Representations of quantum algebras

Inventiones mathematicae -     

Representations of quantum affine superalgebras

Mathematische Zeitschrift - We study the quantum affine superalgebra $$U_q(mathcal {L}mathfrak {sl}(M,N))$$ and its finite-dimensional representations. We prove a triangular decomposition and...     

Representations of quantum permutation algebras

We develop a combinatorial approach to the quantum permutation algebras, as Hopf images of representations of type π:A_s(n)→B(H). We discuss several general problems, including the commutativity and cocommutativity ones, the existence of tensor product or free wreath product decompositions, and the Tannakian aspects of the construction. The main motivation comes from the quantum invariants of the complex Hadamard matrices: we show here that, under suitable regularity assumptions, the computations can be performed up to n=6.     

Representations of homogeneous quantum Lévy fields

We study homogeneous quantum Lévy processes and fields with independent additive increments over a noncommutative *-monoid. These are described by infinitely divisible generating state functionals, invariant with respect to an endomorphic injective action of a symmetry semigroup. A strongly covariant GNS representation for the conditionally positive logarithmic functionals of these states is constructed in the complex Minkowski space in terms of canonical quadruples and isometric representations on the underlying pre-Hilbert field space. This is of much use in constructing quantum stochastic representations of homogeneous quantum Lévy fields on Itô monoids, which is a natural algebraic way of defining dimension free, covariant quantum stochastic integration over a space-time indexing set.     

Representations of deformed preprojective algebras and quantum groups

Let (Γ,I) be the bound quiver of a cyclic quiver whose vertices correspond to the Abelian group Zd. In this paper, we list all indecomposable representations of (Γ,I) and give the conditions that those representations of them can be extended to representations of deformed preprojective algebra Πλ(Γ,I). It is shown that those representations given by extending indecomposable representations of (Γ,I) are all simple representations of Πλ(Γ,I). Therefore, it is concluded that all simple representa-tions of rest...     

Representations of quantum groups defined over commutative rings

In this article we study the structure of highest weight modules for quantum groups defined over a commutative ring with particular emphasis on the structure theory for invariant bilinear forms on these modules. The notion of an induced invariant form is introduced and a setting in described where all invariant forms are induced     

Representations of quantum groups defined over commutative rings II

In this article we study the structure of highest weight modules for quantum groups defined over a commutative ring with particular emphasis on the structure theory for invariant bilinear forms on these modules.     

Representations of pointed Hopf algebras and their Drinfel'd quantum doubles

We study representations of nontrivial liftings of nilpotent type of quantum linear spaces and their Drinfel'd quantum doubles. We construct a family of Verma-type modules in both cases and prove a parametrization theorem for simple modules. We compute the Loewy and socle series of Verma modules under a mild restriction on the datum of a lifting. We find bases and dimensions of simple modules.     

Representations of the compact quantum groupSU q(2) and geometrical quantization

The Souriau—Kostant method of geometrical quantization is used to construct infinite-dimensional irreducible unitary representations of the algebra of functions of the compact quantum groupSU _q(2). The generalization to the case of the quantum groupSU _q (n) is discussed.V. A. Steklov Mathematics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 2, pp. 163–172, August, 1994.     

Interval orders and circle orders

A finite poset is an interval order if its point can be mapped into real intervals so that x in the poset precisely when x's interval lies wholly to the left of y's; the poset is a circle order if its points can be mapped into circular disks in the plane so that x precisely when x's circular disk is properly included in y's. This note proves that every finite interval order is a circle order.     

Circle orders and angle orders

A poset is a circle order if its points can be mapped into circular disks in the plane so that x in the poset precisely when x's circular disk is properly included in y's; the poset is an angle order if its points can be mapped into unbounded angular regions that preserve < by proper inclusion. It is well known that many finite angle orders are not circle orders, but has been open as to whether every finite circle order is an angle order. This paper proves that there are finite circle orders that are not angle orders.     

Interval orders based on weak orders

One definition of an interval order is as an order isomorphic to that of a family of nontrivial intervals of a linearly ordered set with [a,b] < [c,d] if b c. Fishburn's theorem states that an order is an interval order if and only if it has no four-element restriction isomorphic to the ordered set (shown in Fig. 1) “ ”. We show that an order is isomorphic to a family of nontrivial intervals of a weak order, ordered as above, if and only if it has no restriction to one of the four ordered sets (shown in Fig. 2) “ ”, a six-element crown or a six-element fence.     

Tilted symmetric orders are symmetric orders

We prove in this note that a ring which is derived equivalent to a symmetric order is again a symmetric order. Group rings of finite groups over an integral domain of characteristic 0 are symmetric orders.     

Weil Representations as Globally Irreducible Representations

The notion of globally irreducible representations of finite groups was introduced by B.H. Gross, in order to explain new series of Euclidean lattices discovered recently by N. Elkies and T. Shioda using Mordell–Weil lattices of elliptic curves. It has been observed by R. Gow and Gross that irreducible Weil representations of certain finite classical groups lead to globally irreducible representations. In this paper we classify all globally irreducible representations coming from Weil representations of finite classical groups.     

Quasi-hereditary orders

Quasi-hereditary rings are defined as generalizations of quasi-hereditary algebras; it is shown that some of the main properties of quasi-hereditary algebras carry over to quasi-hereditary rings. In particular, a generalization of the concept of highest weight categories gives a class of classical orders which satisfy our definition of quasi-heredity. For this class of orders the category of good lattices is investigated and it is shown that this category has almost split sequences.