The tetrahedron algebra and its finite-dimensional irreducible modules

Recently Terwilliger and the present author found a presentation for the three-point sl₂ loop algebra via generators and relations. To obtain this presentation we defined a Lie algebra ? by generators and relations and displayed an isomorphism from ? to the three-point sl₂ loop algebra. In this paper we classify the finite-dimensional irreducible ?-modules.     

Structure and Classification of Superconformal Nets

We study the general structure of Fermi conformal nets of von Neumann algebras on S ¹ and consider a class of topological representations, the general representations, that we characterize as Neveu–Schwarz or Ramond representations, in particular a Jones index can be associated with each of them. We then consider a supersymmetric general representation associated with a Fermi modular net and give a formula involving the Fredholm index of the supercharge operator and the Jones index. We then consider the net associated with the super-Virasoro algebra and discuss its structure. If the central charge c belongs to the discrete series, this net is modular by the work of F. Xu and we get an example where our setting is verified by taking the Ramond irreducible representation with lowest weight c/24. We classify all the irreducible Fermi extensions of any super-Virasoro net in the discrete series, thus providing a classification of all superconformal nets with central charge less than 3/2. Sebastino Carpi: Supported by MIUR, GNAMPA-INDAM and EU network “Noncommutative Geometry” MRTN-CT-2006-0031962 Yasuyuki Kawahigashi: Supported in part by the Grants-in-Aid for Scientific Research, JSPS. Submitted: March 3, 2008. Accepted: May 5, 2008.     

Yangians: Their representations and characters

The finite-dimensional irreducible representations of the Yangian of sl₂ are parametrized by their highest weights, which are monic polynomials in one variable. In this paper, we give a formula for the character of such a representation which depends only on its highest weight, and is an analogue of the classical Weyl character formula.     

Some conjectures on modular representations of affine sl<Subscript>2</Subscript> and Virasoro algebra

We conjecture an explicit bound on the prime characteristic of a field, under which theWeyl modules of affine sl₂ and the minimal series modules of Virasoro algebra remain irreducible, and Goddard-Kent-Olive coset construction for affine sl₂ is valid.     

The Polynomial Behavior of Weight Multiplicities for the Affine Kac–Moody Algebras A(1)r

We prove that the multiplicity of an arbitrary dominant weight for an irreducible highest weight representation of the affine Kac–Moody algebra A ⁽¹⁾ _r is a polynomial in the rank r. In the process we show that the degree of this polynomial is less than or equal to the depth of the weight with respect to the highest weight. These results allow weight multiplicity information for small ranks to be transferred to arbitrary ranks.     

Rational Functions with Prescribed Critical Points

A rational function is the ratio of two complex polynomials in one variable without common roots. Its degree is the maximum of the degrees of the numerator and the denominator. Rational functions belong to the same class if one turns into the other by postcomposition with a linear-fractional transformation. We give an explicit formula for the number of classes having a given degree d and given multiplicities m₁,..., m_n of given n critical points, for generic positions of the critical points. This number is the multiplicity of the irreducible sl₂ representation with highest weight in the tensor product of the irreducible sl₂ representations with highest weights The classes are labeled by the orbits of critical points of a remarkable symmetric function which first appeared in the XIX century in studies of Fuchsian differential equations, and then in the XX century in the theory of KZ equations.     

Decomposition ofq-deformed Fock spaces

A decomposition of the level-oneq-deformed Fock space representations ofU _q(sl _n ) is given. It is found that the action ofU _q(sl _n ) on these Fock spaces is centralized by a Heisenberg algebra, which arises from the center of the affine Hecke algebra _N in the limitN . Theq-deformed Fock space is shown to be isomorphic as aU _q(sl _n )-Heisenberg-bimodule to the tensor product of a level-one irreducible highest weight representation ofU _q(sl _n ) and the Fock representation of the Heisenberg algebra. The isomorphism is used to decompose theq-wedging operators, which are intertwiners between theq-deformed Fock spaces, into constituents coming fromU _q(sl _n ) and from the Heisenberg algebra.     

Ideals of identities of representations of nilpotent lie algebras

Let L be a Lie algebra, nilpotent of class 2, over an infinite field K, and suppose that the centre C of L is one dimensional; such Lie algebras are called Heisenberg algebras. Let ρ:L→hom _KV be a finite dimensional representation of the Heisenberg algebra L such that ρ(C) contains non-singular linear transformations of V, and denote l(ρ) the ideal of identities for the representation ρ. We prove that the ideals of identities of representations containing I(ρ) and generated by multilinear polynomials satisfy the ACC. Let sl ₂(L) be the Lie algebra of the traceless 2×2 matrices over K, and suppose the characteristic of K equals 2. As a corollary we obtain that the ideals of identities of representations of Lie algebras containing that of the regular representation of sl ₂(K) and generated by multilinear polynomials, are finitely based. In addition we show that one cannot simply dispense with the condition of multilinearity. Namely, we show that the ACC is violated for the ideals of representations of Lie algebras (over an infinite field of characteristic 2) that contain the identities of the regular representation of sl ₂(K).     

Analytic actions on compact surfaces and fixed points

 Consider an effective real analytic action of a connected Lie group G on a compact connected surface of Euler characteristic χ≠0. We show that if the action has no fixed point then χ≥1 and the Lie algebra 𝒢 of G is isomorphic either to a subalgebra of the affine algebra of ℝ², which is the extension of the ideal of constant vector fields by an irreducible linear subalgebra, or to sl(2,ℝ), o(3), sl(2,ℂ) and sl(3,ℝ). Received: 7 August 2001 Published online: 24 January 2003     

Quantum differential operators on the quantum plane

The universal enveloping algebra of a Lie algebra acts on its representation ring R through D(R), the ring of differential operators on R. A quantised universal enveloping algebra (or quantum group) is a deformation of a universal enveloping algebra and acts not through the differential operators of its representation ring but through the quantised differential operators of its representation ring. We present this situation for the quantum group of sl₂.     

A basis for the symplectic group branching algebra

The symplectic group branching algebra, B\mathcal {B}, is a graded algebra whose components encode the multiplicities of irreducible representations of Sp_2n−2(ℂ) in each finite-dimensional irreducible representation of Sp_2n (ℂ). By describing on B\mathcal {B} an ASL structure, we construct an explicit standard monomial basis of B\mathcal {B} consisting of Sp_2n−2(ℂ) highest weight vectors. Moreover, B\mathcal {B} is known to carry a canonical action of the n-fold product SL₂×⋯×SL₂, and we show that the standard monomial basis is the unique (up to scalar) weight basis associated to this representation. Finally, using the theory of Hibi algebras we describe a deformation of Spec(B)\mathrm{Spec}(\mathcal {B}) into an explicitly described toric variety.     

Tensor Products and Whittaker Vectors for Quantum Groups

We describe the action of the center of the quantum group U_q () on the tensor product V ? L(λ) of an infinite-dimensional representation V having an infinitesimal character χ_τ and an irreducible finite-dimensional U_q () representation L(λ) of highest weight λ. We apply this result in order to describe the tensor product of a Whittaker module and a finite-dimensional simple module for the algebra U_q(l₂).     

Quantum groups and quantum shuffles

Let U _q ⁺ be the “upper triangular part” of the quantized enveloping algebra associated with a symetrizable Cartan matrix. We show that U _q ⁺ is isomorphic (as a Hopf algebra) to the subalgebra generated by elements of degree 0 and 1 of the cotensor Hopf algebra associated with a suitable Hopf bimodule on the group algebra of Z ⁿ . This method gives supersymetric as well as multiparametric versions of U _q ⁺ in a uniform way (for a suitable choice of the Hopf bimodule). We give a classification result about the Hopf algebras which can be obtained in this way, under a reasonable growth condition. We also show how the general formalism allows to reconstruct higher rank quantized enveloping algebras from U _q sl(2) and a suitable irreducible finite dimensional representation. Oblatum 21-III-1997 & 12-IX-1997     

Lattice construction of logarithmic modules for certain vertex algebras

A general method for constructing logarithmic modules in vertex operator algebra theory is presented. By utilizing this approach, we give explicit vertex operator construction of certain indecomposable and logarithmic modules for the triplet vertex algebra W(p){\mathcal{W}(p)} and for other subalgebras of lattice vertex algebras and their N = 1 super extensions. We analyze in detail indecomposable modules obtained in this way, giving further evidence for the conjectural equivalence between the category of W(p){\mathcal{W}(p)}-modules and the category of modules for the restricted quantum group [`(U)]_q(sl₂){\overline{\mathcal{U}}_q(sl_2)} , q = e ^{π i/p} . We also construct logarithmic representations for a certain affine vertex operator algebra at admissible level realized in Adamović (J. Pure Appl. Algebra 196:119–134, 2005). In this way we prove the existence of the logarithmic representations predicted in Gaberdiel (Int. J. Modern Phys. A 18, 4593–4638, 2003). Our approach enlightens related logarithmic intertwining operators among indecomposable modules, which we also construct in the paper.     

CLASSIFICATION OF INFINITE DIMENSIONAL WEIGHT MODULES OVER THE LIE SUPERALGEBRA sl(2/1)

We give a complete classification of infinite dimensional indecomposable weight modules over the Lie superalgebra sl(2/1).     

Representations of the Quantum Algebra Uq(un, 1)

The main aim of the paper is to study infinite-dimensional representations of the real form U _q (u _{n, 1}) of the quantized universal enveloping algebra U _q (gl _{n + 1}). We investigate the principal series of representations of U _q (u _{n, 1}) and calculate the intertwining operators for pairs of these representations. Some of the principal series representations are reducible. The structure of these representations is determined. Then we classify irreducible representations of U _q (u _{n, 1}) obtained from irreducible and reducible principal series representations. All *-representations in this set of irreducible representations are separated. Unlike the classical case, the algebra U _q (u _{n, 1}) has finite-dimensional irreducible *-representations.     

On simple semiabelian p-adic lie algebras

A quantization of a non-standard rational solution of CYBE for sl ₂ is given explicitly. We obtain the quantization with the help of a twisting of the usual Yangian Y{sl ₂). This quantum object (deformed Yangian Y _ηξ( sl ₂)) is a two-parametric deformation of the universal enveloping algebra U(sl ₂[u]) of the polynomial current algebra sl ₂[u]. We consider the pseudotriangular structure on Y _ηξ( sl ₂), the quantum double DY _ηξ(sl ₂), its the universal R-matrix and also the RTT-realization of Y _ηξ( sl ₂).