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In this paper we give a detailed proof of the classification of extremal (=massless) unitary highest weight representations in the Neveu Schwarz and Ramond sectors of the big $N=4$ superconformal algebra which can be found in [5]. Our…

Let $G \leq \operatorname{SL}_3(\mathbb{C})$ be a non-trivial finite group, acting on $R = \mathbb{C}[x_1, x_2, x_3]$. The resulting skew-group algebra $R \ast G$ is $3$-Calabi-Yau, and can sometimes be endowed with the structure of a…

In this paper we study unitary Ramond twisted representations of minimal $W$-algebras. We classify all such irreducible highest weight representations with a non-Ramond extremal highest weight (unitarity in the Ramond extremal case, as well…

We explain (following V. Drinfeld) how the equivariant derived category of the affine Grassmannian can be described in terms of coherent sheaves on the Langlands dual Lie algebra equivariant with respect to the adjoint action, due to some…

With an eye to applications to type A and Schur-Weyl duality, we study Kazhdan-Lusztig bases for a general parabolic Hecke algebra. Parabolic Hecke algebras are idempotent subalgebras of Hecke algebras corresponding to parabolic subgroups,…

We study the pro-$p$ Iwahori-Hecke algebra and its Gelfand-Graev modules for the $p$-adic general linear group and its metaplectic covers. We develop the theory of quantum wreath products of skew polynomial type and use it to provide…

We revisit an identification of the quantum Toda lattice for $\mathrm{GL}_N$ and the truncated shifted Yangian of $\mathfrak{sl}_2$, as well as related constructions, from a purely algebraic point of view, bypassing the topological medium…

In this paper we investigate equivariant recollements of abelian (resp. triangulated) categories. We first characterize when a recollement of abelian (resp. triangulated) categories induces an equivariant recollement, i.e. a recollement…

We investigate the structure of Kazhdan-Lusztig polynomials of the symmetric group by leveraging computational approaches from big data, including exploratory and topological data analysis, applied to the polynomials for symmetric groups of…

We introduce a class of dg-algebras which generalize the classical Brauer graph algebras. They are constructed from mixed-angulations of surfaces and often admit a (relative) Calabi--Yau structure. We discovered these algebras through two…

We give a new construction of tensor product gamma factors for a pair of irreducible representations of $\operatorname{GL}_c\left(\mathbb{F}_q\right)$ and $\operatorname{GL}_k\left(\mathbb{F}_q\right)$. This construction is a finite field…

For the module category of an Artin algebra, we generalize the notion of torsion pairs to ideal torsion pairs. Instead of full subcategories of modules, ideals of morphisms of the ambient category are considered. We characterize the…

In this paper, we obtain a full classification of reductive dual pairs in a, real or complex, Lie superalgebra $\mathfrak{spo}(E)$ and Lie supergroup $\textbf{SpO}(E)$. Moreover, by looking at the natural action of the orthosymplectic Lie…

As conjectured by T. Przebinda, the transfer of characters in the Howe's correspondence should be obtained via the Cauchy-Harish-Chandra integral. In this paper, we prove that the conjecture holds for the dual pair $(G = U(p, q), G' = U(r,…

Let $\mathfrak{g}$ be a simple Lie algebra over~$\mathbb{C}$ with root system~$\Phi$. In the simply laced case, Frenkel and Kac found a particularly simple construction of~$\mathfrak{g}$, together with a Chevalley basis and explicitly given…

Let $\mathcal{H} = \mathcal{H}(W,S)$ be the Hecke algebra of the Coxeter system $(W,S)$ over $\mathbb{Z}[q^{\pm1}]$, where $W$ is the Weyl group of a symmetrizable Kac-Moody algebra. In this paper, we show that the matrix of Kazhdan-Lusztig…

We initiate the study of non-semisimple algebras in fusion categories by establishing the framework of $\mathcal{C}$-species -- analogous to the framework of species and quivers used in the study of Artin algebras. Under the (necessary)…

We study the relationship between Serre duality and the Mukai pairing for smooth and proper dg-algebras. We introduce an alternative definition of the Mukai pairing and prove that it coincides with the Mukai pairings defined by…

For the Coxeter groups of ADE type, we provide a construction of their Coxeter planes as fixed points of actions of hypergroups associated to Verlinde fusion rings. This builds upon the well-known ADE classification of…

For any ring $R$, we investigate balanced pairs of classes of modules and their relations to cotorsion triples. We characterize the case when a balanced pair generates a tilting cotorsion pair, and dually, when it cogenerates a cotilting…