Isoclinism of Algebras with Bracket

Abstract. In this paper we give an introduction to the theory of uni-versal central extensions of perfect Lie algebras. In particular, we willprovide a model for the universal coverings of Lie tori and we show thatautomorphisms and derivations lift to the universal coverings. We alsoprove that the universal covering of a Lie Λ-torus of type ∆ is again a LieΛ-torus of type ∆. 0. IntroductionCentral extensions play an important role in the theory of Lie algebras.Universal central extension of Lie algebras over rings were described in [23, §1],later references on universal central extensions are [12, §1], [15, 1.9], [24, 7.9] or[21]. Central extensions in the category of certain topological Lie algebras arestudied in [17]. Garland studies universal central extensions of Lie algebrasoverfields [12, §1]. In particular, he constructsa model ofuniversalcentralextensionof a perfect Lie algebra, using the universal 2-cocycle, which is different fromthe Van der Kallen , s model (see [23, §1]). Our construction for universal centralextensions of Lie tori is essentially the Van der Kallen’s model.Lie tori play a critical role in the theory of extended affine Lie algebras whichare natural generalization of finite dimensional simple Lie algebras and affineLie algebras. Yoshii and Neher were interested in Lie tori primarily becauseof the connection between Lie tori and extended affine Lie algebras ([26] and[19]). The centerless core of an extended affine Lie algebra is a centerless ofLie torus and conversely any centerless Lie torus is the centerless core of anextended affine Lie algebra [26, Theorem 7.3]. Extended affine Lie algebrasare defined axiomatically by Alison, Azam, Berman, Gao and Pianzola in [1].The various classes of these Lie algebras have been investigated in many papers(see [4, 6, 13, 14, 20, 25, 27] and [16]). Lie tori as well as extended affine Lie

In the category of Hom-Leibniz algebras we introduce the notion of Hom-co-representation as adequate coefficients to construct the chain complex from which we compute the Leibniz homology of Hom-Leibniz algebras. We study universal central extensions of Hom-Leibniz algebras and generalize some classical results, nevertheless it is necessary to introduce new notions of α-central extension, universal α-central extension and α-perfect Hom-Leibniz algebra due to the fact that the composition of two central extensions of Hom-Leibniz algebras is not central. We also provide the recognition criteria for these kind of universal central extensions. We prove that an α-perfect Hom-Lie algebra admits a universal α-central extension in the categories of Hom-Lie and Hom-Leibniz algebras and we obtain the relationships between both of them. In case α = Id we recover the corresponding results on universal central extensions of Leibniz algebras.

We study the derivations, the central extensions and the automorphism group of the extended Schrödinger–Virasoro Lie algebra [Formula: see text], introduced by Unterberger in the context of two-dimensional conformal field theory and statistical physics. Moreover, we show that [Formula: see text] is an infinite-dimensional complete Lie algebra, and the universal central extension of [Formula: see text] in the category of Leibniz algebras is the same as that in the category of Lie algebras.

The search for elliptic quantum groups leads to a modified quantum Yang–Baxter relation and to a special class of quasi-triangular quasi-Hopf algebras. This Letter calculates deformations of standard quantum groups (with or without spectral parameter) in the category of quasi-Hopf algebras. An earlier investigation of the deformations of quantum groups, in the category of Hopf algebras, showed that quantum groups are generically rigid: Hopf algebra deformations exist only under some restrictions on the parameters. In particular, affine Kac–Moody algebras are more rigid than their loop algebra quotients and only the latter (in the case of sl(n)) can be deformed to elliptic Hopf algebras. The generalization to quasi-Hopf deformations lifts this restriction. The full elliptic quantum groups (with central extension) associated with sl(n) are thus quasi-Hopf algebras. The universal R-matrices satisfy a modified Yang–Baxter relation and are calculated more or less explicitly. The modified classical Yang–Baxter relation is obtained and the elliptic solutions are worked out explicitly.

The Initial Algebra Theorem by Trnkov\'a et al.~states, under mild assumptions, that an endofunctor has an initial algebra provided it has a pre-fixed point. The proof crucially depends on transfinitely iterating the functor and in fact shows that, equivalently, the (transfinite) initial-algebra chain stops. We give a constructive proof of the Initial Algebra Theorem that avoids transfinite iteration of the functor. For a given pre-fixed point $A$ of the functor, it uses Pataraia's theorem to obtain the least fixed point of a monotone function on the partial order formed by all subobjects of $A$. Thanks to properties of recursive coalgebras, this least fixed point yields an initial algebra. We obtain new results on fixed points and initial algebras in categories enriched over directed-complete partial orders, again without iteration. Using transfinite iteration we equivalently obtain convergence of the initial-algebra chain as an equivalent condition, overall yielding a streamlined version of the original proof.

Two characterisations of groups amongst monoids

The method to obtain massive non-relativistic states from the Poincaré algebra is twofold. First, following İnönü and Wigner, the Poincaré algebra has to be contracted to the Galilean one. Second, the Galilean algebra has to be extended to include the central mass operator. We show that the central extension might be properly encoded in the non-relativistic contraction. In fact, any İnönü–Wigner contraction of one algebra to another corresponds to an infinite tower of Abelian extensions of the latter. The proposed method is straightforward and holds for both central and non-central extensions. Apart from the Bargmann (non-zero mass) extension of the Galilean algebra, our list of examples includes the Weyl algebra obtained from an extension of the contracted SO(3) algebra, the Carrollian (ultrarelativistic) contraction of the Poincaré algebra, the exotic Newton–Hooke algebra and some others.This paper is dedicated to the memory of Laurent Houart (1967–2011).

The main theme of the thesis is to present and compare three different viewpoints on semi-abelian homology, resulting in three ways of defining and calculating homology objects. Any two of these three homology theories coincide whenever they are both defined, but having these different approaches available makes it possible to choose the most appropriate one in any given situation, and their respective strengths complement each other to give powerful homological tools. The oldest viewpoint, which is borrowed from the abelian context where it was introduced by Barr and Beck, is comonadic homology, generating projective simplicial resolutions in a functorial way. This concept only works in monadic semi-abelian categories, such as semi-abelian varieties, including the categories of groups and Lie algebras. Comonadic homology can be viewed not only as a functor in the first entry, giving homology of objects for a particular choice of coefficients, but also as a functor in the second variable, varying the coefficients themselves. As such it has certain universality properties which single it out amongst theories of a similar kind. This is well-known in the setting of abelian categories, but here we extend this result to our semi-abelian context. Fixing the choice of coefficients again, the question naturally arises of how the homology theory depends on the chosen comonad. Again it is well-known in the abelian case that the theory only depends on the projective class which the comonad generates. We extend this to the semi-abelian setting by proving a comparison theorem for simplicial resolutions. This leads to the result that any two projective simplicial resolutions, the definition of which requires slightly more care in the semi-abelian setting, give rise to the same homology. Thus again the homology theory only depends on the projective class. The second viewpoint uses Hopf formulae to define homology, and works in a non-monadic setting; it only requires a semi-abelian category with enough projectives. Even this slightly weaker setting leads to strong results such as a long exact homology sequence, the Everaert sequence, which is a generalised and extended version of the Stallings-Stammbach sequence known for groups. Hopf formulae use projective presentations of objects, and this is closer to the abelian philosophy of using any projective resolution, rather than a special functorial one generated by a comonad. To define higher Hopf formulae for the higher homology objects the use of categorical Galois theory is crucial. This theory allows a choice of Birkhoff subcategory to generate a class of central extensions, which play a big role not only in the definition via Hopf formulae but also in our third viewpoint. This final and new viewpoint we consider is homology via satellites or pointwise Kan extensions. This makes the universal properties of the homology objects apparent, giving a useful new tool in dealing with statements about homology. The driving motivation behind this point of view is the Everaert sequence mentioned above. Janelidze's theory of generalised satellites enables us to use the universal properties of the Everaert sequence to interpret homology as a pointwise Kan extension, or limit. In the first instance, this allows us to calculate homology step by step, and it removes the need for projective objects from the definition. Furthermore, we show that homology is the limit of the diagram consisting of the kernels of all central extensions of a given object, which forges a strong connection between homology and cohomology. When enough projectives are available, we can interpret homology as calculating fixed points of endomorphisms of a given projective presentation.

A universal central extension of a group G is a central extension, G, of G satisfying the additional conditions (i) G = [G, G], and (ii) every central extension of G is split. If, by analogy with topology, we call a group G connected if H1(G, Z) = 0 and simply connected if H1(G, Z) H(G, Z) = 0, the above conditions are equivalent to saying G is simply connected. In this language, a universal central extension of G is called a universal covering. Now suppose b is a reduced irreducible root system, A is a commutative ring with unit, and (cD, A) is the elementary subgroup of the points in A of a Chevalley-Demazure group scheme with root system (. Define the Steinberg group, St ((, A), by generators and relations as in [19]. Then the main concern of this paper is to determine under what conditions ((, A) is connected and St ((, A) is simply connected. When A is a field, these questions were posed and resolved in a well-known paper of Steinberg [19]. For the groups of type Al, 1 ? 2, they have been treated by Kervaire [10] and Steinberg [20]. The results of this paper were announced in [16]. Connectivity is discussed in Section 4. For groups of rank > 2, the main result is most suggestively stated E ((, A) and St ((F, A)) are connected if and only if ((, A/m) is connected for every maximal ideal m C Thus these groups are connected unless ( C= or G2 and A has a residue field with. two elements. The above statement is also true for groupsof rank 1, provided A is semi-local. In fact, a stronger result holds in the rank 1 case: (A1, A) and St (A,, A) are connected if the ideal generated by {u2 _1, u C A*} is all of A. Although this implies that A has no residue field with 2 or 3 elements, I do not know whether these are, in general, equivalent conditions. Despite the relationship with the case of fields in the formulation. of these theorems, their proofs do not resemble the proofs for fields. They are based instead on careful exploitation of commutator relations in the groups of rank 2. To decide when St ((, A) is simply connected, it remains to determine when every central extension of St ((, A) is split. The answer, roughly

Basing ourselves on the categorical notions of central extensions and commutators in the framework of semi-abelian categories relative to a Birkhoff subcategory, we study central extensions of Leibniz algebras with respect to the Birkhoff subcategory of Lie algebras, called $$\mathsf {Lie}$$ -central extensions. We obtain a six-term exact homology sequence associated to a $$\mathsf {Lie}$$ -central extension. This sequence, together with the relative commutators, allows us to characterize several classes of $$\mathsf {Lie}$$ -central extensions, such as $$\mathsf {Lie}$$ -trivial extensions, $$\mathsf {Lie}$$ -stem extensions and $$\mathsf {Lie}$$ -stem covers, and to introduce and characterize $$\mathsf {Lie}$$ -unicentral, $$\mathsf {Lie}$$ -capable, $$\mathsf {Lie}$$ -solvable and $$\mathsf {Lie}$$ -nilpotent Leibniz algebras.

Let G be a finite nilpotent group and K a number field with torsion relatively prime to the order of G. By a sequence of central group extensions with cyclic kernel we obtain an upper bound for the minimum number of prime ideals of K ramified in a Galois extension of K with Galois group isomorphic to G. This sharpens and extends results of Geyer and Jarden and of Plans. Alternatively, we show how to use Frohlich’s result on realizing the Schur multiplicator in order to realize a family of groups given by central extensions with minimal ramification.

This book is an introduction to the use of triangulated categories in the study of representations of finite-dimensional algebras. In recent years representation theory has been an area of intense research and the author shows that derived categories of finite-dimensional algebras are a useful tool in studying tilting processes. Results on the structure of derived categories of hereditary algebras are used to investigate Dynkin algebras and interated tilted algebras. The author shows how triangulated categories arise naturally in the study of Frobenius categories. The study of trivial extension algebras and repetitive algebras is then developed using the triangulated structure on the stable category of the algebra's module category. With a comprehensive reference section, algebraists and research students in this field will find this an indispensable account of the theory of finite-dimensional algebras.

Let $R$ be an Artin algebra and $e$ an idempotent of $R$. Assume that ${\rm Tor}_i^{eRe}(Re,G)=0$ for any $G\in{\rm GProj} eRe$ and $i$ sufficiently large. Necessary and sufficient conditions are given for the Schur functor $S_e$ to induce a triangle-equivalence $\mathbb{D}_{def}(R)\simeq\mathbb{D}_{def}(eRe)$. Combine this with a result of Psaroudakis-Skartsaterhagen-Solberg [29], we provide necessary and sufficient conditions for the singular equivalence $\mathbb{D}_{sg}(R)\simeq\mathbb{D}_{sg}(eRe)$ to restrict to a triangle-equivalence $\underline{{\rm GProj} R}\simeq\underline{{\rm GProj} eRe}$. Applying these to the triangular matrix algebra $T=\left( \begin{array}{cc} A & M \quad 0 & B \end{array} \right)$, corresponding results between candidate categories of $T$ and $A$ (resp. $B$) are obtained. As a consequence, we infer Gorensteinness and CM-freeness of $T$ from those of $A$ (resp. $B$). Some concrete examples are given to indicate one can realise the Gorenstein defect category of a triangular matrix algebra as the singularity category of one of its corner algabras.

In this article we introduce the notion of the equivalence relation, isoclinism, on the central extensions of Lie algebras, and determine all central extensions occurring in an isoclinism class of a given central extension. We also show that under some conditions, the concepts of isoclinism and isomorphism between the central extensions of finite dimensional Lie algebras are identical. Finally, the connection between isoclinic extensions and the Schur multiplier of Lie algebras are discussed.

We investigate possibility of central extension for non-relativistic conformal algebras in 1+1 dimension. Three different forms of charges can be suggested. A trivial charge for temporal part of the algebra exists for all elements of l-Galilei algebra class. In attempt to find a central extension as of conformal Galilean algebra for other elements of the l-Galilei class, possibility for such extension was excluded. For integer and half integer elements of the class, we can have an infinite extension of the generalized mass charge for the Virasoro-like extended algebra. For finite algebras, a regular charge inspired by Schrödinger central extension is possible.