Stable rigged configurations for quantum affine algebras of nonexceptional types

For an affine algebra of nonexceptional type in the large rank we show the fermionic formula depends only on the attachment of the node 0 of the Dynkin diagram to the rest, and the fermionic formula of not type A can be expressed as a sum of that of type A with Littlewood–Richardson coefficients. Combining this result with Kirillov et al. (2002) [13] and Lecouvey et al. (2011) [18] we settle the X=M conjecture under the large rank hypothesis.     

Finite-dimensional crystals B 2,s for quantum affine algebras of type D (1) n

The Kirillov–Reshetikhin modules W^r,s are finite-dimensional representations of quantum affine algebras U’_q labeled by a Dynkin node r of the affine Kac–Moody algebra and a positive integer s. In this paper we study the combinatorial structure of the crystal basis B^2,s corresponding to W^2,s for the algebra of type D⁽¹⁾_n. 2000 Mathematics Subject Classification Primary—17B37; Secondary—81R10 Supported in part by the NSF grants DMS-0135345 and DMS-0200774.     

Level one representations of quantum affine algebras U_q(C^{(1)}_n)

We give explicit constructions of quantum symplectic affine algebras at level one using vertex operators.     

Vertex operator algebras associated to modified regular representations of affine Lie algebras

Let G be a simply-connected complex Lie group with simple Lie algebra g and let be its affine Lie algebra. We use intertwining operators and Knizhnik-Zamolodchikov equations to construct a family of N-graded vertex operator algebras (VOAs) associated to g. These vertex operator algebras contain the algebra of regular functions on G as the conformal weight 0 subspaces and are -modules of dual levels in the sense that , where h^∨ is the dual Coxeter number of g. This family of VOAs was previously studied by Arkhipov-Gaitsgory and Gorbounov-Malikov-Schechtman from different points of view. We show that when k is irrational, the vertex envelope of the vertex algebroid associated to G and the level k is isomorphic to the vertex operator algebra we constructed above. The case of rational levels is also discussed.     

Tubular algebras and affine Kac-Moody algebras

The purpose of this paper is to construct quotient algebras L(A)1C/I(A) of complex degenerate composition Lie algebras L(A)1C by some ideals, where L(A)1C is defined via Hall algebras of tubular algebras A, and to prove that the quotient algebras L(A)1C/I(A) are isomorphic to the corresponding affine Kac-Moody algebras. Moreover, it is shown that the Lie algebra Lre(A)1C generated by A-modules with a real root coincides with the degenerate composition Lie algebra L(A)1C generated by simple A-modules.     

Certain categories of modules for twisted affine Lie algebras

In this paper, using generating functions, we study two categories ? and ? of modules for twisted affine Lie algebras g^[σ], which were firstly introduced and studied for untwisted affine Lie algebras by H. -S. Li [Math Z, 2004, 248: 635-664]. We classify integrable irreducible g^[σ]-modules in categories ? and ?, where ? is proved to contain the well-known evaluation modules and ? to unify highest weight modules, evaluation modules and their tensor product modules. We determine also the isomorphism classes of those irreducible modules.     

Elementary operators on algebras of unbounded operators

Summary Structural results about elementary operators of length one, local elementary operators and injectivity preserving maps are proved. These are generalizations of results concerning algebras of bounded operators on Banach spaces to algebras of unbounded operators on Hilbert spaces.     

Varieties of affine Kac-Moody algebras

In this paper the identities of the complex affine Kac-Moody algebras are studied. It is proved that the identities of twisted affine algebras coincide with those of the corresponding nontwisted algebras. Moreover, in the class of nontwisted affine Kac-Moody algebras, each of these algebras is uniquely defined by its identities. It is shown that the varieties of affine algebras, as well as the varieties defined by finitely generated three-step solvable Lie algebras, have exponential growth. Translated fromMatematicheskie Zametki, Vol. 62 No. 1, pp. 95–102, July 1997. Translated by A. I. Shtern     

Pointed torsion free modules of affine lie algebras

It is shown that all pointed torsion free modules for affine Lie algebras belong to C⁽¹⁾ _n and A⁽¹⁾ _n-1 and are the result of the natural construction of tensoring the Laurent polynomials with a torsion free module of the “underlying” simple finite dimensional Lie Algebra. These latter modules have been completely determined by Britten and Lemire [1].     

Projection method and a universal weight function for the quantum affine algebra

We calculate the projection of the product of the Drinfeld currents on the intersection of the different Borel subalgebras in the current realization of the quantum affine algebra . This projection yields a universal weight function and has the structure of nested Bethe vectors. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 2, pp. 286–303, February, 2007.     

Filtrations and completions of certain positive level modules of affine algebras

We define a filtration indexed by the integers on the tensor product of a simple highest weight module and a loop module for a quantum affine algebra. We prove that such a filtration is either trivial or strictly decreasing and give sufficient conditions for this to happen. In the first case we prove that the tensor product is simple and in the second case we prove that the intersection of all the modules in the filtration is zero, thus allowing us to define the completed tensor product. In certain special cases, we identify the subsequent quotients of filtration.     

Hypercyclic algebras for convolution and composition operators

We provide an alternative proof to those by Shkarin and by Bayart and Matheron that the operator D of complex differentiation supports a hypercyclic algebra on the space of entire functions. In particular we obtain hypercyclic algebras for many convolution operators not induced by polynomials, such as $\text{cos}(D)$, $D{e}^D$, or $e^D?aI$, where $0<a\le 1$. In contrast, weighted composition operators on function algebras of analytic functions on a plane domain fail to support supercyclic algebras.     

On algebras of polynomially compact operators

In this paper, we find sufficient and necessary conditions for a triangularizable closed algebra of polynomially compact operators to be commutative modulo the radical. We also prove that an algebraic algebra of operators of a bounded degree on a Banach space is triangularizable under some mild additional conditions. As a special case we obtain a result stating that every algebraic algebra of operators of bounded degree is triangularizable whenever its commutators are nilpotent operators.     

Multiplets of representations, twisted Dirac operators and Vogan's conjecture in affine setting

We extend classical results of Kostant et al. on multiplets of representations of finite-dimensional Lie algebras and on the cubic Dirac operator to the setting of affine Lie algebras and twisted affine cubic Dirac operator. We prove in this setting an analogue of Vogan's conjecture on infinitesimal characters of Harish-Chandra modules in terms of Dirac cohomology. For our calculations we use the machinery of Lie conformal and vertex algebras.     

Perron-Frobenius operators and representations of the Cuntz-Krieger algebras for infinite matrices

In this paper we extend the work of Kawamura, see [K. Kawamura, The Perron-Frobenius operators, invariant measures and representations of the Cuntz-Krieger algebras, J. Math. Phys. 46 (2005)], for Cuntz-Krieger algebras OA for infinite matrices A. We generalize the definition of branching systems, prove their existence for any given matrix A and show how they induce some very concrete representations of OA. We use these representations to describe the Perron-Frobenius operator, associated to a nonsingular transformation, as an infinite sum and under some hypothesis we find a matrix representation for the operator. We finish the paper with a few examples.     

On finite rank operators in CSL algebras II

For finite rank operators in a commutative subspace lattice algebra algℒ we introduce the concept of correlation matrices, basing on which we prove that a finite rank operator in algℒ can be written as a finite sum of rank-one operators in algℒ, if it has only finitely many different correlation matrices. Thus we can recapture the results of J.R. Ringrose, A. Hopenwasser and R.Moore as corollaries of our theorems. Research supported by NSF of China     

Spectral conditions on Lie and Jordan algebras of compact operators

We investigate the properties of bounded operators which satisfy a certain spectral additivity condition, and use our results to study Lie and Jordan algebras of compact operators. We prove that these algebras have nontrivial invariant subspaces when their elements have sublinear or submultiplicative spectrum, and when they satisfy simple trace conditions. In certain cases we show that these conditions imply that the algebra is (simultaneously) triangularizable.