SciPost Phys. 8, 033 (2020) ·
published 2 March 2020
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The algebraic structure underlying the classical $r$-matrix formulation of
the complex sine-Gordon model is fully elucidated. It is characterized by two
matrices $a$ and $s$, components of the $r$ matrix as $r=a-s$. They obey a
modified classical reflection/Yang--Baxter set of equations, further deformed
by non-abelian dynamical shift terms along the dual Lie algebra $su(2)^*$. The
sign shift pattern of this deformation has the signature of the twisted
boundary dynamical algebra. Issues related to the quantization of this
algebraic structure and the formulation of quantum complex sine-Gordon on those
lines are introduced and discussed.
SciPost Phys. 6, 054 (2019) ·
published 7 May 2019
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We construct $q$-deformations of quantum $\mathcal{W}_N$ algebras with
elliptic structure functions. Their spin $k+1$ generators are built from $2k$
products of the Lax matrix generators of
${\mathcal{A}_{q,p}(\widehat{gl}(N)_c)}$). 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 $\mathcal{W}_N$
algebras and quantum $\mathcal{W}_{q,p}(\mathfrak{sl}_N)$.
