The representations of nested composition algebras

In this paper, we investigate the basis graph of the monoid algebra of a submonoid of the monoid of mappings from N = {1,…,n} to itself, defined by a nested sequence of compositions of N. Each such monoid is a left regular band (LRB), that is, a semigroup S satisfying x² = x and xyx = xy for all x,y∈S. This class is su?ciently rich that every path algebra of an acyclic quiver can be embedded in such a monoid algebra. The multiplication in the monoid algebra has a particularly simple quasi-multiplicative form, allowing definition over the integers. Combining this with a formula for Ext-groups for LRBs due to Margolis et al. [6 Margolis, S., Saliola, F., Steinberg, B. (2015). Combinatorial topology and the global dimension of algebras arising in combinatorics. J. Eur. Math Soc. 17(12):3037–3080.[Crossref], [Web of Science ®] , [Google Scholar]], we get a simple criterion for the nested composition algebras to be hereditary.     

Microwave-assisted rapid cyclization of lapachol, a main constituent of Heterophragma adenophyllum

Cyclization reactions of lapachol (1) isolated from Heterophragma adenophyllum have been studied under microwave irradiation under different conditions using alumina (acidic, basic and neutral)/silica gel/montmorillonite (KSF and K-10) as solid support along with neat reaction using 2-3 drops of DMF giving naturally occurring dehydro-alpha-lapachone (2), alpha-lapachone (3), beta-lapachone (4) depending upon the nature of support and irradiation time. A novel naphthoquinone derivative adenophyllone (5) can be synthesized from lapachol using DMF under microwaves.     

Reaction of β‐Lapachone with 1,2‐Diamines: Facile Synthesis of Novel Tetracyclic Pyrazines

The reactions of β‐lapachone (1) with 1,2‐diaminoethane (2) and 1,2‐diaminopropane (4) gave the tetracyclic pyrazine derivatives (3) and mixtures of (5) and (6), respectively, while reaction with o‐phenylene diamine (7) gave the quinoxaline derivative (8). The structure of pyrazine derivative (3) is confirmed by single‐crystal X‐ray diffraction study.