Algebra Seminar: Ragoucy -- On elliptic deformation of W_N algebras

Eric Ragoucy (CNRS) 

Friday 30 August, 12-1pm, Place: Carslaw 375 

On elliptic deformation of W_N algebras 

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