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Twisted generalized Weyl algebras (TGWAs) $A(R,\sigma,t)$ are defined over a base ring $R$ by parameters $\sigma$ and $t$, where $\sigma$ is an $n$-tuple of automorphisms, and $t$ is an $n$-tuple of elements in the center of $R$. We show that, for fixed $R$ and $\sigma$, there is a natural algebra map $A(R,\sigma,tt')\to A(R,\sigma,t)\otimes_R A(R,\sigma,t')$. This gives a tensor product operation on modules, inducing a ring structure on the direct sum (over all $t$) of the Grothendieck groups of the categories of weight modules for $A(R,\sigma,t)$. We give presentations of these Grothendieck rings for $n=1,2$, when $R=\mathbb{C}[z]$. As a consequence, for $n=1$, any indecomposable module for a TGWA can be written as a tensor product of indecomposable modules over the usual Weyl algebra. In particular, any finite-dimensional simple module over $\mathfrak{sl}_2$ is a tensor product of two Weyl algebra modules., Comment: 31 pages, 3 figures

In this paper, we report on the unexplored reaction mechanisms of bimolecular homolytic substitution (SH2) between GeH3 radicals and the nitrogen atom of NF3. The SH2 reactions are studied both exp...

Kato's exotic nilpotent cone was introduced as a substitute for the ordinary nilpotent cone of type C with cleaner properties. The geometric Robinson-Schensted correspondence is obtained by parametrizing the irreducible components of the Steinberg variety (the conormal variety for the action of a semisimple group on two copies of its flag variety); in type A the bijection coincides with the classical Robinson-Schensted algorithm for the symmetric group. Here we give a combinatorial description of the bijection obtained by using the exotic nilpotent cone instead of ordinary type C nilpotent cone in the geometric Robinson-Schensted correspondence; we refer this as the "exotic Robinson-Schensted bijection". This is interesting from a combinatorial perspective, and not a naive extension of the type A Robinson-Schensted bijection., Comment: 33 pages

In this paper, we report on the unexplored reaction mechanisms of bimolecular homolytic substitution (SH2) between GeH3 radicals and the nitrogen atom of NF3. The SH2 reactions are studied both experimentally and theoretically with ab initio and density functional theory (DFT) calculations. The experimental results of X-ray irradiation of mixtures of GeH4 and NF3 show the formation of GeH3–NF2 and GeH3–F. The trend of product yields as a function of the increase in GeH4 partial pressure in the irradiated mixtures evidences the predominant role of GeH3 radicals. Particularly, the SH2 mechanism can be hypothesized for the reaction between GeH3 radicals and NF3 molecules leading to GeH3–NF2. This mechanism is further confirmed by the increase in GeH3–NF2 yield observed if O2 is added, as a radical scavenger, to the reaction mixture. In agreement with the experimental data, from the calculations performed at the CCSD(T) and G3B3 levels of theory, we observe that the GeH3–NF2 product actually occurs from a bimolecular homolytic substitution by the GeH3 radical, which attacks the N atom of NF3, and this reaction is in competition with the fluorine abstraction reaction leading to GeH3F, even if other mechanisms may be involved in the formation of this product.

Kato introduced the exotic nilpotent cone to be a substitute for the ordinary nilpotent cone of type C with cleaner properties. Here we describe the irreducible components of exotic Springer fibres (the fibres of the resolution of the exotic nilpotent cone), and prove that they are naturally in bijection with standard bitableaux. As a result, we deduce the existence of an exotic Robinson–Schensted bijection, which is a variant of the type C Robinson–Schensted bijection between pairs of same-shape standard bitableaux and elements of the Weyl group; this bijection is described explicitly in the sequel to this paper. Note that this is in contrast with ordinary type C Springer fibres, where the parametrisation of irreducible components, and the resulting geometric Robinson–Schensted bijection, are more complicated. As an application, we explicitly describe the structure in the special cases where the irreducible components of theexotic Springer fibre have dimension 2, and show that in those cases one obtains Hirzebruch surfaces.

Starting from a graded Frobenius superalgebra $B$, we consider a graphical calculus of $B$-decorated string diagrams. From this calculus we produce algebras consisting of closed planar diagrams and of closed annular diagrams. The action of annular diagrams on planar diagrams can be used to make clockwise (or counterclockwise) annular diagrams into an inner product space. Our main theorem identifies this space with the space of symmetric functions equipped with the Jack inner product at Jack parameter $\operatorname{dim} B_\mathrm{even} - \operatorname{dim} B_\mathrm{odd}$. In this way, we obtain a graphical realization of that inner product space., Comment: 29 pages. v2: Minor corrections, published version

The convolution algebra of GL d -invariant functions over the space of pairs of partial n-step flags over a finite field was studied by Beilinson-Lusztig-MacPherson. They used it to give a construction of the quantum Schur algebras and, using a stabilization procedure, of the idempotented version of the quantum enveloping algebra of $$ {\mathfrak{gl}}_n $$ . In this paper we expand the construction to the mirabolic setting of triples of two partial flags and a vector, and examine the resulting convolution algebra. In the case of n = 2, we classify the finite-dimensional irreducible representations of the mirabolic quantum algebra and we prove that the category of such representations is semisimple. Finally, we describe a mirabolic version of the quantum Schur-Weyl duality, which involves the mirabolic Hecke algebra.

Let $R$ be a polynomial ring in $m$ variables over a field of characteristic zero. We classify all rank $n$ twisted generalized Weyl algebras over $R$, up to $\mathbb{Z}^n$-graded isomorphisms, in terms of higher spin 6-vertex configurations. Examples of such algebras include infinite-dimensional primitive quotients of $U(\mathfrak{g})$ where $\mathfrak{g}=\mathfrak{gl}_n$, $\mathfrak{sl}_n$, or $\mathfrak{sp}_{2n}$, algebras related to $U(\widehat{\mathfrak{sl}}_2)$ and a finite W-algebra associated to $\mathfrak{sl}_4$. To accomplish this classification we first show that the problem is equivalent to classifying solutions to the binary and ternary consistency equations. Secondly, we show that the latter problem can be reduced to the case $n=2$, which can be solved using methods from previous work by the authors. As a consequence we obtain the surprising fact that (in the setting of the present paper) the ternary consistency relation follows from the binary consistency relation., 18 pages, 2 figures, v2: fixed minor typos

Soluble biobased lignin-like polymeric substances (SBO) isolated from the alkaline hydrolysates of composted biowastes are promising chemical auxiliaries for multiple uses. They have good surfactant properties. Their black color spoils their performance in detergency and dyeing. Ozonization of SBO is reported now to yield bleached biosurfactants with improved performance. The work was performed, with SBO obtained from composted gardening residues, alone or mixed with kitchen wastes. They were dissolved in water at 3.3 % concentration. Oxygen containing 4 mol/mol % ozone was flown at 60 L h−1 for 48 h through the SBO solution. The crude ozonized products in ca. 80 % yield were filtered through different molecular cut off membranes and characterized for chemical features, molecular weight, and surface tension in water. The ozonized fractions with 200–500 kDa molecular weights accounted for 12–29 % of the total ozonized fractions's organic matter. They lowered water surface tension to 48–52 mN m−1. The lower molecular weight ozonized fractions had no surfactant activity, but their molecular features suggest other potentially valuable uses.

We classify all solutions (p,q) to the equation p(u)q(u)=p(u+b)q(u+a) where p and q are complex polynomials in one indeterminate u, and a and b are fixed but arbitrary complex numbers. This equation is a special case of a system of equations which ensures that certain algebras defined by generators and relations are non-trivial. We first give a necessary condition for the existence of non-trivial solutions to the equation. Then, under this condition, we use combinatorics of generalized Dyck paths to describe all solutions and a canonical way to factor each solution into a product of irreducible solutions., 21 pages

peer-reviewed The aim of this work was to address the issue of processed vs. non-processed biowastes for agriculture, by comparing materials widely differing for the amount of process energy consumption. Thus, residual post harvest tomato plants (TP), the TP hydrolysates obtained at pH 13 and 60 degrees C, and two known biochar products obtained by 650 degrees C pyrolysis were prepared. All products were characterized and used in a cultivation of radish plants. The chemical composition and molecular nature of the materials was investigated by solid state C-13 NMR spectrometry, elemental analysis and potentiometric titration. The plants were analysed for growth and content of chlorophyll, carotenoids and soluble proteins. The results show that the TP and the alkaline hydrolysates contain lignin, hemicellulose, protein, peptide and/or amino acids moieties, and several mineral elements. The biochar samples contain also similar mineral elements, but the organic fraction is characterized mainly by fused aromatic rings. All materials had a positive effect on radish growth, mainly on the diameter of roots. The best performances in terms of plant growth were given by miscanthus originated biochar and TP. The most significant effect was the enhancement of soluble protein content in the plants treated with the lowest energy consumption non processed TP. The significance of these findings for agriculture and the environment is discussed. PUBLISHED peer-reviewed

The hydrolysis of five fermented water insoluble urban wastes is shown to convert the insoluble recalcitrant organic lignin-like fraction to added value soluble lignin-like products in high yields. The hydrolysate products are a mixture of complex polymeric molecules which maintain the memory of the proximates of the pristine materials. For a composted biowaste, taken as the case study, a process feasibility study comparing the microwave (MW) assisted versus the conventional (CON) reaction is reported. Product yields and quality are investigated as a function of four process parameters: i.e. pH (8–13), temperature (60–200 °C), and liquid–solid w/w ratio (4 and 10) and contact time (1 min–4 h). Microwave heating allows obtaining the same products, and in the same yields, as conventional heating in 1–2 orders of magnitude lower time. It is possible to achieve 50–60% yields of soluble lignin-like products at relatively low temperatures ≤100 °C. The obtained yield vs. temperature trend indicates that even higher yields may be obtained at higher temperatures. The potential scalability of MW assisted versus CON hydrolysis to the industrial level is discussed in relation to the reaction rate and reactor cost. The results offer worthwhile research scope to compare MW and CON heating for the hydrolysis of other fermented and non-fermented material as collected biowastes.

In this paper we first generalize to the case of partial flags a result proved both by Spaltenstein and by Steinberg that relates the relative position of two complete flags and the irreducible components of the flag variety in which they lie, using the Robinson-Schensted-Knuth correspondence. Then we use this result to generalize the mirabolic Robinson-Schensted-Knuth correspondence defined by Travkin, to the case of two partial flags and a line., Comment: 27 pages, slightly rewritten to combine two papers into one and clarify some sections

We associate a monoidal category $\mathcal{H}_B$, defined in terms of planar diagrams, to any graded Frobenius superalgebra $B$. This category acts naturally on modules over the wreath product algebras associated to $B$. To $B$ we also associate a (quantum) lattice Heisenberg algebra $\mathfrak{h}_B$. We show that, provided $B$ is not concentrated in degree zero, the Grothendieck group of $\mathcal{H}_B$ is isomorphic, as an algebra, to $\mathfrak{h}_B$. For specific choices of Frobenius algebra $B$, we recover existing results, including those of Khovanov and Cautis--Licata. We also prove that certain morphism spaces in the category $\mathcal{H}_B$ contain generalizations of the degenerate affine Hecke algebra. Specializing $B$, this proves an open conjecture of Cautis--Licata., Comment: 46 pages. v2: Several sign errors and other minor typos corrected. v3: Minor corrections, published version

We show that the Grothendieck groups of the categories of finitely-generated graded supermodules and finitely-generated projective graded supermodules over a tower of graded superalgebras satisfying certain natural conditions give rise to twisted Hopf algebras that are twisted dual. Then, using induction and restriction functors coming from such towers, we obtain a categorification of the twisted Heisenberg double and its Fock space representation. We show that towers of wreath product algebras (in particular, the tower of Sergeev superalgebras) and the tower of nilcoxeter graded superalgebras satisfy our axioms. In the latter case, one obtains a categorification of the quantum Weyl algebra., 29 pages; v2: Minor changes (corrected typos, etc.) throughout

We introduce a twisted version of the Heisenberg double, constructed from a twisted Hopf algebra and a twisted Hopf pairing. We state a Stone--von Neumann type theorem for a natural Fock space representation of this twisted Heisenberg double and deduce the effect on the algebra of shifting the product and coproduct of the original twisted Hopf algebra. We conclude by showing that the quantum Weyl algebra, quantum Heisenberg algebras, and lattice Heisenberg algebras are all examples of the general construction., Comment: 22 pages; v2: minor corrections

The Iwahori-Hecke algebra of the symmetric group is the convolution algebra of $\gl_n$-invariant functions on the variety of pairs of complete flags over a finite field. Considering convolution on the space of triples of two flags and a vector we obtain the mirabolic Hecke algebra $R_n$, which had originally been described by Solomon. In this paper we give a new presentation for $R_n$ which shows that it is a quotient of a cyclotomic Hecke algebra, as defined by Ariki and Koike. From this we recover the results of Siegel about the representations of $R_n$. We use Jucys-Murphy elements to describe the center of $R_n$ and to give a $\mathfrak{gl}_\infty$-structure on the Grothendieck group of the category of its representations, giving `mirabolic' analogues of classical results about the Iwahori-Hecke algebra. We also outline a strategy towards a proof of the conjecture that the mirabolic Hecke algebra is a cellular algebra., 27 pages