Algebras of quotients of Jordan algebras

We define a Jordan analogue of Lambek and Utumi's associative algebra of quotients and we construct the maximal algebra of quotients for nondegenerate Jordan algebras. We apply those results to other classes of algebras of quotients appearing in the literature.     

Algebras of quotients of graded Lie algebras

In this paper we explore graded algebras of quotients of Lie algebras with special emphasis on the 3-graded case and answer some natural questions concerning its relation to maximal Jordan systems of quotients.     

On the homology of quotients of path algebras

We extend the results and techniques of [Al] to find a method of constructing projective resolutions for certain simple modules over homomorphic images of path algebras. We provide a number of applications in the case when the image algebra is finite dimensional.     

Rings of quotients of some infinite dimensional path algebras

We compute the maximal symmetric ring of quotients of infinite dimensional path algebras that can be approximated by finite dimensional path algebras.     

Martindale quotients of Jordan algebras

In this paper we introduce Martindale quotients of Jordan algebras over arbitrary rings of scalars with respect to denominator filters of ideals. For any denominatored algebra, we show the existence of maximal Martindale quotients naturally containing all Martindale quotients of the algebra with respect to the given denominator filter.     

Multi-algebras as tolerance quotients of algebras

If A is an algebra and τ is a tolerance on A, then A/τ is a multi-algebra in a natural way. We give an example to show that not every multi-algebra arises in this manner. We slightly generalize the construction of A/τ and prove that every multi-algebra arises from this modified construction.     

The rings of quotients of free algebras

It is proved that a symmetric Utumi ring of quotients, U, of a free associative (noncommutative) algebra F(X) with unity coincides with the algebra itself, U=F(X). From this, we obtain a similar statement concerning a symmetric Martindale ring of quotients, Q(F(X))=F(X), which is well known. In addition, it is shown that a left Martindale ring of quotients, F(X)_F, of a free algebra is a prime algebra and, moreover, every homogeneous element in a free algebra has the right inverse in F(X)_F but does not have the left one (unless, of course, r belongs to an underlying field). Since a left Utumi ring of quotients and a left Martindale ring of quotients for a free algebra both appear prime, an interesting question arises as to whether or not they coincide. Supported by RFFR grant No. 95-01-01356 and by ISF grant RPS000-RPS300. Translated fromAlgebra i Logika, Vol. 35, No. 6, pp. 655–662, November–December, 1996.     

On certain homomorphisms of quotients of group algebras

LetA be the algebra of functions on the circle groupT={z: |z|=1} having absolutely convergent Fourier series. For a subsetE ofA, the algebra of restrictions toE of functions inA is denoted byA(E). It is shown that for sets that are “thick” in one of several senses, algebra homomorphisms of theA(E) having norm 1 must arise from point mappings having a certain amount of rigidity.     

Hecke group algebras as quotients of affine Hecke algebras at level 0

The Hecke group algebra of a finite Coxeter group , as introduced by the first and last authors, is obtained from by gluing appropriately its 0-Hecke algebra and its group algebra. In this paper, we give an equivalent alternative construction in the case when is the finite Weyl group associated to an affine Weyl group W. Namely, we prove that, for q not a root of unity of small order, is the natural quotient of the affine Hecke algebra H(W)(q) through its level 0 representation.The proof relies on the following core combinatorial result: at level 0 the 0-Hecke algebra H(W)(0) acts transitively on . Equivalently, in type A, a word written on a circle can be both sorted and antisorted by elementary bubble sort operators. We further show that the level 0 representation is a calibrated principal series representation M(t) for a suitable choice of character t, so that the quotient factors (non-trivially) through the principal central specialization. This explains in particular the similarities between the representation theory of the 0-Hecke algebra and that of the affine Hecke algebra H(W)(q) at this specialization.     

Finite-dimensional Leavitt path algebras

We classify the directed graphs E for which the Leavitt path algebra L(E) is finite dimensional. In our main results we provide two distinct classes of connected graphs from which, modulo the one-dimensional ideals, all finite-dimensional Leavitt path algebras arise.