We revisit the construction of deformed Virasoro algebras from elliptic quantum algebras of vertex type, generalizing the  bilinear trace procedure proposed in the 90's. It allows us to make contact with the vertex operator techniques that were introduced separately at the same period by Shiraishi et al.

The case of dynamical elliptic quantum algebras is also studied.

 

Then, we extend our method to q-deformations of quantum W_N algebras with elliptic structure functions. 

Their generators of spin k+1 are built from 2k products of the Lax matrix generators of A_{q,p}(gl(N)_c). The closure of the algebras is insured by a critical surface condition relating the parameters p,q and the central charge c.

Further abelianity conditions are determined, either as c=-N or as a second condition on p,q,c. When abelianity is achieved, a Poisson bracket can be defined, that we determine explicitly. One connects these structures with previously built classical q-deformed W_N algebras  and quantum W_{q,p}(sl_N).

 

Based on joint works with J. Avan and L. Frappat, see arXiv:1607.05050, arXiv:1703.05223, arXiv:1810.11410.

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