0. Introduction. The quantum Yang-Baxter equation (QYBE) is one of the basic equations in mathematical physics that lies in the foundation of the theory of quantum groups. This equation involves a linear operator R : V ⊗ V → V ⊗ V , where V is a vector space, and has the form

In this note we define geometric classical r-matrices and quantum Rmatrices, and show how any geometric classical r-matrix can be quantized to a geometric quantum R-matrix. This is one of the simplest nontrivial examples of quantization of solutions of the classical Yang-Baxter equation, which can be explicitly computed. The idea of the above quantization was inspired by the results in [ESS]. We note that a construction similar to ours was obtained in [KLM].

Increasing the effectiveness of known, well-tested drugs is a promising low-cost alternative to the search for new drug molecular forms. Powerful approaches to solve this problem are (a) an active drug particle size reduction down to the nanoscale and (b) thermodynamically metastable but kinetically stable crystal modifications of drug acquisition. The combined cryochemical method has been used for size and structural modifications of the antibacterial drug 2,3-quinoxalinedimethanol-1,4-dioxide (dioxidine). The main stage of the proposed technique includes the formation of a molecular vapor of the drug substance, combined with a carrier gas (CO2) flow, followed by a fast condensation of the drug substance and CO2 molecules on a cooled-by-liquid nitrogen surface of preparative cryostate. It was established that the molecular chemical structure of the drug substance remained unchanged during cryochemical modification; however, it led to a significant decrease of the drug particles’ size down to nanosizes and changes in the crystal structures of the solid drug nanoforms obtained. Varying carrier gas (CO2) flow led to changes in their solid phase composition. A higher dissolution rate and changes in antibacterial activity were demonstrated for cryomodified dioxidine samples in comparison to the properties of the initial pharmacopeia dioxidine.

We give a new realization of Y (sl 3 ) via Cartan-Weyl elements. An algebraic description of Yangian Double DY (sl 3 ), explicit comultiplication formulas and universal R-matrix are obtained in these terms. *

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