Mathematics

The "odd transgression" introduced by the authors in an earlier article is applied to construct and study the inverse image functor in the theory of Courant algebroids.

For a finite group G , Turaev introduced the notion of a braided G -crossed fusion category. The classification of braided G -crossed extensions of braided fusion categories was studied by Etingof, Nikshych and Ostrik in terms of certain group cohomological data. In this paper we will define the notion of a G -crossed Frobenius ⋆ -algebra and give a classification of (strict) G -crossed extensions of a commutative Frobenius ⋆ -algebra R equipped with a given action of G , in terms of the second group cohomology H 2 (G, R × ) . Now suppose that B is a non-degenerate braided fusion category equipped with a braided action of a finite group G . We will see that the associated G -graded fusion ring is in fact a (strict) G -crossed Frobenius ⋆ -algebra. We will describe this G -crossed fusion ring in terms of the classification of braided G -actions by Etingof, Nikshych, Ostrik and derive a Verlinde formula to compute its fusion coefficients.

We associate a formal power series with integer coefficients to a positive real number, we interpret this series as a " q -analogue of a real." The construction is based on the notion of q -deformed rational number introduced in arXiv:1812.00170. Extending the construction to negative real numbers, we obtain certain Laurent series.

For a quantum group, we study those right coideal subalgebras, for which all irreducible representations are one-dimensional. If a right coideal subalgebra is maximal with this property, then we call it a Borel subalgebra. Besides the positive part of the quantum group and its reflections, we find new unfamiliar Borel subalgebras, for example ones containing copies of the quantum Weyl algebra. Given a Borel subalgebra, we study its induced (Verma-)modules and prove among others that they have all irreducible finite-dimensional modules as quotients. We then give structural results using the graded algebra, which in particular leads to a conjectural formula for all triangular Borel subalgebras, which we partly prove. As examples, we determine all Borel subalgebras of U q ( sl 2 ) and U q ( sl 3 ) and discuss the induced modules.

In this paper, we study an impact of Leibniz algebras on the algebraic structure of N -graded vertex algebras. We provide easy ways to characterize indecomposable non-simple N -graded vertex algebras ⊕ ∞ n=0 V (n) such that dim V (0) ≥2 . Also, we examine the algebraic structure of N -graded vertex algebras V= ⊕ ∞ n=0 V (n) such that dim V (0) ≥2 and V (1) is a (semi)simple Leibniz algebra that has s l 2 as its Levi factor. We show that under suitable conditions this type of vertex algebra is indecomposable but not simple. Along the way we classify vertex algebroids associated with (semi)simple Leibniz algebras that have s l 2 as their Levi factor.

Let A be a finite dimensional unital commutative associative algebra and let B be a finite dimensional vertex A -algebroid such that its Levi factor is isomorphic to s l 2 . Under suitable conditions, we construct an indecomposable non-simple N -graded vertex algebra V B ¯ ¯ ¯ ¯ ¯ ¯ from the N -graded vertex algebra V B associated with the vertex A -algebroid B . We show that this indecomposable non-simple N -graded vertex algebra V B ¯ ¯ ¯ ¯ ¯ ¯ is C 2 -cofinite and has only two irreducible modules.

This paper is based on a talk given at the 14-th International Workshop on Differential Geometry and Its Applications, hosted by the Petroleum Gas University from Ploiesti, between July 9-th and July 11-th, 2019. After presenting some historical facts, we will consider some geometry problems related to unification approaches. Jordan algebras and Lie algebras are the main non-associative structures. Attempts to unify non-associative algebras and associative algebras led to UJLA structures. Another algebraic structure which unifies non-associative algebras and associative algebras is the Yang-Baxter equation. We will review topics relared to the Yang-Baxter equation and Yang-Baxter systems, with the goal to unify constructions from Differential Geometry.

We prove the injectivity of the Miura maps for W-superalgberas and the isomorphisms between the Poisson vertex superalgebras obtained as the associated graded of the W-superalgebras in terms of the Li's filtration and the level 0 Poisson vertex superalgebras associated with the arc spaces of the corresponding Slodowy slices in full generality.

In this paper, we study the family of vertex operator algebras SF(d ) + , known as symplectic fermions. This family is of a particular interest because these VOAs are irrational and C 2 -cofinite. We determine the Zhu's algebra A(SF(d ) + ) and show that the equality of dimensions of A(SF(d ) + ) and the C 2 --algebra P(SF(d ) + ) holds for d≥2 (the case of d=1 was treated by T. Abe). We use these results to prove a conjecture by Y. Arike and K. Nagatomo on the dimension of the space of one-point functions on SF(d ) + .

In arXiv:1910.12059 Liu, Palcoux and Wu proved a remarkable necessary condition for a fusion ring to admit a unitary categorification, by constructing invariants of the fusion ring that have to be positive if it is unitarily categorifiable. The main goal of this note is to provide a somewhat more direct proof of this result. In the last subsection we discuss integrality properties of the Liu-Palcoux-Wu invariants.