Cohomology of complements of toric arrangements associated with root systems

We develop an algorithm for computing the cohomology of complements of toric arrangements. In the case a finite group $$\Gamma $$ Γ is acting on the arrangement, the algorithm determines the cohomology groups as representations of $$\Gamma $$ Γ . As an important application, we determine the cohomology groups of the complements of the toric arrangements associated with root systems of exceptional type as representations of the corresponding Weyl groups.

Cohomologies of n-Lie Algebras with Derivations

The goal of this paper is to study cohomological theory of n-Lie algebras with derivations. We define the representation of an n-LieDer pair and consider its cohomology. Likewise, we verify that a cohomology of an n-LieDer pair could be derived from the cohomology of associated LeibDer pair. Furthermore, we discuss the (n−1)-order deformations and the Nijenhuis operator of n-LieDer pairs. The central extensions of n-LieDer pairs are also investigated in terms of the first cohomology groups with coefficients in the trivial representation.

Cohomology of Lie Conformal Algebra Vir⋉Cur g

In this article, we compute cohomology groups of the semisimple Lie conformal algebra [Formula: see text] with coefficients in its irreducible modules for a finite-dimensional simple Lie algebra [Formula: see text].

Landau–Ginzburg/Calabi–Yau Correspondence for a Complete Intersection via Matrix Factorizations

By generalizing the Landau–Ginzburg/Calabi–Yau correspondence for hypersurfaces, we can relate a Calabi–Yau complete intersection to a hybrid Landau–Ginzburg model: a family of isolated singularities fibered over a projective line. In recent years Fan, Jarvis, and Ruan have defined quantum invariants for singularities of this type, and Clader and Clader–Ross have provided an equivalence between these invariants and Gromov–Witten invariants of complete intersections, in this way quantum cohomology yields an identification of the cohomology groups of the Calabi–Yau and of the hybrid Landau–Ginzburg model. It is not clear how to relate this to the known isomorphism descending from derived equivalences (due to Segal and Shipman, and Orlov and Isik). We answer this question for Calabi–Yau complete intersections of two cubics.

The Controlling $$L_\infty $$-Algebra, Cohomology and Homotopy of Embedding Tensors and Lie–Leibniz Triples

In this paper, we first construct the controlling algebras of embedding tensors and Lie–Leibniz triples, which turn out to be a graded Lie algebra and an $$L_\infty $$ L ∞ -algebra respectively. Then we introduce representations and cohomologies of embedding tensors and Lie–Leibniz triples, and show that there is a long exact sequence connecting various cohomologies. As applications, we classify infinitesimal deformations and central extensions using the second cohomology groups. Finally, we introduce the notion of a homotopy embedding tensor which will induce a Leibniz$$_\infty $$ ∞ -algebra. We realize Kotov and Strobl’s construction of an $$L_\infty $$ L ∞ -algebra from an embedding tensor, as a functor from the category of homotopy embedding tensors to that of Leibniz$$_\infty $$ ∞ -algebras, and a functor further to that of $$L_\infty $$ L ∞ -algebras.