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Register a new userFinite Abelian algebras are fully dualizable
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We show that every finite Abelian algebra A from congruence-permutable varieties admits a full duality. In the process, we prove that A also allows a strong duality, and that the duality may be induced by a dualizing structure of finite type. We give an explicit bound on the arities of the partial and total operations appearing in the dualizing structure. In addition, we show that the enriched partial hom-clone of A is finitely generated as a clone.
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A finite algebra $bA=alg{A;cF}$ is emph{dualizable} if there exists a discrete topological relational structure $BA=alg{A;cG;cT}$, compatible with $cF$, such that the canonical evaluation map $e_{bB}colon bBto Hom( Hom(bB,bA),BA)$ is an isomorphism for every $bB$ in the quasivariety generated by $bA$. Here, $e_{bB}$ is defined by $e_{bB}(x)(f)=f(x)$ for all $xin B$ and all $fin Hom(bB,bA)$. We prove that, given a finite congruence-modular Abelian algebra $bA$, the set of all relations compatible with $bA$, up to a certain arity, emph{entails} the whole set of all relations compatible with $bA$. By using a classical compactness result, we infer that $bA$ is dualizable. Moreover we can choose a dualizing alter-ego with only relations of arity $le 1+alpha^3$, where $alpha$ is the largest exponent of a prime in the prime decomposition of $card{A}$. This improves Kearnes and Szendrei result that modules are dualizable, and Bentz and Mayrs result that finite modules with constants are dualizable. This also solves a problem stated by Bentz and Mayr in 2013.
Let g be a free brace algebra. This structure implies that g is also a prelie algebra and a Lie algebra. It is already known that g is a free Lie algebra. We prove here that g is also a free prelie algebra, using a description of g with the help of planar rooted trees, a permutative product, and anipulations on the Poincare-Hilbert series of g.
We extend the classification of solvable Lie algebras with abelian nilradicals to classify solvable Leibniz algebras which are one dimensional extensions of an abelian nilradicals.
We prove that free pre-Lie algebras, when considered as Lie algebras, are free. Working in the category of S-modules, we define a natural filtration on the space of generators. We also relate the symmetric group action on generators with the structure of the anticyclic PreLie operad.
In this paper, we introduce the notion of a derivation of a Hom-Lie algebra and construct the corresponding strict Hom-Lie 2-algebra, which is called the derivation Hom-Lie 2-algebra. As applications, we study non-abelian extensions of Hom-Lie algebras. We show that iso- morphism classes of diagonal non-abelian extensions of a Hom-Lie algebra g by a Hom-Lie algebra h are in one-to-one correspondence with homotopy classes of morphisms from g to the derivation Hom-Lie 2-algebra DER(h).
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