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Register a new userThe deformed Virasoro algebra at roots of unity
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We discuss some aspects of the representation theory of the deformed Virasoro algebra $virpq$. In particular, we give a proof of the formula for the Kac determinant and then determine the center of $virpq$ for $q$ a primitive N-th root of unity. We derive explicit expressions for the generators of the center in the limit $t=qp^{-1}to infty$ and elucidate the connection to the Hall-Littlewood symmetric functions. Furthermore, we argue that for $q=sqrtN{1}$ the algebra describes `Gentile statistics of order $N-1$, i.e., a situation in which at most $N-1$ particles can occupy the same state.
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n-ary algebras have played important roles in mathematics and mathematical physics. The purpose of this paper is to construct a deformation of Virasoro-Witt n-algebra based on an oscillator realization with two independent parameters (p, q) and investigate its n-Lie subalgebra.
Motivated by recent advances in the categorification of quantum groups at prime roots of unity, we develop a theory of 2-representations for 2-categories enriched with a p-differential which satisfy finiteness conditions analogous to those of finitary or fiat 2-categories. We construct cell 2-representations in this setup, and consider 2-categories stemming from bimodules over a p-dg category in detail. This class is of particular importance in the categorification of quantum groups, which allows us to apply our results to cyclotomic quotients of the categorifications of small quantum group of type $mathfrak{sl}_2$ at prime roots of unity by Elias-Qi [Advances in Mathematics 288 (2016)]. Passing to stable 2-representations gives a way to construct triangulated 2-representations, but our main focus is on working with p-dg enriched 2-representations that should be seen as a p-dg enhancement of these triangulated ones.
We explicitly construct two classes of infinitly many commutative operators in terms of the deformed Virasoro algebra. We call one of them local integrals and the other nonlocal one, since they can be regarded as elliptic deformations of the local and nonlocal integrals of motion obtained by V.Bazhanov, S.Lukyanov and Al.Zamolodchikov.
To describe the representation theory of the quantum Weyl algebra at an $l$th primitive root $gamma$ of unity, Boyette, Leyk, Plunkett, Sipe, and Talley found all nonsingular irreducible matrix solutions to the equation $yx-gamma xy=1$, assuming $yx eq xy$. In this note, we complete their result by finding and classifying, up to equivalence, all irreducible matrix solutions $(X, Y)$, where $X$ is singular.
The solution of Delignes conjecture on Hochschild cochains and the formality of the operad of little disks provide us with a natural homotopy Gerstenhaber algebra structure on the Hochschild cochains of an associative algebra. In this paper we construct a natural chain of quasi-isomorphisms of homotopy Gerstenhaber algebras between the Hochschild cochain complex C(A) of a regular commutative algebra A over a field of characteristic zero and the Gerstenhaber algebra of multiderivations of A. Unlike the original approach of the second author based on the computation of obstructions our method allows us to avoid the bulky Gelfand-Fuchs trick and prove the formality of the homotopy Gerstenhaber algebra structure on the sheaf of polydifferential operators on a smooth algebraic variety, a complex manifold, and a smooth real manifold.
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