GRADED TWISTED CALABI–YAU ALGEBRAS ARE GENERALIZED ARTIN–SCHELTER REGULAR

This is a general study of twisted Calabi–Yau algebras that are $\mathbb {N}$ -graded and locally finite-dimensional, with the following major results. We prove that a locally finite graded algebra is twisted Calabi–Yau if and only if it is separable modulo its graded radical and satisfies one of several suitable generalizations of the Artin–Schelter regularity property, adapted from the work of Martinez-Villa as well as Minamoto and Mori. We characterize twisted Calabi–Yau algebras of dimension 0 as separable k-algebras, and we similarly characterize graded twisted Calabi–Yau algebras of dimension 1 as tensor algebras of certain invertible bimodules over separable algebras. Finally, we prove that a graded twisted Calabi–Yau algebra of dimension 2 is noetherian if and only if it has finite GK dimension.

Extending Landau-Ginzburg Models to the Point

We classify framed and oriented 2-1-0-extended TQFTs with values in the bicategories of Landau-Ginzburg models, whose objects and 1-morphisms are isolated singularities and (either $$\mathbb {Z}_2$$ Z 2 - or $$(\mathbb {Z}_2 \times \mathbb {Q})$$ ( Z 2 × Q ) -graded) matrix factorisations, respectively. For this we present the relevant symmetric monoidal structures and find that every object $$W\in \mathbb {k}[x_1,\dots ,x_n]$$ W ∈ k [ x 1 , ⋯ , x n ] determines a framed extended TQFT. We then compute the Serre automorphisms $$S_W$$ S W to show that W determines an oriented extended TQFT if the associated category of matrix factorisations is $$(n-2)$$ ( n - 2 ) -Calabi-Yau. The extended TQFTs we construct from W assign the non-separable Jacobi algebra of W to a circle. This illustrates how non-separable algebras can appear in 2-1-0-extended TQFTs, and more generally that the question of extendability depends on the choice of target category. As another application, we show how the construction of the extended TQFT based on $$W=x^{N+1}$$ W = x N + 1 given by Khovanov and Rozansky can be derived directly from the cobordism hypothesis.

Separability of Nonassociative Algebras with Metagroup Relations

This article is devoted to a class of nonassociative algebras with metagroup relations. This class includes, in particular, generalized Cayley–Dickson algebras. The separability of the nonassociative algebras with metagroup relations is investigated. For this purpose the cohomology theory is utilized. Conditions are found under which such algebras are separable. Algebras satisfying these conditions are described.

ON THE EXISTENCE OF MAXIMAL ORDERS

We generalize the existence of maximal orders in a semi-simple algebra for general ground rings. We also improve several statements in Chaps. 5 and 6 of Reiner's book [Maximal Orders, London Mathematical Society Monographs, Vol. 5 (Academic Press, London, 1975), 395 pp.] concerning separable algebras by removing the separability condition, provided the ground ring is only assumed to be Japanese, a very mild condition. Finally, we show the existence of maximal orders as endomorphism rings of abelian varieties in each isogeny class.