Gorenstein projective bimodules via monomorphism categories and filtration categories

We generalize the monomorphism category from quiver (with monomial relations) to arbitrary finite dimensional algebras by a homological definition. Given two finite dimension algebras A and B, we use the special monomorphism category $\mathsf{Mon}(B,A\text{-}\mathsf{Gproj})$ to describe some Gorenstein projective bimodules over the tensor product of A and B. If one of the two algebras is Gorenstein, we give a sufficient and necessary condition for $\mathsf{Mon}(B,A\text{-}\mathsf{Gproj})$ being the category of all Gorenstein projective bimodules. In addition, if both A and B are Gorenstein, we can describe the category of all Gorenstein projective bimodules via filtration categories. Similarly, in this case, we get the same result for infinitely generated Gorenstein projective bimodules.     

Indecomposable involutive set-theoretic solutions of the Yang–Baxter equation

We describe the indecomposable involutive non-degenerate set-theoretic solutions of the Yang–Baxter equation as dynamical extensions of non-degenerate left cycle sets. Moreover we characterize the indecomposable dynamical extensions and we produce several examples. As an application we construct a family of finite indecomposable solutions whose structure groups have not the unique product property.     

Embedding Banach spaces into the space of bounded functions with countable support

We prove that a WLD subspace of the space 201800308:mana201800308-math-0001" class="section_image" src="/cms/asset/71777a6e-fdf1-40ca-9538-0a7a42d9a48e/mana201800308-math-0001.png"> consisting of all bounded, countably supported functions on a set Γ embeds isomorphically into 201800308:mana201800308-math-0002" class="section_image" src="/cms/asset/eb0c08eb-9208-4e36-959b-0416665b92f2/mana201800308-math-0002.png"> if and only if it does not contain isometric copies of 201800308:mana201800308-math-0003" class="section_image" src="/cms/asset/27533b6f-fe13-4878-8d52-8f452519bcd5/mana201800308-math-0003.png">. Moreover, a subspace of 201800308:mana201800308-math-0004" class="section_image" src="/cms/asset/941943a9-9b8d-4252-ac6a-a03a05981684/mana201800308-math-0004.png"> is constructed that has an unconditional basis, does not embed into 201800308:mana201800308-math-0005" class="section_image" src="/cms/asset/6bfe9958-c29d-4bdb-984d-fc35620aa6e0/mana201800308-math-0005.png">, and whose every weakly compact subset is separable (in particular, it cannot contain any isomorphic copies of 201800308:mana201800308-math-0006" class="section_image" src="/cms/asset/48be3f92-0893-4a7a-8c9a-50b0de42c7bf/mana201800308-math-0006.png">).     

The martingale problem for a class of nonlocal operators of diagonal type

We consider systems of stochastic differential equations of the form for with continuous, bounded and non‐degenerate coefficients. Here are independent one‐dimensional stable processes with . In this article we research on uniqueness of weak solutions to such systems by studying the corresponding martingale problem. We prove the uniqueness of weak solutions in the case of diagonal coefficient matrices.     

Universal deformation rings and self-injective Nakayama algebras

Let k be a field and let Λ be an indecomposable finite dimensional k-algebra such that there is a stable equivalence of Morita type between Λ and a self-injective split basic Nakayama algebra over k. We show that every indecomposable finitely generated Λ-module V has a universal deformation ring $R(\mathrm{\Lambda },V)$ and we describe $R(\mathrm{\Lambda },V)$ explicitly as a quotient ring of a power series ring over k in finitely many variables. This result applies in particular to Brauer tree algebras, and hence to p-modular blocks of finite groups with cyclic defect groups.