The Birman-Murakami-Wenzl algebras of type E n

The Birman-Murakami-Wenzl algebras (BMW algebras) of type E _n for n = 6; 7; 8 are shown to be semisimple and free over the integral domain \mathbbZ[ d^±1,l^±1,m ]

Traces and idempotents in group algebras

Let G be a group and E an idempotent matrix with entries in the complex group algebra C G. In this paper, we study arithmetic properties of the coefficients r _E (g), gG, of the Hattori-Stallings rank r _E of E. Bass proved in [2] that the r _E (g)s are algebraic numbers. Following Zaleskii, we proceed by reduction to positive characteristic and give an alternative proof of that assertion, while obtaining at the same time an upper bound for the degree of the minimum polynomial of r _E (g) over Q.     

On idempotents in reduced enveloping algebras

Explicit constructions are given for idempotents that generate all projective indecomposable modules for certain finite-dimensional quotients of the universal enveloping algebra of the Lie algebra in odd prime characteristic. The program is put in a general context, although constructions are only carried through in the case of .

     

Zero sums of idempotents in Banach algebras

The problem treated in this paper is the following.Let p ₁,...,p _k be idempotents in a Banach algebra B, and assume p ₁+...+p _k =0.Does it follow that p _j =0,j=1,..., k? For important classes of Banach algebras the answer turns out to be positive; in general, however, it is negative. A counterexample is given involving five nonzero bounded projections on infinite-dimensional separable Hilbert space. The number five is critical here: in Banach algebras nontrivial zero sums of four idempotents are impossible. In a purely algebraic context (no norm), the situation is different. There the critical number is four.     

Logarithmic residues, generalized idempotents, and sums of idempotents in Banach algebras

In a commutative Banach algebraB the set of logarithmic residues (i.e., the elements that can be written as a contour integral of the logarithmic derivative of an analyticB-valued function), the set of generalized idempotents (i.e., the elements that are annihilated by a polynomial with non-negative integer simple zeros), and the set of sums of idempotents are all the same. Also, these (coinciding) sets consist of isolated points only and are closed under the operations of addition and multiplication. Counterexamples show that the commutativity condition onB is essential. The results extend to logarithmic residues of meromorphicB-valued functions.     

On idempotents and isomorphisms of multiplication algebras of bernstein algebras

The purpose of this paper is twofold. First of all we characterize Bernstein algebras having a nilpotent kernel using idempotents of their multiplication algebras. Secondly, we describe some properties of Bernstein algebras which are preserved by isomorphisms of their multiplication algebras.     

Linear associative algebras with finitely many idempotents

Let A be a linear (i.e., finite-dimensional) associative algebra with unity defined over K, an algebraically closed field. Then A with respect to its multiplication is an algebraic monoid over k, denoted by A_M, and with respect to the the bracket forms a Lie algebra over K, denoted by A_L. The following theorem is established A_M is nilpotent as an algebraic monoid (equivalentlyA_L is so as a Lie algebra) if and only if the set of idempotents of A is finite if and only if all irreducible closed submonoids of codimension 1 are nilpotent.     

Symmetrization of the symbol in Banach algebras generated by idempotents

It is known [2] that a Banach algebra generated by 2N idempotents with relations (1), (2) is an algebra with matrix symbol of order 2N. This symbol is completely and explicitly defined via the spectrums of two (indicator) elements (3) and (5). But in the case when is a c^*-algebra, the symbol constructed in [2] is notsymmetric. Moreover in order to construct asymmetric symbol, one needs some additional information about the algebra , even for the case whenp _j=p _j ^* (j=1, ..., 2N). These additional conditions and the explicit form ofsymmetric symbol are described in this paper.     

On certain finite-dimensional algebras generated by two idempotents

This paper is concerned with algebras generated by two idempotents P and Q satisfying (PQ)^m=(QP)^m and (PQ)^m-1≠(QP)^m-1. The main result is the classification of all these algebras, implying that for each m?2 there exist exactly eight nonisomorphic copies. As an application, it is shown that if an element of such an algebra has a nondegenerate leading term, then it is group invertible, and a formula for the explicit computation of the group inverse is given.     

On the identities in algebras generated by linearly connected idempotents

We investigate the problem of the existence of polynomial identities (PI) in algebras generated by idempotents whose linear combination is equal to identity. In the case where the number of idempotents is greater than or equal to five, we prove that these algebras are not PI-algebras. In the case of four idempotents, in order that an algebra be a PI-algebra, it is necessary and sufficient that the sum of the coefficients of the linear combination be equal to two. In this case, these algebras are F₄-algebras.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 6, pp. 782–795, June, 2004.     

Similarity implies homotopy of idempotents in Banach algebras of stable rank one

We establish the property stated in the title by means of an elementary construction: in a Banach algebra of stable rank one, any two similar idempotents can be connected by a piecewise affine path of idempotents consisting of at most 3 affine steps. Received: 1 February 2006     

On Coxeter functors for some classes of algebras generated by idempotents

For an algebra generated by linearly connected idempotents, we describe the construction of Coxeter functors on the category of representations of the algebra in the category of Banach spaces.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 7, pp. 996–1001, July, 2004.     

Skein related links in 3-manifolds

This paper addresses two problems in the skein theory of homotopy spheres first posed by P. Traczyk. Solutions to both problems are obtained for a large class of manifolds and, since one of the basic techniques used requires the first homology group of the ambient manifold to be torsion free, the extent to which this hypothesis is actually necessary is further explored.     

Primitive idempotents in central simple algebras over Fq(t) with an application to coding theory

We consider the algorithmic problem of computing a primitive idempotent of a central simple algebra over the field of rational functions over a finite field. The algebra is given by a set of structure constants. The problem is reduced to the computation of a division algebra Brauer equivalent to the central simple algebra. This division algebra is constructed as a cyclic algebra, once the Hasse invariants have been computed. We give an application to skew constacyclic convolutional codes.