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We show that some finite W-superalgebras based on gl(M|N) are truncation of the super-Yangian Y(gl(M|N)). In the same way, we prove that finite W-superalgebras based on osp(M|2n) are truncation of the twisted super-Yangians Y(gl(M|2n))^{+}. Using this homomorphism, we present these W-superalgebras in an R-matrix formalism, and we classify their finite-dimensional irreducible representations.
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We classify the finite-dimensional irreducible representations of the Yangians associated with the orthosymplectic Lie superalgebras ${frak{osp}}_{1|2n}$ in terms of the Drinfeld polynomials. The arguments rely on the description of the representations in the particular case $n=1$ obtained in our previous work.
The deformed $mathcal W$ algebras of type $textsf{A}$ have a uniform description in terms of the quantum toroidal $mathfrak{gl}_1$ algebra $mathcal E$. We introduce a comodule algebra $mathcal K$ over $mathcal E$ which gives a uniform construction of basic deformed $mathcal W$ currents and screening operators in types $textsf{B},textsf{C},textsf{D}$ including twisted and supersymmetric cases. We show that a completion of algebra $mathcal K$ contains three commutative subalgebras. In particular, it allows us to obtain a commutative family of integrals of motion associated with affine Dynkin diagrams of all non-exceptional types except $textsf{D}^{(2)}_{ell+1}$. We also obtain in a uniform way deformed finite and affine Cartan matrices in all classical types together with a number of new examples, and discuss the corresponding screening operators.
We present a connection between W-algebras and Yangians, in the case of gl(N) algebras, as well as for twisted Yangians and/or super-Yangians. This connection allows to construct an R-matrix for the W-algebras, and to classify their finite-dimensional irreducible representations. We illustrate it in the framework of nonlinear Schroedinger equation in 1+1 dimension.
We introduce a subalgebra $overline F$ of the Clifford vertex superalgebra ($bc$ system) which is completely reducible as a $L^{Vir} (-2,0)$-module, $C_2$-cofinite, but it is not conformal and it is not isomorphic to the symplectic fermion algebra $mathcal{SF}(1)$. We show that $mathcal{SF}(1)$ and $overline{F}$ are in an interesting duality, since $overline{F}$ can be equipped with the structure of a $mathcal{SF}(1)$-module and vice versa. Using the decomposition of $overline F$ and a free-field realization from arXiv:1711.11342, we decompose $L_k(mathfrak{osp}(1vert 2))$ at the critical level $k=-3/2$ as a module for $L_k(mathfrak{sl}(2))$. The decomposition of $L_k(mathfrak{osp}(1vert 2))$ is exactly the same as of the $N=4$ superconformal vertex algebra with central charge $c=-9$, denoted by $mathcal V^{(2)}$. Using the duality between $overline{F}$ and $mathcal{SF}(1)$, we prove that $L_k(mathfrak{osp}(1vert 2))$ and $mathcal V^{(2)}$ are in the duality of the same type. As an application, we construct and classify all irreducible $L_k(mathfrak{osp}(1vert 2))$-modules in the category $mathcal O$ and the category $mathcal R$ which includes relaxed highest weight modules. We also describe the structure of the parafermion algebra $N_{-3/2}(mathfrak{osp}(1vert 2))$ as a $N_{-3/2}(mathfrak{sl}(2))$-module. We extend this example, and for each $p ge 2$, we introduce a non-conformal vertex algebra $mathcal A^{(p)}_{new}$ and show that $mathcal A^{(p)}_{new} $ is isomorphic to the doublet vertex algebra as a module for the Virasoro algebra. We also construct the vertex algebra $ mathcal V^{(p)} _{new}$ which is isomorphic to the logarithmic vertex algebra $mathcal V^{(p)}$ as a module for $widehat{mathfrak{sl}}(2)$.
In this paper, we classify the compatible left-symmetric superalgebra structures on the super-Virasoro algebras satisfying certain natural conditions.
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