Extension Of Algebra Research Articles

Relative Rota–Baxter Leibniz algebras, their characterization and cohomology

Recently, relative Rota–Baxter (Lie/associative) algebras are extensively studied in the literature from cohomological points of view. In this paper, we consider relative Rota–Baxter Leibniz algebras (rRB Leibniz algebras) as the object of our study. We construct an L∞-algebra that characterizes rRB Leibniz algebras as its Maurer–Cartan elements. Then we define representations of an rRB Leibniz algebra and introduce cohomology with coefficients in a representation. As applications of cohomology, we study deformations and abelian extensions of rRB Leibniz algebras.

A Point-Free Approach to Canonical Extensions of Boolean Algebras and Bounded Archimedean $$\ell$$-Algebras

Recently W. Holliday gave a choice-free construction of a canonical extension of a boolean algebra B as the boolean algebra of regular open subsets of the Alexandroff topology on the poset of proper filters of B. We make this construction point-free by replacing the Alexandroff space of proper filters of B with the free frame $$\mathcal {L}_B$$ generated by the bounded meet-semilattice of all filters of B (ordered by reverse inclusion) and prove that the booleanization of $$\mathcal {L}_B$$ is a canonical extension of B. Our main result generalizes this approach to the category $$\varvec{ ba \ell }$$ of bounded archimedean $$\ell$$ -algebras, thus yielding a point-free construction of canonical extensions in $$\varvec{ ba \ell }$$ . We conclude by showing that the algebra of normal functions on the Alexandroff space of proper archimedean $$\ell$$ -ideals of A is a canonical extension of $$A\in \varvec{ ba \ell }$$ , thus providing a generalization of the result of Holliday to $$\varvec{ ba \ell }$$ .

Superconformal Algebras and Holomorphic Field Theories

We show that four-dimensional superconformal algebras admit an infinite-dimensional derived enhancement after performing a holomorphic twist. The type of higher symmetry algebras we find is closely related to algebras studied by Faonte–Hennion–Kapranov, Hennion–Kapranov, and the second author with Gwilliam in the context of holomorphic QFT. We show that these algebras are related to the two-dimensional chiral algebras extracted from four-dimensional superconformal theories by Beem and collaborators; further deforming by a superconformal element induces the Koszul resolution of a plane in mathbb {C}^2 cong {mathbb {R}}^4. The central charges at the level of chiral algebras arise from central extensions of the higher symmetry algebras.

Exact Hochschild extensions and deformed Calabi-Yau completions

We introduce the Hochschild extensions of dg algebras, which are -algebras. We show that all exact Hochschild extensions are symmetric Hochschild extensions, more precisely, every exact Hochschild extension of a finite dimensional complete typical dg algebra is a symmetric -algebra. Moreover, we prove that the Koszul dual of trivial extension is Calabi-Yau completion and the Koszul dual of exact Hochschild extension is deformed Calabi-Yau completion, more precisely, the Koszul dual of the trivial extension of a finite dimensional complete dg algebra is the Calabi-Yau completion of its Koszul dual, and the Koszul dual of an exact Hochschild extension of a finite dimensional complete typical dg algebra is the deformed Calabi-Yau completion of its Koszul dual.

Irreducible modules of toroidal Lie algebras arising from ϕϵ-coordinated modules of vertex algebras

In this paper, for every ϵ∈Z, we introduce an extension of the 2-toroidal Lie algebra by certain derivations. Based on the ϕϵ-coordinated modules theory for vertex algebras, we give an explicit realization of a class of irreducible highest weight modules for this extended toroidal Lie algebra. When ϵ=1, this affords a realization of certain irreducible modules for the toroidal extended affine Lie algebras first constructed by Billig.

Two remarks on Narkiewicz’s property (P)

Due to Narkiewicz a field F has property (P) if for no polynomial fin F[x] of degree at least two there is an infinite f-invariant subset of F. We present a new example of an algebraic extension of {mathbb {Q}} satisfying (P). This is the first example in which we can find points of arbitrarily small positive Weil-height. Moreover, we study the possibility of property (P) for the field generated by all symmetric Galois extensions of {mathbb {Q}}. In particular we prove that there are no infinite backward orbits of non linear polynomials in this field.

Commutative bidifferential algebra

Motivated by the Poisson Dixmier-Moeglin equivalence problem, a systematic study of commutative unitary rings equipped with a biderivation, namely a binary operation that is a derivation in each argument, is here begun, with an eye toward the geometry of the corresponding B-varieties. Foundational results about extending biderivations to localisations, algebraic extensions and transcendental extensions are established. Resolving a deficiency in Poisson algebraic geometry, a theory of base extension is achieved, and it is shown that dominant B-morphisms admit generic B-fibres. A bidifferential version of the Dixmier-Moeglin equivalence problem is articulated.

Symmetric Zinbiel superalgebras

The notion of symmetric Zinbiel superalgebras is introduced. We prove that the nilpotency index of a symmetric Zinbiel superalgebra is not greater than 4 and describe two-generated symmetric Zinbiel algebras and odd generated superalgebras. We discuss identities of mono and binary symmetric Zinbiel and Leibniz algebras. It is proven that each quadratic Zinbiel algebra is 2-step nilpotent. Also, we study double extensions of symmetric Zinbiel algebras.

Tilting Modules and Exceptional Sequences for a Family of Dual Extension Algebras

We provide a classification of generalized tilting modules and full exceptional sequences for a certain family of quasi-hereditary algebras, namely dual extension algebras of path algebras of uniformly oriented linear quivers, modulo the ideal generated by paths of length two, with their opposite algebra. An important step in the classification is to prove that all indecomposable self-orthogonal modules (with respect to extensions of positive degree) admit a filtration with standard subquotients or a filtration with costandard subquotients. Furthermore, we prove that that every generalized tilting module, not equal to the characteristic tilting modules, admits either a filtration with standard subquotients or a filtration with costandard subquotients. Since the algebras studied in this article admit a simple-preserving duality, combining these two statements reduces the problem to classifying generalized tilting modules admitting a filtration with standard subquotients. To finalize the classification of generalized tilting modules we develop a combinatorial model for the poset of indecomposable self-orthogonal modules with standard filtration with respect to the relation arising from higher extensions. This model is also used for the classification of full exceptional sequences.

Homological mirror symmetry for Milnor fibers via moduli of A∞$A_\infty$‐structures

We show that the base spaces of the semiuniversal unfoldings of some weighted homogeneous singularities can be identified with moduli spaces of A ∞ $A_\infty$ -structures on the trivial extension algebras of the endomorphism algebras of the tilting objects. The same algebras also appear in the Fukaya categories of their mirrors. Based on these identifications, we discuss applications to homological mirror symmetry for Milnor fibers, and give a proof of homological mirror symmetry for an n $n$ -dimensional affine hypersurface of degree n + 2 $n+2$ and the double cover of the n $n$ -dimensional affine space branched along a degree 2 n + 2 $2n+2$ hypersurface. Along the way, we also give a proof of a conjecture of Seidel (Proceedings of the International Congress of Mathematicians, 2002) which may be of independent interest.

Quantum coordinate ring in WZW model and affine vertex algebra extensions

In this paper, we construct various simple vertex superalgebras which are extensions of affine vertex algebras, by using abelian cocycle twists of representation categories of quantum groups. This solves the Creutzig and Gaiotto conjectures (Creutzig and Gaiotto in Comm Math Phys 379:785–845, 2020, Conjecture 1.1 and 1.4) in the case of type ABC. If the twist is trivial, the resulting algebras correspond to chiral differential operators in the chiral case, and to WZW models in the non-chiral case.

Representations of Hopf-Ore Extensions of Group Algebras

In this paper, we study the representations of the Hopf-Ore extensions kG(χ− 1,a,0) of group algebra kG, where k is an algebraically closed field. We classify all finite dimensional simple kG(χ− 1,a,0)-modules under the assumption $|\chi |=\infty $ and $|\chi |=|\chi (a)|<\infty $ respectively, and all finite dimensional indecomposable kG(χ− 1,a,0)-modules under the assumption that kG is finite dimensional and semisimple, and |χ| = |χ(a)|. Moreover, we investigate the decomposition rules for the tensor product modules over kG(χ− 1,a,0) when char(k) = 0. Finally, we consider the representations of some Hopf-Ore extension of the dihedral group algebra kDn, where n = 2m, m > 1 odd, and char(k) = 0. The Grothendieck ring and the Green ring of the Hopf-Ore extension are described respectively in terms of generators and relations.

Current Hom-Lie algebras

In this paper, we study Hom-Lie structures on tensor products. In particular, we consider current Hom-Lie algebras and discuss their representations. We determine faithful representations of minimal dimension of current Heisenberg Hom-Lie algebras. Moreover derivations, including generalized derivations, and centroids are studied. Furthermore, cohomology and extensions of current Hom-Lie algebras are also considered.

On isoclinism of Hom-Lie algebras

Hom-isoclinism of central extensions of Hom-Lie algebras is introduced and studied. This concept is used to describe the Schur multiplier of Hom-Lie algebras, to determine the structure of Hom-stem covers, and to establish a relation between Hom-stem covers and the universal central extensions of perfect Hom-Lie algebras.

The Ext-algebra of standard modules over dual extension algebras

We exhibit an isomorphism of associative algebras between the Ext-algebra ExtΛ⁎(Δ,Δ) of standard modules over the dual extension algebra Λ of two directed algebras B and A and the dual extension algebra of the Ext-algebra ExtB⁎(L,L) with A. There are natural A∞-structures on these Ext-algebras, and, under certain technical assumptions on B, we describe that on ExtΛ⁎(Δ,Δ) completely in terms of that on ExtB⁎(L,L). As an example, we compute these A∞-structures explicitly in the case where B=A=KAn/(radKAn)ℓ.

The theory of j-operators with application to (weak) liftings of DG modules

A major part of this paper is devoted to an in-depth study of j-operators and their properties. This study enables us to obtain several results on liftings and weak liftings of DG modules along simple extensions of DG algebras and unify the proofs of the existing results obtained by the authors on these subjects. Finally, we provide a new characterization of the (weak) lifting property of DG modules along simple extensions of DG algebras.

A boson-fermion correspondence in cohomological Donaldson–Thomas theory

AbstractWe introduce and study a fermionisation procedure for the cohomological Hall algebra $\mathcal{H}_{\Pi_Q}$ of representations of a preprojective algebra, that selectively switches the cohomological parity of the BPS Lie algebra from even to odd. We do so by determining the cohomological Donaldson–Thomas invariants of central extensions of preprojective algebras studied in the work of Etingof and Rains, via deformed dimensional reduction. Via the same techniques, we determine the Borel–Moore homology of the stack of representations of the $\unicode{x03BC}$ -deformed preprojective algebra introduced by Crawley–Boevey and Holland, for all dimension vectors. This provides a common generalisation of the results of Crawley-Boevey and Van den Bergh on the cohomology of smooth moduli schemes of representations of deformed preprojective algebras and my earlier results on the Borel–Moore homology of the stack of representations of the undeformed preprojective algebra.

Representations and cohomologies of relative Rota-Baxter Lie algebras and applications

In this paper, first we give the notion of a representation of a relative Rota-Baxter Lie algebra and introduce the cohomologies of a relative Rota-Baxter Lie algebra with coefficients in a representation. Then we classify abelian extensions of relative Rota-Baxter Lie algebras using the second cohomology group, and classify skeletal relative Rota-Baxter Lie 2-algebras using the third cohomology group as applications. At last, using the established general framework of representations and cohomologies of relative Rota-Baxter Lie algebras, we give the notion of representations of Rota-Baxter Lie algebras, which is consistent with representations of Rota-Baxter associative algebras in the literature, and introduce the cohomologies of Rota-Baxter Lie algebras with coefficients in a representation. Applications are also given to classifying abelian extensions of Rota-Baxter Lie algebras and skeletal Rota-Baxter Lie 2-algebras.

Amalgamated duplication of Banach algebras from homological point of view

In a paper from 2002, Zhang [Weak amenability of module extensions of Banach algebras, Trans. Amer. Math. Soc. 354 (2002) 4131–4151] used module extensions to construct an example of a weakly amenable Banach algebra which is not 3-weakly amenable. Following his approach, we show that for amalgamated duplication of dual Banach algebras, we are able to choose appropriate conditions for weak Connes amenability of those Banach algebras which extend the Zhang’s result in a much more general setting. Moreover, apart from the characterization of minimal idempotents, we restrict our investigation to the study of injectivity, projectivity and flatness of amalgamated duplication of Banach algebras. Particularly, it is worth mentioning that our result paves the way for studying of homological properties and weak Connes amenability of those dual Banach algebras that have a semidirect product-like structure including Lau algebras.

The section conjecture over large algebraic extensions of finitely generated fields

AbstractLet K be a finitely generated extension of its prime field and let be an integer. We prove the injectivity part of the section conjecture of Grothendieck for almost all and for all smooth geometrically integral projective curves of genus ≥1 over the field .