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1.
The following analog of the characterization of flat modules has been obtained
for the variety of semimodules over a semiring R: A semimodule
RA is flat (i.e., the tensor
product functor – A preserves all finite limits) iff
A is L-flat (i.e., A
is a filtered colimit of finitely generated free semimodules). We also give new (homological) characterizations of
Boolean algebras and complete Boolean algebras within the classes of distributive lattices
and Boolean algebras, respectively, which solve two problems left open in [14]. It is also
shown that, in contrast with the case of modules over rings, in general for semimodules over
semirings the notions of flatness and mono-.atness (i.e., the tensor product functor – A
preserves monomorphisms) are different. 相似文献
2.
The Golod-Shafarevich examples show that not every finitely generated nil algebra A is nilpotent. On the other hand, Kaplansky proved that every finitely generated nil PI-algebra is indeed nilpotent. We generalise
Kaplansky’s result to include those algebras that are only infinitesimally PI. An associative algebra A is infinitesimally PI whenever the Lie subalgebra generated by the first homogeneous component of its graded algebra gr( A)=⊕
t⩾1
A
i
/ A
i+1 is a PI-algebra. We apply our results to a problem of Kaplansky’s concerning modular group algebras with radical augmentation
ideal.
The author is supported by NSERC of Canada. 相似文献
3.
We study the Hilbert series of finitely generated prime PI algebras. We show that given such an algebra A there exists some finite dimensional subspace V of A which contains 1
A
and generates A as an algebra such that the Hilbert series of A with respect to the vector space V is a rational function. 相似文献
4.
Given a symmetrizable generalized Cartan matrix A, for any index k, one can define an automorphism associated with A, of the field Q(u1,…, un) of rational functions of n independent indeterminates u1,…,un.It is an isomorphism between two cluster algebras associated to the matrix A (see sec. 4 for the precise meaning). When A is of finite type, these isomorphisms behave nicely; they are compatible with the BGP-reflection functors of cluster categories defined in a previous work if we identify the indecomposable objects in the categories with cluster variables of the corresponding cluster algebras, and they are also compatible with the "truncated simple reflections" defined by Fomin-Zelevinsky. Using the construction of preprojective or preinjective modules of hereditary algebras by DIab-Ringel and the Coxeter automorphisms (i.e. a product of these isomorphisms), we construct infinitely many cluster variables for cluster algebras of infinite type and all cluster variables for finite types. 相似文献
5.
We consider quiver algebras A q over a field k defined by two cycles and a quantum-like relation depending on a nonzero element q in k, and describe the minimal projective bimodule resolution of A q . In particular, in the case q = 1, we determine the Hochschild cohomology ring of A 1 and show that it is a finitely generated k-algebra. Moreover the Hochschild cohomology ring of A 1 modulo nilpotence is isomorphic to the polynomial ring of two variables. 相似文献
6.
To every von Neumann algebra, one can associate a (multiplicative) determinant defined on the invertible elements of the algebra with range a subgroup of the Abelian group of the invertible elements of the center of the von Neumann algebra. This determinant is a normalization of the usual determinant for finite von Neumann algebras of type I, for the type II 1-case it is the Fuglede-Kadison determinant, and for properly infinite von Neumann algebras the determinant is constant equal to 1. It is proved that every invertible element of determinant 1 is a product of a finite number of commutators. This extends a result of T. Fack and P. de la Harpe for II 1-factors. As a corollary, it follows that the determinant induces an injection from the algebraic K
1-group of the von Neumann algebra into the Abelian group of the invertible elements of the center. Its image is described. Another group, K
1
w
( A), which is generated by elements in matrix algebras over A that induce injective right multiplication maps, is also computed. We use the Fuglede-Kadison determinant to detect elements in the Whitehead group Wh( G).Partially supported by NSF Grant DMS-9103327. 相似文献
7.
We consider quiver algebras A q over a field k defined by two cycles and a quantum-like relation depending on a nonzero element q in k. We determine the Hochschild cohomology ring of A q and give necessary and sufficient conditions for A q to have the finitely generated Hochschild cohomology ring. 相似文献
8.
A sufficient condition is proved for the Specht property of varieties of right alternative metabelian algebras over a field
of characteristic distinct from 2. As a consequence, the Specht property of some varieties generated by right alternative
metabelian algebras A satisfying a commutator identity is stated. In particular, it is proved that if A
(−) is a binary Lie algebra, then var( A) is Spechtian.
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 2, pp. 89–100, 2006. 相似文献
9.
In this article we give the classification of compact exceptional simple Kantor triple systems defined on tensor products of composition algebras A = 1? 2 such that their Kantor algebras ?(φ, A) are real forms of exceptional simple Lie algebras. 相似文献
10.
The present paper is devoted to the classification of infinite-dimensional naturally graded Lie algebras that are narrow in the sense of Zelmanov and Shalev [9]. Such Lie algebras are Lie algebras of slow linear growth. In the theory of nonlinear hyperbolic partial differential equations the notion of the characteristic Lie algebra of equation is introduced [3]. Two graded Lie algebras n1 and n2 from our list, that are positive parts of the affine Kac–Moody algebras A1(1) and A2(2), respectively, are isomophic to the characteristic Lie algebras of the sinh-Gordon and Tzitzeika equations [6]. We also note that questions relating to narrow and slowly growing Lie algebras have been extensively studied in the case of a field of positive characteristic [2]. 相似文献
11.
For a smashing localisation L of the derived category of a differential graded (dg) algebra A we construct a dg algebra A
L
and a morphism of dg algebras A→ A
L
that induces the canonical map in cohomology. As a first application we obtain a localisations of a dg algebra A with graded commutative homology at a prime ideal in the homology H
*
A, namely a morphism of dg algebras. As a second application we can use results of Keller to “model” every smashing localisation of compactly
generated algebraic triangulated categories by a morphism of dg algebras.
相似文献
12.
Let F ⊂ K be a field extension, A be a K-algebra. It is proved that, in general, GK dim
F
A≥GK dim
K
A+tr
F
( K). For commutative algebras or Noetherian P.I. algebras, the equality holds. Two examples are also constructed to show that:
(i) there exists an algebra A such that GK dim
F
A=GK dim
K
A+tr
F
( K)+1; (ii) there exists an algebraic extension F ⊂ K and a K-algebra A such that GK dim
F
A=∞, but GK dim
K
A<∞. 相似文献
13.
Let G be the infinite dimensional Grassmann algebra over a field F of characteristic zero and UT
2 the algebra of 2 × 2 upper triangular matrices over F. The relevance of these algebras in PI-theory relies on the fact that they generate the only two varieties of almost polynomial
growth, i.e., they grow exponentially but any proper subvariety grows polynomially. In this paper we completely classify,
up to PI-equivalence, the associative algebras A such that A ∈ Var( G) or A ∈ Var( UT
2). 相似文献
14.
It is a small step toward the Koszul-type algebras.The piecewise-Koszul algebras are, in general,a new class of quadratic algebras but not the classical Koszul ones,simultaneously they agree with both the classical Koszul and higher Koszul algebras in special cases.We give a criteria theorem for a graded algebra A to be piecewise-Koszul in terms of its Yoneda-Ext algebra E(A),and show an A_∞-structure on E(A).Relations between Koszul algebras and piecewise-Koszul algebras are discussed.In particular,our results are related to the third question of Green-Marcos. 相似文献
16.
We study the K-theory of unital C*-algebras A satisfying the condition that all irreducible representations are finite and of some bounded dimension. We construct
computational tools, but show that K-theory is far from being able to distinguish between various interesting examples. For example, when the algebra A is n-homogeneous, i.e., all irreducible representations are exactly of dimension n, then K*( A) is the topological K-theory of a related compact Hausdorff space, this generalises the classical Gelfand-Naimark theorem, but there are many inequivalent
homogeneous algebras with the same related topological space. For general A we give a spectral sequence computing K*( A) from a sequence of topological K-theories of related spaces. For A generated by two idempotents, this becomes a 6-term long exact sequence. 相似文献
17.
Let Abe a PI-algebra over a field F. We study the asymptotic behavior of the sequence of codimensions cn( A) of A. We show that if Ais finitely generated over Fthen Inv( A)=lim n→∞
always exists and is an integer. We also obtain the following characterization of simple algebras: Ais finite dimensional central simple over Fif and only if Inv( A)=dim= A. 相似文献
18.
In this note we classify the derived-tame tree algebras up to derived equivalence. A tree algebra is a basic algebra A = kQ/ I whose quiver Q is a tree. The algebra A is said to be derived-tame when the repetitive category  of A is tame. We show that the tree algebra A is derived-tame precisely when its Euler form
A
is non-negative. Moreover, in this case, the derived equivalence class of A is determined by the following discrete invariants: The number of vertices, the corank and the Dynkin type of
A
. Representatives of these derived equivalence classes of algebras are given by the following algebras: the hereditary algebras of finite or tame type, the tubular algebras and a certain class of poset algebras, the so-called semichain-algebras which we introduce below. 相似文献
19.
Self-dual algebras are ones with an A bimodule isomorphism A → A ∨op, where A ∨ = Hom k ( A, k) and A ∨op is the same underlying k-module as A ∨ but with left and right operations by A interchanged. These are in particular quasi self-dual algebras, i.e., ones with an isomorphism H*( A,A) ≌ H*( A,A ∨ op). For all such algebras H*( A,A) is a contravariant functor of A. Finite-dimensional associative self-dual algebras over a field are identical with symmetric Frobenius algebras; an example of deformation of one is given. (The monoidal category of commutative Frobenius algebras is known to be equivalent to that of 1+1 dimensional topological quantum field theories.) All finite poset algebras are quasi self-dual. 相似文献
20.
Riassunto Sia X una classe di strutture algebriche simili, AA
i) i∈I una famiglia di algebre di X e L
i il reticolo delle congruenze di A
i. In modo naturale ad ogni ideale di
resta associata una congruenza di
. Il concetto di ?congruenza ideale? cioè di congruenza di A legata a un ideale di L estende il concetto di ?congruenza filtrale? di congruenza cioè legata a un filtro su I.
Ciò conduce a introdurre un concetto di ?classe ideale? di algebre che estende quello di ?classe filtrale? di [3]. Vengono
estesi alle classi ideali i risultati di [3].
Summary LetX be a class of similar algebras, (A
i)i∈I a family of algebras inX and letL
i be the lattice of the congruences ofA
i.
In a natural way we can link each ideal ofL=IIL
i with a congruence ofA=IIA
i. So we are brought to consider ?ideal classes? of algebras wich are an extension of ?filtral classes? [3]. The results of
[3] are extended to ?ideal classes?.
Lavoro eseguito nell'ambito dell'attività del Comitato Nazionale per la Matematica del C.N.R., contratto 115218205174, anno
1970. 相似文献
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