Indecomposable Algebras Research Articles

LEFT INVARIANT COMPLEX STRUCTURES ON NILPOTENT SIMPLY CONNECTED INDECOMPOSABLE 6-DIMENSIONAL REAL LIE GROUPS

Integrable complex structures on indecomposable 6-dimensional nilpotent real Lie algebras have been computed in a previous paper, along with normal forms for representatives of the various equivalence classes under the action of the automorphism group. Here we go to the connected simply connected Lie group G0 associated to such a Lie algebra 𝔤. For each normal form J of integrable complex structures on 𝔤, we consider the left invariant complex manifold G = (G0, J) associated to G0 and J. We explicitly compute a global holomorphic chart for G and we write down the multiplication in that chart.

COMPLEX STRUCTURES ON INDECOMPOSABLE 6-DIMENSIONAL NILPOTENT REAL LIE ALGEBRAS

We show how resorting to dependable computer calculations makes it possible to compute all integrable complex structures on indecomposable 6-dimensional nilpotent real Lie algebras and their equivalence classes under the automorphism group of the Lie algebra. We also prove that the set comprised of all integrable complex structures on such a Lie algebra is a smooth submanifold of ℝ36.

The inverse problem for six-dimensional codimension two nilradical lie algebras

Ado’s theorem asserts that every real Lie Algebra g of dimension n has a faithful representation as a subalgebra of gl(p,R) for some p. The theorem offers no practical information about the size of p in relation to n and in principle p may be very large compared to n. This article is concerned with finding representations for a certain class of six-dimensional Lie algebras, specifically, real, indecomposable algebras that have a codimension two nilradical. These algebras were classified by Turkowski and comprise of 40 cases, some of which contain up to four parameters. Linear representations are found for each algebra in these classes: More precisely, a matrix Lie group is given whose Lie algebra corresponds to each algebra in Turkowski’s list and can be found by differentiating and evaluating at the identity element of the group. In addition a basis for the right-invariant vector fields that are dual to the Maurer-Cartan forms are given thereby providing an effective realization of Lie’s third theorem. The geodesic spray of the canonical symmetric connection for each of the 40 linear Lie group G is given. Thereafter the inverse problem of the calculus of variations for each of the geodesic sprays is investigated. In all cases it is determined whether the spray is of Euler-Lagrange type and in the affirmative case at least one concrete Lagrangian is written down. In none of the cases is there a Lagrangian of metric type.

Free nilpotent minimum algebras

AbstractIn the present paper we give a description of the free algebra over an arbitrary set of generators in the variety of nilpotent minimum algebras. Such description is given in terms of a weak Boolean product of directly indecomposable algebras over the Boolean space corresponding to the Boolean subalgebra of the free NM‐algebra. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Indecomposable Lie algebras with nontrivial Levi decomposition cannot have filiform radical

Let g = s n r be an indecomposable Lie algebra with nontrivial semisimple Levi subalgebra s and nontrivial solvable radical r. In this note it is proved that r cannot be isomorphic to a filiform nilpotent Lie algebra. The proof uses the fact that any Lie algebra g = snr with filiform radical would degenerate (even contract) to the Lie algebra snfn, where fn is the standard graded filiform Lie algebra of dimension n = dim r. This leads to a contradiction, since no such indecomposable algebra snr with r = fn exists

The number of simple modules of a cellular algebra

Let n be a natural number, and let A be an indecomposable cellular algebra such that the spectrum of its Cartan matrix C is of the form { n , 1, …, 1}. In general, not every natural number could be the number of non-isomorphic simple modules over such a cellular algebra. Thus, two natural questions arise: (1) which numbers could be the number of non-isomorphic simple modules over such a cellular algebra A ? (2) Given such a number, is there a cellular algebra such that its Cartan matrix has the desired property ? In this paper, we shall completely answer the first question, and give a partial answer to the second question by constructing cellular algebras with the pre-described Cartan matrix.

The Krull―Schmidt Theorem for Henselian Rings

A necessary and sufficient condition on a local ring over which all indecomposable finite-dimensional algebras are local is found. The Krull―Schmidt theorem for a class of rings that includes both the Henselian valuation rings and the rings of integers of multidimensional fields is proved. Bibliography: 2 titles.

GROUPS OF AUTOMORPHISMS OF CHEVALLEY ALGEBRAS

In the paper we present an extension of Segal theorem on Lie algebra automorphisms of upper triangular nilpotent matrices to the whole class of nilpotent subalgebras generated by positive parts of indecomposable Chevalley algebras.

Quasi-Identities of Congruence-Distributive Quasivarieties of Algebras

LetR be an arbitrary quasivariety of algebras and A an algebra not necessarily inR. A congruence θ ∈ ConA is an R-congruence if A/θ ∈ R. The set ConRA of all R-congruences of A makes an algebraic lattice under inclusion. A quasivariety R of algebras is congruence-distributive if the lattice of Rcongruences is distributive for every algebra A. In the case when R is a variety this definition coincides with that of a congruence-distributive variety. An algebra A in a quasivariety R is called finitely subdirectly R-indecomposable if, for every representation of A in the form of a finite subdirect product ∏ {Ai ∈ R : i < n}, one of the projections πi : A → Ai is an isomorphism. Note that if R is congruence-distributive then every finitely subdirectly R-indecomposable algebra is (absolutely) finitely subdirectly indecomposable (see [1, 2]). It is well known [3] that every congruence-distributive variety of algebras of finite signature with the finitely axiomatizable class of finitely subdirectly indecomposable algebras has a finite basis of identities. It was asked in some articles (see [1, 4, 5]) whether any congruence-distributive quasivariety R of algebras with the finitely axiomatizable class of finitely subdirectly R-indecomposable algebras has a finite basis of quasi-identities. A positive answer to this question is known in two particular cases: for finitely generated quasivarieties [1] and for quasivarieties with equationally definable intersections of principal congruences [5]. We say that a lattice L has at most countable width if every antichain in L is at most countable. A lattice L satisfies the minimality condition if every chain in L has a least element. A lattice with the maximality condition is defined by reversal. We say that a class R of algebras is finitely axiomatizable relative to a class K if there exists a finite set Ψ of propositions such that R = K ∩ Mod(Ψ). In this article we prove the following two theorems.

Fields of representatives of commutative local rings and maximal scalar fields of finite-dimensional algebras

The structure of scalar fields for a directly indecomposable finite-dimensional algebra treated as a ring is studied. Scalar fields are assumed similar if their action on a ring is identical modulo an annihilator. The criterion for a class of maximal scalar fields to be unique under a similitude relation is established.

Definability of directly indecomposable congruence modular algebras

It is proved that the directly indecomposable algebras in a congruence modular equational class ν form a ∀∃∀ first-order class provided that ν fulfils some two natural assumptions.

Conditions for the spatial flatness and spatial injectivity of an indecomposable CSL algebra of finite width

This paper is concerned with the connection between the geometric properties of the latticeL of subspaces of a Hilbert spaceH and homological properties (flatness and injectivity) ofH regarded as a natural module over the reflexive algebra AlgL that consists of all operators leaving invariant each element of the latticeL. It follows from these results that the cohomology groups with coefficients inB(H) are trivial for a broad class of reflexive algebras.

Embeddability and epimorphism relations in undecidable locally finite varieties

Some of the results obtained by A. Pinus for congruence distributive varieties are generalized to the case of congruence-nodular varieties. Most attention is paid to congruence-modular varieties containing a subdirectly indecomposable algebra with non-Abelian monolith.

On the Structure of Bernstein Algebras

Journal of the London Mathematical SocietyVolume 51, Issue 1 p. 41-52 Notes and Papers On the Structure of Bernstein Algebras Teresa Cortés, Teresa Cortés Departamento de Matemáticas, Facultad de Ciencias, Universidad de Oviedo, Calvo Sotelo s/n, 33007 Oviedo, SpainSearch for more papers by this authorFernando Montaner, Fernando Montaner Departamento de Matemáticas, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, SpainSearch for more papers by this author Teresa Cortés, Teresa Cortés Departamento de Matemáticas, Facultad de Ciencias, Universidad de Oviedo, Calvo Sotelo s/n, 33007 Oviedo, SpainSearch for more papers by this authorFernando Montaner, Fernando Montaner Departamento de Matemáticas, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, SpainSearch for more papers by this author First published: February 1995 https://doi.org/10.1112/jlms/51.1.41Citations: 10 Research partially supported by the DGICYT (PS 90-0129). AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volume51, Issue1February 1995Pages 41-52 RelatedInformation

Three remarks on the modular commutator

First a problem of Ralph McKenzie is answered by proving that in a finitely directly representable variety every directly indecomposable algebra must be finite. Then we show that there is no local proof of the fundamental theorem of Abelian algebras nor of H. P. Gumm's permutability results. This part may also be of interest for those investigating non-modular Abelian algebras. Finally we provide a Gumm-type characterization of the situation when twonot necessarily comparable congruences centralize each other. In doing this, we introduce a four variable version of the difference term in every modular variety. A “two-terms condition” is also investigated.

On decidable varieties of Heyting algebras

In this paper we present a new proof of a decidability result for the firstorder theories of certain subvarieties of Heyting algebras. By a famous result of Grzegorczyk, the full first-order theory of Heyting algebras is undecidable. In contrast, the first-order theory of Boolean algebras and of many interesting subvarieties of Boolean algebras is decidable by a result of Tarski [8]. In fact, Kozen [6] gives a comprehensive quantitative classification of the complexities of the first-order theories of various subclasses of Boolean algebras (including the full variety).This stark contrast may be reconciled from the standpoint of universal algebra as arising out of the byplay between structure and decidability: A good structure theory entails positive decidability results. Boolean algebras have a well-developed structure theory [5], while the corresponding theory for Heyting algebras is quite meagre. Viewed in this way, we may hope to obtain decidability results if we focus attention on subclasses of Heyting algebras with good structural properties.K. Idziak and P. M. Idziak [4] have considered an interesting subvariety of Heyting algebras, , which is the variety generated by all linearly-ordered Heyting algebras. This variety is shown to be the largest subvariety of Heyting algebras with a decidable theory of its finite members. However their proof is rather indirect, proceeding via semantic interpretation into the monadic second order theory of trees. The latter is a powerful theory—it interprets many other theories—but is computationally highly infeasible. In fact, by a celebrated theorem of Rabin, its complexity is not bounded by any elementary recursive function. Consequently, the proof of [4], besides being indirect, also gives no information on the quantitative computational complexity of the theory of .Here we pursue the theme of structure and decidability. We isolate the indecomposable algebras in and use this to prove a theorem on the structure of if -algebras. This theorem relates the -algebras structurally to Boolean algebras. This enables us to bootstrap the known decidability results for Boolean algebras to the variety if .

The Green correspondence and normal subgroups

In a recent paper, Harris and Knorr [6] have discovered a connection between the Brauer correspondence and the covering relation between blocks of a finite group and blocks of its normal subgroups. Green’s module point of view of blocks [4] suggests that this result has a moduletheoretic proof and that there is a more general result on indecomposable modules. We shall show that this is the case by giving a short moduletheoretic proof of the Harris-Kniirr theorem and then use the ideas of the proof to formulate and establish a result relating the Green correspondence [IS] with normal subgroups. Let us establish some notation. Let G be a finite group and H a normal subgroup. Let k be a field of prime characteristic p and b a block of H so b is an indecomposable algebra summand of kH. We shall regard b as a k[H x HI-module in the usual way. Let D be a defect group of b, K = N,(D) and b, equal the block of K with bf’ = b, that is, the Brauer correspondent of b. Finally, set L = N,(D) so K is a normal subgroup of L.

Spectral duality for block cohomology

Throughout this paper k denotes an algebraically closed field of characteristic p > 0, G denotes a finite group, B denotes a p-block of G corresponding to the indecomposable algebra summand kB of kG, D denotes a defect group of B (it is unique up to conjugacy in G), p denotes the p-rank of D (that is, the maximum of the ranks of its elementary abelian p-subgroups) and S, T denote kB-modules. A graded polynomial subalgebra R of H ∗(G, k) is specified by taking p algebraically independent canonical homogeneous elements, which are certain Chern classes, to be its generators. A spectral sequence is constructed with Ext R ∗∗ ( Ext kB ∗(S, T), R ) as its E 2 term and with Ext kB ∗(T, S) , with a displaced grading, as its abutment. The latter is thus naturally filtered. A brief account of the relevant algebraic topology, commutative algebra and homological algebra is given.

Narrowness implies uniformity

This paper is about varietiesV of universal algebras which satisfy the following numerical condition on the spectrum: there are only finitely many prime integersp such thatp is a divisor of the cardinality of some finite algebra inV. Such varieties are callednarrow. The variety (or equational class) generated by a classK of similar algebras is denoted by V(K)=HSPK. We define Pr (K) as the set of prime integers which divide the cardinality of a (some) finite member ofK. We callK narrow if Pr (K) is finite. The key result proved here states that for any finite setK of finite algebras of the same type, the following are equivalent: (1) SPK is a narrow class. (2) V(K) has uniform congruence relations. (3) SK has uniform congruences and (3) SK has permuting congruences. (4) Pr (V(K))= Pr(SK). A varietyV is calleddirectly representable if there is a finite setK of finite algebras such thatV= V(K) and such that all finite algebras inV belong to PK. An equivalent definition states thatV is finitely generated and, up to isomorphism,V has only finitely many finite directly indecomposable algebras. Directly representable varieties are narrow and hence congruence modular. The machinery of modular commutators is applied in this paper to derive the following results for any directly representable varietyV. Each finite, directly indecomposable algebra inV is either simple or abelian.V satisfies the commutator identity [x,y]=x·y·[1,1] holding for congruencesx andy over any member ofV. The problem of characterizing finite algebras which generate directly representable varieties is reduced to a problem of ring theory on which there exists an extensive literature: to characterize those finite ringsR with identity element for which the variety of all unitary leftR-modules is directly representable. (In the terminology of [7], the condition is thatR has finite representation type.) We show that the directly representable varieties of groups are precisely the finitely generated abelian varieties, and that a finite, subdirectly irreducible, ring generates a directly representable variety iff the ring is a field or a zero ring.

Classification of some 2-graded lie algebras

We continue the study of the structure of 2-graded Lie algebras (see next section for the definition) which was begun by Hochschild [6] and Hochschild and the author [4]. AU semi-simple 2-graded Lie algebras over an algebraically closed field of characteristic 0 have been determined in [4]. In this paper we introduce the analog of the Killing form for 2-graded Lie algebras. AS in the case of ordinary Lie algebras, this form plays an important role in the structure theory. For semi-simple 2-graded Lie algebras this form is non-degenerate although the converse is not true. Semi-simplicity of 2-graded Lie algebras is preserved under both extension and restriction of scalars. Our main result is the classification of indecomposable 2-graded Lie algebras L over an algebraically closed field of characteristic 0 under the condition that the zero component Lo of L be simple (Theorem 8). There are seven infinite series of such algebras. This classification is accomplished in two steps. The first step is the classification of those 2-graded Lie algebras L which are simple and have Lo simple (Theorem 6). In the proof of this theorem we need the following fact: If‘ L, is a simple Lie algebra then the multiplicity 172 r) (resp. nzl ) of L, in the symmetric power S2(Lo) (resp. the exterior power A-(Lo)) is nzo = I if Lo =: rl ,t (112 2),