The Principal Element of a Frobenius Lie Algebra

We introduce the notion of the principal element of a Frobenius Lie algebra ${\frak{f}}$ . The principal element corresponds to a choice of ${F \in \frak{f}^{*}}$ such that F[–, –] non-degenerate. In many natural instances, the principal element is shown to be semisimple, and when associated to sl _n , its eigenvalues are integers and are independent of F. For certain “small” functionals F, a simple construction is given which readily yields the principal element. When applied to the first maximal parabolic subalgebra of sl _n , the principal element coincides with semisimple element of the principal three-dimensional subalgebra. We also show that Frobenius algebras are stable under deformation.     

Higher operations on the Hochschild complex

University of Pennsylvania and Princeton University. University of Pennsylvania. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 29, No. 1, pp. 1–6, January–March, 1995.     

Deformation of Towers of Algebras and Quantized Intersection Cardinality Functions

The inductive limit of a tower of separable algebras is unchanged, up to isomorphism, by consistent deformations but the inductive limit of a corresponding tower of modules may be nontrivially deformed, thereby quantizing the limit module. In the case of the inductive limit of the complex group algebras of the symmetric groups and their deformations, the Hecke algebras, this quantization preserves properties of the finite case which disappear in the absence of quantization.     

Enantioselective conjugate allylation of cyclic enones

Enantioselective organocatalytic 1,2-allylation of a cyclic enone followed by anionic oxy-Cope rearrangement delivered the ketone as a mixture of diastereomers. This appears to be a general method for the net enantioselective conjugate allylation of cyclic enones.     

On the cohomology of the Weyl algebra,the quantum plane,and the q-Weyl algebra

Deformation theory can be used to compute the cohomology of a deformed algebra with coefficients in itself from that of the original. Using the invariance of the Euler–Poincaré characteristic under deformation, it is applied here to compute the cohomology of the Weyl algebra, the algebra of the quantum plane, and the q-Weyl algebra. The behavior of the cohomology when q is a root of unity may encode some number theoretic information.     

Boundary Solutions of the Quantum Yang–Baxter Equation and Solutions in Three Dimensions

Boundary solutions to the quantum Yang–Baxter (qYB) equation are defined to be those in the boundary of (but not in) the variety of solutions to the modified qYB equation, the latter being analogous to the modified classical Yang–Baxter (cYB) equation. We construct, for a large class of solutions r to the modified cYB equation, explicit boundary quantizations, i.e., boundary solutions to the qYB equation of the form I + tr + t²r₂ +. In the last section we list and give quantizations for all classical r-matrices in sl(3) sl(3).     

Path algebras,wave-particle duality,and quantization of phase space

Semigroup algebras admit certain ‘coherent’ deformations which, in the special case of a path algebra, may associate a periodic function to an evolving path; for a particle moving freely on a straight line after an initial impulse, the wave length is that hypothesized by de Broglie’s wave-particle duality. This theory leads to a model of “physical” phase space of which mathematical phase space, the cotangent bundle of configuration space, is a projection. This space is singular, quantized at the Planck level, its structure implies the existence of spin, and the spread of a packet can be described as a random walk. The wavelength associated to a particle moving in this space need not be constant and its phase can change discontinuously.