Combinatorics of vertex operators and deformed $W$algebra of type D$(2,1;\alpha)$
Abstract
We consider sets of screening operators with fermionic screening currents. We study sums of vertex operators which formally commute with the screening operators assuming that each vertex operator has rational contractions with all screening currents with only simple poles. We develop and use the method of $qq$characters which are combinatorial objects described in terms of deformed Cartan matrix. We show that each qqcharacter gives rise to a sum of vertex operators commuting with screening operators and describe ways to understand the sum in the case it is infinite. We discuss combinatorics of the qqcharacters and their relation to the qcharacters of representations of quantum groups. We provide a number of explicit examples of the qqcharacters with the emphasis on the case of $D(2,1;\alpha)$. We describe a relationship of the examples to various integrals of motion.
 Publication:

arXiv eprints
 Pub Date:
 March 2021
 DOI:
 10.48550/arXiv.2103.15247
 arXiv:
 arXiv:2103.15247
 Bibcode:
 2021arXiv210315247F
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematical Physics;
 Mathematics  Combinatorics;
 Mathematics  Representation Theory
 EPrint:
 Latex, 44 pages. We made some corrections