Hereditary Algebras

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Derived equivalence classification of generalized multifold extensions of piecewise hereditary algebras of tree type

Derived equivalence classification of generalized multifold extensions of piecewise hereditary algebras of tree type

algebras; it requires that the algebras in consideration have the form of orbit categories (usually of repetitive categories of some algebras hav- ing no oriented cycles in their ordinary quivers). In fact, it was applied in [3] to give the classification of twisted multifold extensions of piecewise hereditary algebras of tree type by giving a complete invariant. Here an algebra is called a twisted multifold extension of an algebra A if it has the form

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Ringel duality for certain strongly quasi-hereditary algebras

Ringel duality for certain strongly quasi-hereditary algebras

Abstract We study quasi-hereditary endomorphism algebras defined over a new class of finite dimensional monomial algebras with a special ideal structure. The main result is a uniform formula describing the Ringel duals of these quasi-hereditary algebras. As special cases, we obtain a Ringel duality formula for a family of strongly quasi- hereditary algebras arising from a type A configuration of projective lines in a rational, projective surface as recently introduced by Hille and Ploog, for certain Auslander– Dlab–Ringel algebras, and for Eiriksson and Sauter’s nilpotent quiver algebras when the quiver has no sinks and no sources. We also recover Tan’s result that the Auslander algebras of self-injective Nakayama algebras are Ringel self-dual.
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Radically filtered quasi-hereditary algebras and rigidity of tilting modules

Radically filtered quasi-hereditary algebras and rigidity of tilting modules

weight λ let ∆pλq be the Weyl module of highest weight λ, ∇pλq its contravariant dual, and Lpλq the simple head of ∆pλq. For any finite saturated set π of dominant weights, the full subcategory of rational G-modules whose composition factors are indexed by weights in π is equivalent to a module category Spπq´mod, where Spπq is a finite-dimensional algebra called a generalized Schur algebra [10]. The algebra Spπq is quasi-hereditary (in fact a BGG algebra) with standard and costandard modules ∆pλq and ∇pλq respectively. When necessary we will deal with Spπq-modules instead of rational G-modules for a sufficiently large set π.
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Radically filtered quasi-hereditary algebras and rigidity of tilting modules

Radically filtered quasi-hereditary algebras and rigidity of tilting modules

Let A be a finite-dimensional K-algebra. We recall the notion of a quasi-hereditary algebra. Suppose the irreducible A-modules Lpλq are indexed by a poset Λ. Let P pλq and Ipλq denote the projective cover and injective hull of Lpλq respectively. Let ∆pλq be the maximal quotient of P pλq whose composition factors are among tLpµq | µ ď λu. These are the Weyl or standard modules. Define ∇ pλq (the good or costandard modules) dually. We say that A is quasi-hereditary if for all λ P Λ

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The generalized path algebras over standardly stratified algebras

The generalized path algebras over standardly stratified algebras

∆ = {∆(1), · · · , ∆(n)}. We denote by F(Θ) the full subcategory of ModΛ consisting of modules which have a filtration with factors in Θ, where ModΛ denotes the category of f.g. left modules over Λ and Θ is a set of modules. These modules are said to be Θ-good. The algebra (Λ, E ) is called standardly stratified with respect to the ordering of simple mod- ules if P (i) ∈ F(∆), for all i = 1, · · · , n [7]; if in addition, EndΛ(∆(i)) is a division ring, for all i = 1, · · · , n, then (Λ, E) is quasi-hereditary [11].Standardly stratified algebras as a generalization of quasi-hereditary algebras have been studied recently by some authors in various aspects [1] [7] [9] [15].
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Quasi-hereditary twisted category algebras

Quasi-hereditary twisted category algebras

do not yield quasi-hereditary algebras. Let k be a field, and let M = {1, a, e} be a monoid consisting of three elements, with identity element 1, such that the product of any two non- identity elements is e. Then M is abelian, e is an idempotent in M , and a is the unique nonsplit element in M . In particular, the set of nonsplit elements in M is not an ideal. The algebra kM is not quasi-hereditary: an easy calculation shows that kM ∼ = kM e × kM(1 − e), and kM e ∼ = k, while kM(1 − e) is a local 2-dimensional algebra (with basis {1 − e, a − e}). In particular, kM is a symmetric non-semisimple k-algebra, hence any non-projective kM -module has infinite projective dimension, and in particular, kM is not quasi-hereditary.
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Knots and algebras

Knots and algebras

We use similar methods, starting with Kauman's polynomial in its Dubrovnik form [K] , again keeping a close watch on the ring used, to construct algebras whose specialisations can be identied with Brauer's algebras [B] in a very natural way. Such algebras have been studied in more detail by Birman and Wenzl [BW] , and it was from them that we got to hear of Brauer's algebras. While their algebras are essentially isomorphic to the ones constructed here, our choice of ring and the use of the Dubrovnik variant of the polynomial allow us to make their conjectured connection with Brauer's algebras very directly both algebraically and geometrically. For this reason, and also because of the form of recently discovered connections of both polynomials with invariants derived from Lie algebras [T] , we feel that the Dubrovnik version, and the version of P which we use, both have a particularly appropriate choice of variables.
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On The Solvable Length of Associative Algebras, Matrix Groups, and Lie Algebras

On The Solvable Length of Associative Algebras, Matrix Groups, and Lie Algebras

to groups and Lie algebras would result with: if G (or L) is abelian, then G 1 = e (L 1 = 0), and if |G| = p 2 (dimension of L = 2), then G 1 =e (L 1 = 0). So if G (or L) is to have length 2 then |G| ≥ 3 (dimension of L ≥ 3). (Such an L is L = <x, y, z> where [x, y] = z and [x, z] =[y, z] =0. Then L 1 = <z> and L 2 = 0 leaving the length of L to be 2). In [6] it is shown that for G to have length t and |G| = p n , then n ≥ 2 t-1 + t -1 leaving the following table:

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On BRK Algebras

On BRK Algebras

In 1996, Imai and Is´eki 1 introduced two classes of abstract algebras: BCK-algebras and BCI-algebras. These algebras have been extensively studied since their introduction. In 1983, Hu and Li 2 introduced the notion of a BCH-algebra which is a generalization of the notion of BCK and BCI-algebras and studied a few properties of these algebras. In 2001, Neggers et al. 3 introduced a new notion, called a Q-algebra and generalized some theorems discussed in BCI/BCK-algebras. In 2002, Neggers and Kim 4 introduced a new notion, called a B-algebra, and obtained several results. In 2007, Walendziak 5 introduced a new notion, called a BF-algebra, which is a generalization of B-algebra. In 6, C. B. Kim and H. S. Kim introduced BG-algebra as a generalization of B-algebra. We introduce a new notion, called a BRK-algebra, which is a generalization of BCK/BCI/BCH/Q/QS/BM-algebras. The concept of G-part, p-radical, and medial of a BRK-algebra are introduced and studied their properties.
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Affine Lie Algebras, Vertex Operator Algebras and Combinatorial Identities

Affine Lie Algebras, Vertex Operator Algebras and Combinatorial Identities

In this chapter, we will give the definitions of Lie algebras and their representations (over the complex numbers, C ). Then we will define the special class of Lie algebras called affine (Kac-Moody) Lie algebras and an important class of representations associated with affine Lie algebras called highest-weight representations. For more details about Lie algebras (especially finite dimensional theory), see [14]. A detailed introduction to Kac-Moody algebras can be found in [16].

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Relationship between Symmetric Brace Algebras and Pre-Lie Algebras

Relationship between Symmetric Brace Algebras and Pre-Lie Algebras

Although pre-Lie algebras are more familiar to mathematicians than symmet- ric brace algebras, applications of these algebras seem to be more easily given by symmetric brace formulas [3]. It seems more difficult, in general, to phrase these applications in terms of pre-Lie multiplication. Perhaps, this explicit correspondence between symmetric braces and the pre-Lie product will remedy this situation and cause present and future applications to be solved more clearly.

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On Liu algebras: a new composite structure of the BCL⁺ algebras and the semigroups

On Liu algebras: a new composite structure of the BCL⁺ algebras and the semigroups

It turns out that BCL⁺ algebras are a very interesting area of research in the theory of algebraic systems in mathematics. In the present paper, the BCL⁺ algebras plays a central role in Liu algebras, and the semigroup is actually helping one important aspect of Liu algebras, so we put some useful definitions and properties into Section 4, Main Results.

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On hereditary interval algebras

On hereditary interval algebras

1. Introduction. Boolean algebras that are generated by subchains, that is, subsets that are linearly ordered under the Boolean partial order, were introduced in 1939 by Mostowski and Tarski [7] and have been extensively studied since then. Nowadays they are called interval algebras. All basic facts about these algebras can be found in [6, Section 15]. We remark, at this stage, that a subalgebra of an interval algebra need not be an interval algebra. For instance, one can consider the algebra of finite and co-finite subsets of the first uncountable cardinal. This leads us to the study of hereditary interval algebras, that is, those algebras of which any subalgebra is an in- terval algebra. The main concern of this note is to shed more light on these algebras. This note is organized as follows. Section 2 deals with definitions. Section 3 is a pre- sentation of the main theorem. In Section 4, some consequences of this theorem are given.
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The global dimension of the algebras of polynomial integro-differential operators In and the Jacobian algebras An

The global dimension of the algebras of polynomial integro-differential operators In and the Jacobian algebras An

obstacles in finding the groups of automorphisms for the polynomial and Weyl algebras are the Jacobian Conjecture (for polynomials) and the Dixmier Conjecture (for the Weyl algebras) that are still open. The Dixmier Conjecture states that every algebra endomorphism of the Weyl algebra A n is an automorphism, [22]. In [10], it is shown that an analogue of the Dixmier Conjecture

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Approximate n Jordan ∗ derivations on C∗ algebras and JC∗ algebras

Approximate n Jordan ∗ derivations on C∗ algebras and JC∗ algebras

1. Ulam, SM: Problems in Modern Mathematics. Science ed. Wiley, New York (1940) (Chapter VI) 2. Hyers, DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222-224 (1941) 3. Aoki, T: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 2, 64-66 (1950) 4. Rassias, TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297-300 (1978) 5. Cho, Y, Saadati, R, Vahidi, J: Approximation of homomorphisms and derivations on non-Archimedean Lie C * -algebras

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Crossed modules of Hopf algebras and of associative algebras and two-dimensional holonomy

Crossed modules of Hopf algebras and of associative algebras and two-dimensional holonomy

There exists a functor from the category of unital crossed modules of bare algebras A = (∂ : A → B, ⋗, ⋖), to the category of crossed modules of groups. In the finite dimensional case, this is related to the composition of the functors, sending a crossed module of bare algebras to a differential crossed module, and sending a differential crossed module to a crossed module of simply connected Lie groups. In order to work with explicit models, and in the infinite dimensional case, we will do all constructions mostly over C[[h]] (except for the initial construction, only done for motivating purposes), making this relation appear less clear Definition 7 (From A to A • : adjoining an identity). Let A be bare algebra. We denote by A • the algebra
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Sarcomas in hereditary retinoblastoma

Sarcomas in hereditary retinoblastoma

Soft tissue sarcomas (STS) are also one of the most common subsequent cancers following hereditary Rb accounting for 12% up to 32% of all second cancers [6,7,16]. In one large cohort study, an increased risk for STS was first observed within 10 years of Rb diagnosis and continued through adult life up to 50 years after Rb, with specific subtypes occurring at similar ages as in the general population [22,23]. Fifty years after radiation treatment for hereditary Rb, the cumulative risk of developing a STS was 13.1%, and the cumulative inci- dence for a STS inside the radiation field was higher than outside the field (8.9% vs. 5.1%) [22]. Table 2 pre- sents the incidence and mortality due to STS after Rb in cohort studies of at least 100 hereditary Rb survivors.
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A family with hereditary ataxia

A family with hereditary ataxia

A family with hereditary ataxia Med J Malaysia Vol 35 No 2 December 1980 A FAMILY WITH HEREDITARY ATAXIA C T TAN SUMMARY An Indian family with four members having hereditary ataxia was presented The i[.]

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Hereditary ovalocytosis in Malays

Hereditary ovalocytosis in Malays

Hereditary ovalocytosis in Malays Med J Malaysia Vol 43 No 4 December 1988 Hereditary ovalocytosis in Malays E George FRCPA*, N Mohandas DSc**, G Duraisamy FRCPA**, N Adeeb MRCOG*, Z A Zainuddin MP*,[.]

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SOME RESULT FIXED POINT THEOREMS IN C*-ALGEBRAS AND BANACH ALGEBRAS

SOME RESULT FIXED POINT THEOREMS IN C*-ALGEBRAS AND BANACH ALGEBRAS

ABSTRACT: In this paper, the existence and uniqueness of coupled fixed point for mapping having the mixed monotone property in partially ordered metric spaces which endowed with vec[r]

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