Instanton R-matrix and $$ \mathcal{W} $$-symmetry

Abstract

We study the relation between W 1+∞ algebra and Arbesfeld-Schiffmann-Tsymbaliuk Yangian using the Maulik-Okounkov R-matrix. The central object linking these two pictures is the Miura transformation. Using the results of Nazarov and Sklyanin we find an explicit formula for the mixed R-matrix acting on two Fock spaces associated to two different asymptotic directions of the affine Yangian. Using the free field representation we propose an explicit identification of Arbesfeld-Schiffmann-Tsymbaliuk generators with the generators of Maulik-Okounkov Yangian. In the last part we use the Miura transformation to give a conformal field theoretic construction of conserved quantities and ladder operators in the quantum mechanical rational and trigonometric Calogero-Sutherland models on which a vector representation of the Yangian acts.

Gluing two affine Yangians of 𝔤𝔩1

“…In this paper, we will mostly use the OPE-like relations in terms of fields; it is straightforward to obtain the relations in terms of modes by mode expansion. 9 For the map between formulations of [8] and [18] see section 5.1. of [12]; for the translation between formulations of [9] and [18] see [26]. 10 We adopt the term "OPE-like" here because, similar to the OPEs of 2D CFT, they are defined up to regular terms ∼ z n w m with m, n > 0, and they are merely rewritings of the equivalent relations in terms of modes.…”

“…where h = 1 + ρ is the conformal dimension of the gluing operator 26 and we have only written the leading terms, omitting descendants.…”

“…The effect of x on the removable box can be directly seen from its contribution to the (ψ,ψ) charge functions. An x(z) action with the pole at 26) contributes to the (ψ,ψ) charge function by…”

A Robust and Scalable Continuous Flow Process for Glycerol Carbonate

“…These processes are therefore less economically feasible for industrial-scale preparation [20]. Dimethyl carbonate (DMC) is widely studied as a carbonyl source to synthesize 2-GLC since it is considered an environmentally benign chemical, and has also been used in industrial settings to synthesize 2-GLC [21][22][23][24][25][26]. Many catalysts are known for this transesterification, e.g., inorganic bases [27][28][29][30], tertiary amines [31][32][33][34], lipases [35][36][37], and N-heterocyclic carbenes [38].…”

“…However, only a few continuous processes are known for this transformation [39][40][41][42] while, in general, continuous processes more often meet the basic criteria for potential industrial feasibility and scale-up. To the best of our knowledge, inorganic bases are the preferred catalysts of use for the industrial preparation of 2-GLC prepared from glycerol and DMC, mainly based upon cost [23][24][25][26].…”