Stereoselective [4+2]‐Cycloaddition with Chiral Alkenylboranes

A method for the stereoselective [4+2]‐cycloaddition of alkenylboranes and dienes is presented. This transformation was accomplished through the introduction of a new strategy that involves the use of chiral N‐protonated alkenyl oxazaborolidines as dieneophiles. The reaction leads to the formation of products that can be readily derivatized to more complex structural motifs through stereospecific transformations of the C?B bond such as oxidation and homologation. Detailed computation evaluation of the reaction has uncovered a surprising role of the counterion on stereoselectivity.     

A gas-insulated railgun

A railgun concept is discussed in which a high-pressure prefill of hydrogen serves as the close-in insulator between the rails. The gas-insulated railgun (GIRG) approach can reduce or eliminate the velocity-limiting problem of growth and separation of the current-carrying armature region behind the projectile, and thus allow higher launch velocities, to be achieved. A hybrid armature appears appropriate for this launcher     

Stereoselective [4+2]-Cycloaddition with Chiral Alkenylboranes

A method for the stereoselective [4+2]-cycloaddition of alkenylboranes and dienes is presented. This transformation was accomplished through the introduction of a new strategy that involves the use of chiral N-protonated alkenyl oxazaborolidines as dieneophiles. The reaction leads to the formation of products that can be readily derivatized to more complex structural motifs through stereospecific transformations of the C−B bond such as oxidation and homologation. Detailed computation evaluation of the reaction has uncovered a surprising role of the counterion on stereoselectivity.     

Examples of support varieties for Hopf algebras with noncommutative tensor products

The representations of some Hopf algebras have curious behavior: Nonprojective modules may have projective tensor powers, and the variety of a tensor product of modules may not be contained in the intersection of their varieties. We explain a family of examples of such Hopf algebras and their modules, and classify left, right, and two-sided ideals in their stable module categories.     

Hochschild cohomology and quantum Drinfeld Hecke algebras

Quantum Drinfeld Hecke algebras are generalizations of Drinfeld Hecke algebras in which polynomial rings are replaced by quantum polynomial rings. We identify these algebras as deformations of skew group algebras, giving an explicit connection to Hochschild cohomology. We compute the relevant part of Hochschild cohomology for actions of many reflection groups, and we exploit computations from Naidu et al. (Proc Am Math Soc 139:1553–1567, 2011) for diagonal actions. By combining our work with recent results of Levandovskyy and Shepler (Can J Math 66:874–901, 2014) we produce examples of quantum Drinfeld Hecke algebras. These algebras generalize the braided Cherednik algebras of Bazlov and Berenstein (Selecta Math 14(3–4):325–372, 2009).     

Group actions on algebras and the graded Lie structure of Hochschild cohomology

Hochschild cohomology governs deformations of algebras, and its graded Lie structure plays a vital role. We study this structure for the Hochschild cohomology of the skew group algebra formed by a finite group acting on an algebra by automorphisms. We examine the Gerstenhaber bracket with a view toward deformations and developing bracket formulas. We then focus on the linear group actions and polynomial algebras that arise in orbifold theory and representation theory; deformations in this context include graded Hecke algebras and symplectic reflection algebras. We give some general results describing when brackets are zero for polynomial skew group algebras, which allow us in particular to find noncommutative Poisson structures. For abelian groups, we express the bracket using inner products of group characters. Lastly, we interpret results for graded Hecke algebras.     

Products in Hochschild cohomology and Grothendieck rings of group crossed products

We give a general construction of rings graded by the conjugacy classes of a finite group. Some examples of our construction are the Hochschild cohomology ring of a finite group algebra, the Grothendieck ring of the Drinfel'd double of a group, and the orbifold cohomology ring for a global quotient. We generalize the first two examples by deriving product formulas for the Hochschild cohomology ring of a group crossed product and for the Grothendieck ring of an abelian extension of Hopf algebras. Our results account for similarities in the product structures among these examples.     

Skew Derivations and Deformations of a Family of Group Crossed Products

We obtain deformations of a crossed product of a polynomial algebra with a group, under some conditions, from universal deformation formulas. We show that the resulting deformations are nontrivial by a comparison with Hochschild cohomology. The universal deformation formulas arise from actions of Hopf algebras generated by automorphisms and skew derivations, and are universal in the sense that they apply to deform all algebras with such Hopf algebra actions.     

Gerstenhaber brackets on Hochschild cohomology of general twisted tensor products

We present techniques for computing Gerstenhaber brackets on Hochschild cohomology of general twisted tensor product algebras. These techniques involve twisted tensor product resolutions and are based on recent results on Gerstenhaber brackets expressed on arbitrary bimodule resolutions.