效应代数的表示理论

The aim of this paper is to characterize representable and weak representable effect algebras and establish a representation theory of effect algebras. An effect algebra E is said to be representable if there exists a Hilbert space H and a monomorphism π from E into the Hilbert space effect algebra ε(H) and it is said to be weakly representable if there exists an injective morphism from E into some ε(H). It is proved that an effect algebra E with the nonempty state space S(E) is representable if and only if x, y ∈ E, f(x)+f(y) ≤ 1 implies x⊕y is defined; it is weakly representable if and only if the state space S(E) separates the points of E. Some operational properties of representable effect algebras are established, and some applications of the obtained results are listed.     

Representation theory of q-rook monoid algebras

We show that, over an arbitrary field, q-rook monoid algebras are iterated inflations of Iwahori-Hecke algebras, and, in particular, are cellular. Furthermore we give an algebra decomposition which shows a q-rook monoid algebra is Morita equivalent to a direct sum of Iwahori-Hecke algebras. We state some of the consequences for the representation theory of q-rook monoid algebras.Supported by EPSRC grant GR/S18151/01     

Representation theory and the branching graph for the family of Turaev algebras

We consider the family of algebras {H _q ^1,n } _n=1 ^∞ , where H _q ^1,n is obtained by changing the first generator in the group algebra of the symmetric group S_n+1. We describe the irreducible representations of these algebras and construct the branching graph of the family {H _q ^1,n } _n=1 ^∞ . Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 171–180.     

Coloured peak algebras and Hopf algebras

For G a finite abelian group, we study the properties of general equivalence relations on G _n = G ⁿ ⋊ _n , the wreath product of G with the symmetric group _n , also known as the G-coloured symmetric group. We show that under certain conditions, some equivalence relations give rise to subalgebras of G _n as well as graded connected Hopf subalgebras of ⨁ _{n≥ o} G _n . In particular we construct a G-coloured peak subalgebra of the Mantaci-Reutenauer algebra (or G-coloured descent algebra). We show that the direct sum of the G-coloured peak algebras is a Hopf algebra. We also have similar results for a G-colouring of the Loday-Ronco Hopf algebras of planar binary trees. For many of the equivalence relations under study, we obtain a functor from the category of finite abelian groups to the category of graded connected Hopf algebras. We end our investigation by describing a Hopf endomorphism of the G-coloured descent Hopf algebra whose image is the G-coloured peak Hopf algebra. We outline a theory of combinatorial G-coloured Hopf algebra for which the G-coloured quasi-symmetric Hopf algebra and the graded dual to the G-coloured peak Hopf algebra are central objects. 2000 Mathematics Subject Classification Primary: 16S99; Secondary: 05E05, 05E10, 16S34, 16W30, 20B30, 20E22Bergeron is partially supported by NSERC and CRC, CanadaHohlweg is partially supported by CRC     

Representation type of Jordan algebras

The problem of classification of Jordan bimodules over (non-semisimple) finite dimensional Jordan algebras with respect to their representation type is considered. The notions of diagram of a Jordan algebra and of Jordan tensor algebra of a bimodule are introduced and a mapping Qui is constructed which associates to the diagram of a Jordan algebra J the quiver of its universal associative enveloping algebra S(J). The main results are concerned with Jordan algebras of semi-matrix type, that is, algebras whose semi-simple component is a direct sum of Jordan matrix algebras. In this case, criterion of finiteness and tameness for one-sided representations are obtained, in terms of diagram and mapping Qui, for Jordan tensor algebras and for algebras with radical square equals to 0.     

Representation dimension of m-replicated algebras

Let A be a finite-dimensional hereditary algebra over an algebraically closed field and A ^(m) be the m-replicated algebra of A. We prove that the representation dimension of A ^(m) is at most 3, and that the dominant dimension of A ^(m) is at least m.     

Representation dimension of exterior algebras

We determine the representation dimension of exterior algebras. This provides the first known examples of representation dimension > 3. We deduce that the Loewy length of the group algebra over F₂ of a finite group is strictly bounded below by the 2-rank of the group (a conjecture of Benson). A key tool is the use of the concept of dimension of a triangulated category.     

Representation type of Schur algebras

We give a complete classification of the classical Schur algebras and the infinitesimal Schur algebras which have tame representation type. In combination with earlier work of some of the authors on semisimplicity and finiteness, this completes the classification of representation type of all classical and infinitesimal Schur algebras in all characteristics. Received October 17, 1997; in final form March 5, 1998     

Representation theorems for Banach algebras

For a space X   denote by _Cb(X)$C_b(X)$ the Banach algebra of all continuous bounded scalar-valued functions on X   and denote by _C0(X)$C_0(X)$ the set of all elements in _Cb(X)$C_b(X)$ which vanish at infinity.     

Representation type of 0-Hecke algebras

In the present paper we determine the representation type of the 0-Hecke algebra of a finite Coxeter group.     

Representation type of -Schur algebras

In this paper we classify the -Schur algebras having finite, tame or wild representation type and also the ones which are semisimple.

     

Representation of varieties in combinatory algebras

It is shown that the set of completions of algebras in a variety can be represented as the set of solutions of a single equation of the formA · X=B · X in the author's model of combinatory algebra.A andB are determined directly from the equations which present the variety. Conversely, the individual structures are realized as retracts and the algebraic operations as combinatory objects; these are reclaimable by fixed combinators from the individual solutionsX. These results can be extended to universal classes and to algorithmic classes.Presented by Walter Taylor.     

Representation and duality for Hilbert algebras

In this paper we introduce a special kind of ordered topological spaces, called Hilbert spaces. We prove that the category of Hilbert algebras with semi-homomorphisms is dually equivalent to the category of Hilbert spaces with certain relations. We restrict this result to give a duality for the category of Hilbert algebras with homomorphisms. We apply these results to prove that the lattice of the deductive systems of a Hilbert algebra and the lattice of open subsets of its dual Hilbert space, are isomorphic. We explore how this duality is related to the duality given in [6] for finite Hilbert algebras, and with the topological duality developed in [7] for Tarski algebras.