Related papers: The duality of $\mathfrak{gl}_{m|n}$ and $\mathfra…

We prove an equivalence between the actions of the Gaudin algebras with irregular singularities for $\mathfrak{gl}_d$ and $\mathfrak{gl}_{p+m|q+n}$ on the Fock space of $d(p+m)$ bosonic and $d(q+n)$ fermionic oscillators. This establishes a…

We establish $({\mathfrak{gl}}_M, {\mathfrak{gl}}_N)$-dualities between quantum Gaudin models with irregular singularities. Specifically, for any $M, N \in {\mathbb Z}_{\geq 1}$ we consider two Gaudin models: the one associated with the Lie…

We show that under a generic condition, the quadratic Gaudin Hamiltonians associated to $\mathfrak{gl}(p+m|q+n)$ are diagonalizable on any singular weight space in any tensor product of unitarizable highest weight…

Let $M$ be a tensor product of unitarizable irreducible highest weight modules over the Lie (super)algebra $\mathcal{G}$, where $\mathcal{G}$ is $\mathfrak{gl}(m|n)$, $\mathfrak{osp}(2m|2n)$ or $\mathfrak{spo}(2m|2n)$. We show, using super…

We study the Gaudin models associated with $\mathfrak{gl}(1|1)$. We give an explicit description of the algebra of Hamiltonians (Gaudin Hamiltonians) acting on tensor products of polynomial evaluation $\mathfrak{gl}(1|1)[t]$-modules. It…

We extend duality between the quantum integrable Gaudin models with boundary and the classical Calogero-Moser systems associated with root systems of classical Lie algebras $B_N$, $C_N$, $D_N$ to the case of supersymmetric ${\rm gl}(m|n)$…

We show that the images of the Bethe subalgebras of the Yangians $Y(\mathfrak{gl}_{n})$ and $Y(\mathfrak{gl}_{m})$ under the homomorphisms to $U(\mathfrak{gl}_{n+m})$ given by the Olshanski centralizer construction coincide. We use this…

The "Capelli problem" for the symmetric pairs $(\mathfrak{gl}\times \mathfrak{gl},\mathfrak{gl})$ $(\mathfrak{gl},\mathfrak{o})$, and $(\mathfrak{gl},\mathfrak{sp})$ is closely related to the theory of Jack polynomials and shifted Jack…

We consider the trigonometric classical $r$-matrix for $\mathfrak{gl}_N$ and the associated quantum Gaudin model. We produce higher Hamiltonians in an explicit form by applying the limit $q\to 1$ to elements of the Bethe subalgebra for the…

The characteristic identity formalism discussed in our recent articles is further utilized to derive matrix elements of type 2 unitary irreducible $gl(m|n)$ modules. In particular, we give matrix element formulae for all gl(m|n) generators,…

In this work we will use a general procedure to construct higher local Hamiltonians for the affine $\mathfrak{sl}_2$ Gaudin model. We focus on the first non-trivial example, the quartic Hamiltonians. We show by direct calculation that the…

We formulate a general super duality conjecture on connections between parabolic categories O of modules over Lie superalgebras and Lie algebras of type A, based on a Fock space formalism of their Kazhdan-Lusztig theories which was…

In our recent paper we described relationships between integrable systems inspired by the AGT conjecture. On the gauge theory side an integrable spin chain naturally emerges while on the conformal field theory side one obtains some special…

We disclose the Lie algebraic structure of two extended Sutherland models. Their Hamiltonians are BC_N, and A_N Sutherland Hamiltonians with some additional terms. We show that both Hamiltonians can be written in the quadratic forms of…

We apply the super duality formalism recently developed by the authors to obtain new equivalences of various module categories of general linear Lie superalgebras. We establish the correspondence of standard, tilting, and simple modules, as…

We review results on the Knizhnik-Zamolodchikov (KZ) and dynamical equations, both differential and difference, in the context of the $(gl_k,gl_n)$ duality, and their implications for hypergeometric integrals. The KZ and dynamical equations…

We introduce a new approach to the study of finite-dimensional representations of the quantum group of the affine Lie superalgebra $\mathrm{L}\mathfrak{gl}_{M|N}=\mathbb{C}[t,t^{-1}]\otimes\mathfrak{gl}_{M|N}$ ($M\neq N$). We explain how…

Gaudin algebras form a family of maximal commutative subalgebras in the tensor product of $n$ copies of the universal enveloping algebra $U(\g)$ of a semisimple Lie algebra $\g$. This family is parameterized by collections of pairwise…

We show that every Galileon theory admits a dual formulation as a Galileon theory with new operator coefficients. In n dimensions a free scalar field in Minkowski spacetime is dual to a (n+1)-th order Galileon theory which exhibits the…

We present a novel realisation of the $\mathbb{Z}_2\times\mathbb{Z}_2$-graded Lie superalgebra $\mathfrak{gl}(m_1,m_2|n_1,n_2)$ inside an algebraic extension of the enveloping algebra of the $\mathbb{Z}_2$-graded Lie superalgebra…