1, m is an element of Z, the algebra A is graded nil and can be represented as a sum of two locally nilpotent subalgebras. L. Bartholdi [3] andYa. S. Krylyuk [15] proved that for p = m(2) + m + 1 the algebra A is not graded nil. However, we show that the second family of self-similar Lie algebras introduced in [24] and their associative hulls are always Z(p)-graded, graded nil, and are sums of two locally nilpotent subalgebras.

UNIFORM APPROXIMATION ON IDEALS OF MULTILINEAR MAPPINGS

For each ideal of multilinear mappings M we explicitly construct a corresponding ideal (a)M such that multilinear forms in (a)M are exactly those which can be approximated, in the uniform norm, by multilinear forms in M. This construction is then applied to finite type, compact, weakly compact and absolutely summing multilinear mappings. It is also proved that the correspondence M bar right arrow (a)M. IS Aron-Berner stability preserving.

Nonhomogeneous subalgebras of Lie and special Jordan superalgebras

We consider polynomial identities satisfied by nonhomogeneous subalgebras of Lie and special Jordan superalgebras: we ignore the grading and regard the superalgebra as an ordinary algebra. The Lie case has been studied by Volichenko and Baranov: they found identities in degrees 3, 4 and 5 which imply all the identities in degrees <= 6. We simplify their identities in degree 5, and show that there are no new identities in degree 7. The Jordan case has not previously been studied: we find identities in degrees 3, 4, 5 and 6 which imply all the identities in degrees <= 6, and demonstrate the existence of further new identities in degree 7. our proofs depend on computer algebra: we use the representation theory of the symmetric group, the Hermite normal form of an integer matrix, the LLL algorithm for lattice basis reduction, and the Chinese remainder theorem. (C) 2009 Elsevier Inc. All rights reserved.

A Character Formula for the Category (O)over-tilde of Modules for Affine sl(2)

One may construct, for any function on the integers, an irreducible module of level zero for affine sl(2) using the values of the function as structure constants. The modules constructed using exponential-polynomial functions realize the irreducible modules with finite-dimensional weight spaces in the category (O) over tilde of Chari. In this work, an expression for the formal character of such a module is derived using the highest weight theory of truncations of the loop algebra.

Thermo-solvatochromism of merocyanine polarity probes - What are the consequences of increasing probe lipophilicity through annelation?

The question raised in the title has been answered by comparing the solvatochromism of two series of polarity probes, the lipophilicities of which were increased either by increasing the length of an alkyl group (R) attached to a fixed pyridine-based structure or through annelation (i.e., by fusing benzene rings onto a central pyridine-based structure). The following novel solvatochromic probes were synthesized: 2,6-dibromo-4-[(E)-2-(1-methylquinolinium-4-yl)ethenyl]-phenolate (MeQMBr(2)) and 2,6-dibromo-4-[(E)-2-(1-methyl-acridinium-4- yl) ethenyl)]phenolate (MeAMBr(2) The solvatochromic behavior of these probes, along with that of 2,6dibromo-4-[(E)-2-(1-methylpyridinium-4-yl)ethenyl]phenol-ate(MePMBr(2)) was analyzed in terms of increasing probe lipophilicity, through annelation. Values of the empirical solvent polarity scale [E(T)(MePMBr(2))] in kcalmol(-1) correlated linearly with ET(30), the corresponding values for the extensively employed probe 2,6-diphenyl-4-(2,4,6-triphenylpyridinium-1-yl)phenolate (RB). On the other hand, the nonlinear correlations of ET(MeQMBr(2)) or ET(MeAMBr(2)) with E(T)(30) are described by second-order polynomials. Possible reasons for this behavior include: i) self-aggregation of the probe, ii) photoinduced cis/trans isomerization of the dye, and iii) probe structure- and solvent-dependent contributions of the quinonoid and zwitterionic limiting formulas to the ground and excited states of the probe. We show that mechanisms (i) and (ii) are not operative under the experimental conditions employed; experimental evidence (NMR) and theoretical calculations are presented to support the conjecture that the length of the central ethenylic bond in the dye increases in the order MeAMBr(2) > MeQMBr(2) > MePMBr(2), That is, the contribution of the zwitterionic limiting formula predominates for the latter probe, as is also the case for RB, this being the reason for the observed linear correlation between the ET(MePMBr2) and the ET(30) scales. The effect of increasing probe lipophilicity on solvatochromic behavior therefore depends on the strategy employed. Increasing the length of R affects solvatochromism much less than annelation, because the former structural change hardly perturbs the energy of the intramolecular charge-transfer transition responsible for solvatochromism. The thermo-solvatochromic behavior (effect of temperature on solvatochromism) of the three probes was studied in mixtures of water with propanol and/or with DMSO. The solvation model used explicitly considers the presence of three ""species"" in the system: bulk solution and probe solvation shell [namely, water (W), organic solvent (Solv)], and solvent-water hydrogen-bonded aggregate (Solv-W). For aqueous propanol, the probe is efficiently solvated by Solv-W; the strong interaction of DMSO with W drastically decreases the efficiency of Solv-W in solvating the probe, relative to its precursor solvents. Temperature increases resulted in desolvation of the probes, due to the concomitant reduction in the structured characters of the components of the binary mixtures.