Symmetries of the Simply-Laced Quantum Connections and Quantisation of Quiver Varieties

We will exhibit a group of symmetries of the simply-laced quantum connections, generalising the quantum/Howe duality relating KZ and the Casimir connection. These symmetries arise as a quantisation of the classical symmetries of the simply-laced isomonodromy systems, which in turn generalise the Harnad duality. The quantisation of the classical symmetries involves constructing the quantum Hamiltonian reduction of the representation variety of any simply-laced quiver, both in filtered and in deformation quantisation.

𝔤𝔩n-webs, categorification and Khovanov–Rozansky homologies

In this paper, we define an explicit basis for the [Formula: see text]-web algebra [Formula: see text] (the [Formula: see text] generalization of Khovanov’s arc algebra) using categorified [Formula: see text]-skew Howe duality. Our construction is a [Formula: see text]-web version of Hu–Mathas’ graded cellular basis and has two major applications: it gives rise to an explicit isomorphism between a certain idempotent truncation of a thick calculus cyclotomic KLR algebra and [Formula: see text], and it gives an explicit graded cellular basis of the [Formula: see text]-hom space between two [Formula: see text]-webs. We use this to give a (in principle) computable version of colored Khovanov–Rozansky [Formula: see text]-link homology, obtained from a complex defined purely combinatorially via the (thick cyclotomic) KLR algebra and needs only [Formula: see text].

Dual pair correspondence in physics: oscillator realizations and representations

We study general aspects of the reductive dual pair correspondence, also known as Howe duality. We make an explicit and systematic treatment, where we first derive the oscillator realizations of all irreducible dual pairs: (GL(M, ℝ), GL(N, ℝ)), (GL(M, ℂ), GL(N, ℂ)), (U∗(2M), U∗(2N)), (U (M+, M−), U (N+, N−)), (O(N+, N−), Sp (2M, ℝ)), (O(N, ℂ), Sp(2M, ℂ)) and (O∗(2N ), Sp(M+, M−)). Then, we decompose the Fock space into irreducible representations of each group in the dual pairs for the cases where one member of the pair is compact as well as the first non-trivial cases of where it is non-compact. We discuss the relevance of these representations in several physical applications throughout this analysis. In particular, we discuss peculiarities of their branching properties. Finally, closed-form expressions relating all Casimir operators of two groups in a pair are established.

Howe duality of Higgs – Hahn algebra for 8D harmonic oscillator

In the light of the Howe duality, two different, but isomorphic representations of one algebra as Higgs algebra and Hahn algebra are considered in this article. The first algebra corresponds to the symmetry algebra of a harmonic oscillator on a 2-sphere and a polynomially deformed algebra SU(2), and the second algebra encodes the bispectral properties of corresponding homogeneous orthogonal polynomials and acts as a symmetry algebra for the Hartmann and certain ring-shaped potentials as well as the singular oscillator in two dimensions. The realization of this algebra is shown in explicit form, on the one hand, as the commutant O(4) ⊕ O(4) of subalgebra U(8) in the oscillator representation of universal algebra U (u(8)) and, on the other hand, as the embedding of the discrete version of the Hahn algebra in the double tensor product SU(1,1) ⊗ SU(1,1). These two realizations reflect the fact that SU(1,1) and U(8) form a dual pair in the state space of the harmonic oscillator in eight dimensions. The N-dimensional, N-fold tensor product SU(1,1)⊗N аnd q-generalizations are briefly discussed.