Indecomposable Algebras Research Articles

Block theory and Brauer's first main theorem for profinite groups

We develop the local-global theory of blocks for profinite groups. Given a field k of characteristic p and a profinite group G, one may express the completed group algebra k[[G]] as a product ∏i∈IBi of closed indecomposable algebras, called the blocks of G. To each block B of G we associate a pro-p subgroup of G, called the defect group of B, unique up to conjugacy in G. We give several characterizations of the defect group in analogy with defect groups of blocks of finite groups. Our main theorem is a version of Brauer's first main theorem: a correspondence between the blocks of G with defect group D and the blocks of the normalizer NG(D) with defect group D.

Description of partial actions

In this paper, we study partial actions of groups on R-algebras, where R is a commutative ring. We describe the partial actions of groups on the indecomposable algebras with enveloping actions. Then we work on algebras that can be decomposed as product of indecomposable algebras and we give a description of the partial actions of groups on these algebras in terms of global actions.

Darboux Families and the Classification of Real Four-Dimensional Indecomposable Coboundary Lie Bialgebras

This work introduces a new concept, the so-called Darboux family, which is employed to determine coboundary Lie bialgebras on real four-dimensional, indecomposable Lie algebras, as well as geometrically analysying, and classifying them up to Lie algebra automorphisms, in a relatively easy manner. The Darboux family notion can be considered as a generalisation of the Darboux polynomial for a vector field. The classification of r-matrices and solutions to classical Yang–Baxter equations for real four-dimensional indecomposable Lie algebras is also given in detail. Our methods can further be applied to general, even higher-dimensional, Lie algebras. As a byproduct, a method to obtain matrix representations of certain Lie algebras with a non-trivial center is developed.

Lie symmetries of the canonical geodesic equations for six-dimensional nilpotent lie groups

For each of the six-dimensional indecomposable nilpotent Lie algebras, the geodesic equations of the associated canonical Lie group connection are given. In each case, a basis for the associated Li...

Cohomological kernels of non-normal extensions in characteristic two and indecomposable division algebras of index eight

This article investigates the cohomological kernels of field extensions E/F in characteristic two where separable part of E is a quadratic extension and E is quadratic or quartic and purely inseparable over Explicit generators for these kernels are described in both the normal and non-normal cases. The second part of the article gives a construction of an indecomposable division algebra D of index 8 and exponent 2 in characteristic two. The non-normal extensions E/F just considered can arise as maximal subfields of D while in the normal case such a maximal subfield will force a division algebra of exponent 2 to decompose.

On the Loewy length of the center of a block with elementary abelian defect groups

ABSTRACTLet G be a finite group and k an algebraically closed field of characteristic p. In this paper we investigate the Loewy structure of centers of indecomposable group algebras kG, for groups G with a normal elementary abelian Sylow p-subgroup. Furthermore, we show a reduction result for the case that a normal abelian Sylow p-subgroup is acted upon by a subgroup of its automorphism group; this is fundamental in providing generic formulae for the Loewy lengths considered.

$${\mathcal{V}_{SI}}$$ V S I first order implies $${\mathcal{V}_{DI}}$$ V D I first order

We prove that if $${\mathcal{V}}$$ is a modular variety such that the subdirectly irreducible algebras form a first order class in which there are no trivial subalgebras, then the class of directly indecomposable algebras of $${\mathcal{V}}$$ is also first order.

EXT-FINITE MODULES FOR WEAKLY SYMMETRIC ALGEBRAS WITH RADICAL CUBE ZERO

Assume that $A$ is a finite-dimensional algebra over some field, and assume that $A$ is weakly symmetric and indecomposable, with radical cube zero and radical square nonzero. We show that such an algebra of wild representation type does not have a nonprojective module $M$ whose ext-algebra is finite dimensional. This gives a complete classification of weakly symmetric indecomposable algebras which have a nonprojective module whose ext-algebra is finite dimensional. This shows in particular that existence of ext-finite nonprojective modules is not equivalent with the failure of the finite generation condition (Fg), which ensures that modules have support varieties.

Power-Central Elements in Tensor Products of Symbol Algebras

Let A be a central simple algebra over a field F. Let k1,…, kr be cyclic extensions of F such that k1 ⊗F… ⊗Fkr is a field. We investigate conditions under which A is a tensor product of symbol algebras where each field ki lies in a symbol F-algebra factor of the same degree as ki over F. As an application, we give an example of an indecomposable algebra of degree 8 and exponent 2 over a field of 2-cohomological dimension 4.

Matrix Representation for Seven-Dimensional Nilpotent Lie Algebras

This paper is concerned with finding linear representations for seven-dimensional real, indecomposable nilpotent Lie algebras. We consider the first 39 algebras presented in Gong’s classification which was based on the upper central series dimensions. For each algebra, we give a corresponding matrix Lie group, a representation of the Lie algebra in terms of left-invariant vector field and left-invariant one forms.

Classification of solvable real rigid Lie algebras with a nilradical of dimension [formula omitted

The complete classification of real solvable rigid Lie algebras possessing a nilradical of dimension at most six is given. Eleven new isomorphism classes of indecomposable algebras are obtained. It is further shown that the resulting solvable Lie algebras have a vanishing second Chevalley cohomology group, thus correspond to algebraically rigid Lie algebras.

Constructing Nearly Frobenius Algebras

In the first part we study nearly Frobenius algebras. The concept of nearly Frobenius algebras is a generalization of the concept of Frobenius algebras. Nearly Frobenius algebras do not have traces, nor they are self-dual. We prove that the known constructions: direct sums, tensor, quotient of nearly Frobenius algebras admit natural nearly Frobenius structures. In the second part we study algebras associated to some families of quivers and the nearly Frobenius structures that they admit. As a main theorem, we prove that an indecomposable algebra associated to a bound quiver (Q, I) with no monomial relations admits a non trivial nearly Frobenius structure if and only if the quiver Q is linearly oriented of type \(\overrightarrow {\mathbb {A}_{n}}\) and I = 0. We also present an algorithm that determines the number of independent nearly Frobenius structures for gentle algebras without oriented cycles.

Decomposable and indecomposable algebras of degree $$8$$ 8 and exponent 2

We study the decomposition of central simple algebras of exponent 2 into tensor products of quaternion algebras. We consider in particular decompositions in which one of the quaternion algebras contains a given quadratic extension. Let B be a biquaternion algebra over F(√a) with trivial corestriction. A degree 3 cohomological invariant is defined and we show that it determines whether B has a descent to F. This invariant is used to give examples of indecomposable algebras of degree 8 and exponent 2 over a field of 2-cohomological dimension 3 and over a field M(t) where the u-invariant of M is 8 and t is an indeterminate. The construction of these indecomposable algebras uses Chow group computations provided by Merkurjev in “Appendix”.

Varieties of Commutative Integral Bounded Residuated Lattices Admitting a Boolean Retraction Term

Let $${\mathbb{BRL}}$$ denote the variety of commutative integral bounded residuated lattices (bounded residuated lattices for short). A Boolean retraction term for a subvariety $${\mathbb{V}}$$ of $${\mathbb{BRL}}$$ is a unary term t in the language of bounded residuated lattices such that for every $${{\bf A} \in \mathbb{V}, t^{A}}$$ , the interpretation of the term on A, defines a retraction from A onto its Boolean skeleton B(A). It is shown that Boolean retraction terms are equationally definable, in the sense that there is a variety $${\mathbb{V}_{t} \subsetneq \mathbb{BRL}}$$ such that a variety $${\mathbb{V} \subsetneq \mathbb{BRL}}$$ admits the unary term t as a Boolean retraction term if and only if $${\mathbb{V} \subseteq \mathbb{V}_{t}}$$ . Moreover, the equation s(x) = t(x) holds in $${\mathbb{V}_{s} \cap \mathbb{V}_{t}}$$ . The radical of $${{\bf A} \in \mathbb{BRL}}$$ , with the structure of an unbounded residuated lattice with the operations inherited from A expanded with a unary operation corresponding to double negation and a a binary operation defined in terms of the monoid product and the negation, is called the radical algebra of A. To each involutive variety $${\mathbb{V} \subseteq \mathbb{V}_{t}}$$ is associated a variety $${\mathbb{V}^{r}}$$ formed by the isomorphic copies of the radical algebras of the directly indecomposable algebras in $${\mathbb{V}}$$ . Each free algebra in such $${\mathbb{V}}$$ is representable as a weak Boolean product of directly indecomposable algebras over the Stone space of the free Boolean algebra with the same number of free generators, and the radical algebra of each directly indecomposable factor is a free algebra in the associated variety $${\mathbb{V}^{r}}$$ , also with the same number of free generators. A hierarchy of subvarieties of $${\mathbb{BRL}}$$ admitting Boolean retraction terms is exhibited.

On Galois Correspondence of a General Azumaya Galois Extension

Let \(B\) be a Galois extension of \(B^G\) with Galois group \(G\) such that \(B^G\) is a separable \(C^G\)-algebra where \(C\) is the center of \(B\) and \(K=\{g\in G\,|\,g(c)=c\) for all \(c\in C\}\). Then it is shown that \(H\longrightarrow B^H\) is a one-to-one correspondence between the set of subgroups of \(K\) andthe set of separable extensions \(A\) of \(B^K\) such that \(V_B(A)\subset\oplus\sum_{g\in G(A)}J_g\) where \(J_g=\{b\in B\,|\,bx=g(x)b\) �for all \(x\in B\}\) for each \(g\in G\) and \(G(A)=\{g\in G\,|\,g(a)=a\) for all \(a\in A\}\). This generalizes the Galois correspondence between \(K\) and \(C\) for an indecomposable Galois algebra given by F. DeMeyer.

Algèbres de Bernstein régulières

The aim of this paper is to give some further results on regular Bernstein algebras over fields of characteristic other than two. This notion was introduced by Yu. I. Lyubich, who also began the classification of such objects; after Lyubich, many authors worked on this subject. In particular, we study the connection between regular and indecomposable Bernstein algebras.

Applying Universal Algebra to Lambda Calculus

The aim of this article is double. From one side we survey the knowledge we have acquired these last ten years about the lattice of all λ-theories (equational extensions of untyped λ-calculus) and the models of lambda calculus via universal algebra. This includes positive or negative answers to several questions raised in these years as well as several independent results, the state of the art about the long-standing open questions concerning the representability of λ-theories as theories of models, and 26 open problems. On the other side, against the common belief, we show that lambda calculus and combinatory logic satisfy interesting algebraic properties. In fact the Stone representation theorem for Boolean algebras can be generalized to combinatory algebras and λ-abstraction algebras. In every combinatory and λ-abstraction algebra, there is a Boolean algebra of central elements (playing the role of idempotent elements in rings). Central elements are used to represent any combinatory and λ-abstraction algebra as a weak Boolean product of directly indecomposable algebras (i.e. algebras that cannot be decomposed as the Cartesian product of two other non-trivial algebras). Central elements are also used to provide applications of the representation theorem to lambda calculus. We show that the indecomposable semantics (i.e. the semantics of lambda calculus given in terms of models of lambda calculus, which are directly indecomposable as combinatory algebras) includes the continuous, stable and strongly stable semantics, and the term models of all semisensible λ-theories. In one of the main results of the article we show that the indecomposable semantics is equationally incomplete, and this incompleteness is as wide as possible.

Contractions and deformations of quasiclassical Lie algebras preserving a nondegenerate quadratic Casimir operator

By means of contractions of Lie algebras, we obtain new classes of indecomposable quasiclassical Lie algebras that satisfy the Yang-Baxter equations in its reformulation in terms of triple products. These algebras are shown to arise naturally from noncompact real simple algebras with nonsimple complexification, where we impose that a nondegenerate quadratic Casimir operator is preserved by the limiting process. We further consider the converse problem and obtain sufficient conditions on integrable cocycles of quasiclassical Lie algebras in order to preserve nondegenerate quadratic Casimir operators by the associated linear deformations.

Decomposability of the Finitely Generated Free Hoop Residuation Algebra

In this paper we prove that, for n > 1, the n-generated free algebra \(F_{{\mathcal{V}}}(n)\) in any locally finite subvariety \({\mathcal{V}}\) of HoRA can be written in a unique nontrivial way as Ł2 × A′, where A′ is a directly indecomposable algebra in \({\mathcal{V}}\) . More precisely, we prove that the unique nontrivial pair of factor congruences of \(F_{{\mathcal{V}}}(n)\) is given by the filters \([{\mathcal{J}})\) and \(F_\mathcal {V}(n) - (\mathcal {J}]\) , where the element \({\mathcal {J}}\) is recursively defined from the term \(j(x, y) =(((x \rightarrow y) \rightarrow y) \rightarrow x) \rightarrow x\) introduced by W. H. Cornish. As an additional result we obtain a characterization of minimal irreducible filters of \(F_{{\mathcal{V}}}(n)\) in terms of its coatoms.

On some elements of the Brauer group of a conic

The main purpose of the paper is to strengthen previous author’s results. Let k be a field of characteristic ≠ 2, n ≥ 2. Suppose that elements $$\bar a,\overline {b_1 } , \ldots ,\overline {b_n } \in k^* /k^{*2} $$ are linearly independent over ℤ/2ℤ. We construct a field extension K/k and a quaternion algebra D = (u, v) over K such that In particular, the algebra A provides an example of an indecomposable algebra of index 2n+1 over a field, the u-invariant and the 2-cohomological dimension of which equal 2n+3 and n + 3, respectively. Bibliography: 10 titles.