Finite Dimensional Algebras

Drozd, Yuriy A.; Kirichenko, V. V.

doi:10.1007/978-3-642-76244-4

Cited by 168 publications

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Section: Lattices Of Annihilatorsmentioning

confidence: 99%

“…An algeba A is called commutative if ab " ba for any a, b P A and is finite dimensional if the space A is finite dimensional over K. If A is an algebra, then by JpAq we denote the Jacobson radical of A. All other information about algebras used here one can find for example in [7,3].If X Ď A is a subset of an algebra A then let L A pXq " LpXq be the left annihilator of X in A and let R A pXq " RpXq be the right annihilator of X in A :LpXq " ta P A : aX " 0u and RpXq " ta P A : Xa " 0u.Thus, by associativity of A, every left annihilator is a left ideal, and every right annihilator is a right ideal in A.Let A l pAq be the set of all left annihilators in A and let A r pAq be the set of all right annihilators in A. Then A l pAq is a complete lattice with operations: …”

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Lattices of Annihilators in Commutative Algebras Over Fields

Jastrzębska¹,

Krempa²

2015

Demonstratio Mathematica

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Abstract. Let K be any field and L be any lattice. In this note we show that L is a sublattice of annihilators in an associative and commutative K-algebra. If L is finite, then our algebra will be finite dimensional over K. IntroductionThe problem of representing lattices as lattices of subalgebras or congruences in various abstract algebras is quite often investigated. In some cases it is natural to restrict considerations to finite lattices. We refer the reader to [11,13,12] for some results on this problem. Some papers are devoted to representing lattices as lattices of annihilators in associative rings, for example [8,6], or annihilators in semigroups with zero (see [14,9]).In this paper K is any field. In [5] it is shown that every lattice L is embeddable in a lattice of left annihilators of a K-algebra, denoted there by KxLy. If L is finite then this algebra is finite dimensional over K. If L has at most 3 elements then the algebra KxLy is commutative, but for greater number of elements it is noncommutative. Thus there is a natural question, whether every (finite) lattice is isomorphic to a sublattice of the lattice of annihilators in a commutative (finite dimensional) algebra over K. This question was asked by some participants of conferences, where the results of [5] were presented. We solve this problem here by constructing suitable algebras over K.To make the paper more readable and self-contained, in Section 2 we recall some definitions and facts on algebras and some results from [5]. We use this opportunity to add a new consequence of Theorem 3.2 from that paper.In Section 3, using ideas from [5], we show that for every lattice L there exists a local, commutative K-algebra KxxLyy and a lattice embedding of L into the lattice of annihilators in KxxLyy.The cardinality of any set X we denote by |X|. All lattices considered here have the smallest element ω and the largest element Ω ‰ ω. If P is any partially ordered set (poset), then by P op we denote the set P , with the reverse order. Lattices of annihilatorsHere, by an algebra over K we mean a vector space A over K together with a bilinear associative multiplication. In other words, for arbitrary elements a, b, c P A and for arbitrary λ P K the following equalities are satisfied:1. apb`cq " ab`ac; 2. pb`cqa " ba`ca; 3. pabqc " apbcq; 4. pλaqb " apλbq " λpabq.All algebras over K considered here, named simply algebras, are with 1 ‰ 0. An algeba A is called commutative if ab " ba for any a, b P A and is finite dimensional if the space A is finite dimensional over K. If A is an algebra, then by JpAq we denote the Jacobson radical of A. All other information about algebras used here one can find for example in [7,3].If X Ď A is a subset of an algebra A then let L A pXq " LpXq be the left annihilator of X in A and let R A pXq " RpXq be the right annihilator of X in A :LpXq " ta P A : aX " 0u and RpXq " ta P A : Xa " 0u.Thus, by associativity of A, every left annihilator is a left ideal, and every right annihilator is a right ideal in A.Let A l pAq be the se...

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Section: Lattices Of Annihilatorsmentioning

confidence: 99%

mentioning

confidence: 99%

Lattices of Annihilators in Commutative Algebras Over Fields

Jastrzębska¹,

Krempa²

2015

Demonstratio Mathematica

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“…It is well known that an indecomposable elementary algebra is Nakayama if and only if its quiver is a basic cycle or a linear quiver A m (see [4]). Thus H is Nakayama too.…”

Section: Implies This Corollarymentioning

confidence: 99%

Pointed Hopf algebras of finite corepresentation type and their classifications

Liu

2006

Proc. Amer. Math. Soc.

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Abstract. Let k be an algebraically closed field. The main goal of this paper is to classify the finite-dimensional pointed Hopf algebras over k of finite corepresentation type. To do so, we give a necessary and sufficient condition for a basic Hopf algebra over k to be of finite representation type firstly. Explicitly, we prove that a basic Hopf algebra over k is of finite representation type if and only if it is Nakayama. By this conclusion, we classify all finite-dimensional pointed Hopf algebras over k of finite corepresentation type.

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“…In Section 1, we prove that an algebra F λ G is of finite representation type if and only if F λ G p is a uniserial algebra (Theorem 1.1; we use the terminology introduced in [15]). We also establish (Theorem 1.2) that if p = 2, then F λ G p is a uniserial algebra if and only if C p is cyclic and one of the following conditions holds:…”

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confidence: 99%

On indecomposable projective representations of finite groups over fields of characteristic p>0

Barannyk

Sobolewska

2003

Colloq. Math.

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Abstract. Let G be a finite group, F a field of characteristic p with p | |G|, and F λ G the twisted group algebra of the group G and the field F with a 2-cocycle λ ∈ Z 2 (G, F * ). We give necessary and sufficient conditions for F λ G to be of finite representation type. We also introduce the concept of projective F -representation type for the group G (finite, infinite, mixed) and we exhibit finite groups of each type.Introduction. Let F be a field of characteristic p > 0, F * the multiplicative group of the field F , F p = {a p : a ∈ F }, G a finite group of order |G|, where p | |G|, andand Z 2 (G, F * ) the group of all F * -valued normalized 2-cocycles of the group G, where we assume that G acts trivially on F * (see [26, Chapter 1]). Denote by F λ G the twisted group algebra of the group G and the field F with a cocycle λ ∈ Z 2 (G, F * ) and by rad F λ G the radical ofHigman [25] showed that if F is an infinite field and G is an abelian p-group which is neither cyclic nor of order 4, then there exist infinitely many non-isomorphic indecomposable F G-modules of F -dimension n for every natural number n > 1. If G is the non-cyclic group of order 4, then the preceding result is valid for even natural numbers n.

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Finite Dimensional Algebras

Cited by 168 publications

References 0 publications

Lattices of Annihilators in Commutative Algebras Over Fields

Lattices of Annihilators in Commutative Algebras Over Fields

Pointed Hopf algebras of finite corepresentation type and their classifications

On indecomposable projective representations of finite groups over fields of characteristic p>0

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