Heisenberg Subalgebra Research Articles

Feigin–Semikhatov conjecture and related topics

Feigin–Semikhatov conjecture, now established, states algebraic isomorphisms between the cosets of the subregular [Formula: see text]-algebras and the principal [Formula: see text]-superalgebras of type A by their full Heisenberg subalgebras. It can be seen as a variant of Feigin–Frenkel duality between the [Formula: see text]-algebras and also as a generalization of the connection between the [Formula: see text] superconformal algebra and the affine algebra [Formula: see text]. We review the recent developments on the correspondence of the subregular [Formula: see text]-algebras and the principal [Formula: see text]-superalgebras of type A at the level of algebras, modules and intertwining operators, including fusion rules.

Affine Lie algebra representations induced from Whittaker modules

We use induction from parabolic subalgebras with infinite-dimensional Levi factor to construct new families of irreducible representations for arbitrary affine Kac-Moody algebras. Our first construction defines a functor from the category of Whittaker modules over the Levi factor of a parabolic subalgebra to the category of modules over the affine Lie algebra. The second functor sends tensor products of a module over the affine part of the Levi factor (in particular any weight module) and of a Whittaker module over the complement Heisenberg subalgebra to the affine Lie algebra modules. Both functors preserve irreducibility when the central charge is nonzero.

Representations of affine Lie superalgebras and their quantization in type A

We construct a new family of irreducible modules over any basic classical affine Kac-Moody Lie superalgebra which are induced from modules over the Heisenberg subalgebra. We also obtain irreducible deformations of these modules for the quantum affine superalgebra Uqslˆ(m|n).

Cylindric symmetric functions and positivity

We introduce new families of cylindric symmetric functions as subcoalgebras in the ring of symmetric functions Λ (viewed as a Hopf algebra) which have non-negative structure constants. Combinatorially these cylindric symmetric functions are defined as weighted sums over cylindric reverse plane partitions or - alternatively - in terms of sets of affine permutations. We relate their combinatorial definition to an algebraic construction in terms of the principal Heisenberg subalgebra of the affine Lie algebra slˆn and a specialised cyclotomic Hecke algebra. Using Schur–Weyl duality we show that the new cylindric symmetric functions arise as matrix elements of Lie algebra elements in the subspace of symmetric tensors of a particular level-0 module which can be identified with the small quantum cohomology ring of the k-fold product of projective space. The analogous construction in the subspace of alternating tensors gives the known set of cylindric Schur functions which are related to the small quantum cohomology ring of Grassmannians. We prove that cylindric Schur functions form a subcoalgebra in Λ whose structure constants are the 3-point genus 0 Gromov–Witten invariants. We show that the new families of cylindric functions obtained from the subspace of symmetric tensors also share the structure constants of a symmetric Frobenius algebra, which we define in terms of tensor multiplicities of the generalised symmetric group G(n,1,k).

Quasi-particles in the principal picture of 𝔰𝔩̂2 and Rogers–Ramanujan-type identities

In their seminal work Lepowsky and Wilson gave a vertex-operator theoretic interpretation of Gordon–Andrews–Bressoud’s generalization of Rogers–Ramanujan combinatorial identities, by constructing bases of vacuum spaces for the principal Heisenberg subalgebra of standard [Formula: see text]-modules, parametrized with partitions satisfying certain difference 2 conditions. In this paper, we define quasi-particles in the principal picture of [Formula: see text] and construct quasi-particle monomial bases of standard [Formula: see text]-modules for which principally specialized characters are given as products of sum sides of the corresponding analytic Rogers–Ramanujan-type identities with the character of the Fock space for the principal Heisenberg subalgebra.

Quantum affine modules for non-twisted Affine Kac-Moody algebras

We construct new irreducible weight modules over quantum affine algebras of type I with all weight spaces infinite-dimensional. These modules are obtained by parabolic induction from irreducible modules over the Heisenberg subalgebra.

Q, t-CHARACTERS AND THE STRUCTURE OF THE ℓ-WEIGHT SPACES OF STANDARD MODULES OVER SIMPLY LACED QUANTUM AFFINE ALGEBRAS

We establish, for a family of standard modules over simply laced quantum affine algebras, a q, t-character formula, first conjectured by Nakajima. It relates, on one hand, the structure of the l-weight spaces of those standard modules regarded as modules over the Heisenberg subalgebra and, on the other hand, the t-dependence of their q, t-characters, as originally defined in terms of the Poincare polynomials of certain Lagrangian quiver varieties of the corresponding simply laced type. Our proof is essentially geometric and generalizes to arbitrary simply laced types earlier representation theoretical results for standard Uq $$ \left({\widehat{\mathfrak{sl}}}_2\right) $$ -modules.

Quasi-Whittaker modules for the Schrödinger algebra

In this paper, we construct a class of new modules for the Schrödinger algebra S, called quasi-Whittaker modules. Different from [30], the quasi-Whittaker module is not induced by the Borel subalgebra of the Schrödinger algebra related with the triangular decomposition, but by its Heisenberg subalgebra H. We prove that, a simple S-module V is a quasi-Whittaker module if and only if V is a locally finite H-module. Furthermore, we classify simple quasi-Whittaker modules by central character of U(S) and their quasi-Whittaker functions. Finally, we characterize arbitrary quasi-Whittaker modules.

On simple modules over conformal Galilei algebras

We study irreducible representations of two classes of conformal Galilei algebras in 1-spatial dimension. We construct a functor which transforms simple modules with nonzero central charge over the Heisenberg subalgebra into simple modules over the conformal Galilei algebras. This can be viewed as an analogue of oscillator representations. We use oscillator representations to describe the structure of simple highest weight modules over conformal Galilei algebras. We classify simple weight modules with finite dimensional weight spaces over finite dimensional Heisenberg algebras and use this classification and properties of oscillator representations to classify simple weight modules with finite dimensional weight spaces over conformal Galilei algebras.

Free Field Realizations of Induced Modules for Affine Lie Algebras

We construct a free field realization of generalized loop modules induced from admissible diagonal ℤ-graded irreducible modules of nonzero level for a Heisenberg subalgebra of untwisted affine Lie algebra of type A. Such generalized loop modules are irreducible as it was shown in Bekkert et al. [2]. We also extend this construction to modules induced from an arbitrary parabolic subalgebra of whose Levi factor is infinite-dimensional and contains .

Quantum Lie algebra solitons

We construct a special type of quantum soliton solutions for quantized affine Toda models. The elements of the principal Heisenberg subalgebra in the affinised quantum Lie algebra are found. Their eigenoperators inside the quantized universal enveloping algebra for an affine Lie algebra are constructed to generate quantum soliton solutions.

Heisenberg-Type Families in Uq(sl2)

Using the second Drinfeld formulation of the quantized universal enveloping algebra $U_q(\widehat{sl_2})$ we introduce a family of its Heisenberg-type elements which are endowed with a deformed commutator and satisfy properties similar to generators of a Heisenberg subalgebra. Explicit expressions for new family of generators are found.

Some remarks on the coherent-state variational approach to nonlinear boson models

Mean-field pictures based on the standard time-dependent variational approach have widely been used in studying nonlinear many-boson systems such as the Bose–Hubbard model. Mean-field schemes relevant to Gutzwiller-like trial states |F⟩, number-preserving states |ξ⟩ and Glauber-like trial states |Z⟩ are compared to evidence of specific properties of such schemes. After deriving the Hamiltonian picture relevant to |Z⟩ from that based on |F⟩, the latter is shown exhibiting a Poisson algebra equipped with a Weyl–Heisenberg subalgebra which preludes to the |Z⟩-based picture. Then states |Z⟩ are shown to be a superposition of -boson states |ξ⟩, and the similarities/differences between the |Z⟩-based and |ξ⟩-based pictures are discussed. Finally, after proving that the simple, symmetric state |ξ⟩ indeed corresponds to a SU(M) coherent state, a dual version of states |Z⟩ and |ξ⟩ in terms of momentum-mode operators is discussed together with some applications.

Solvable Lie algebras with naturally graded nilradicals and their invariants

The indecomposable solvable Lie algebras with graded nilradical of maximal nilindex and a Heisenberg subalgebra of codimension one are analyzed, and their generalized Casimir invariants calculated. It is shown that rank one solvable algebras have a contact form, which implies the existence of an associated dynamical system. Moreover, due to the structure of the quadratic Casimir operator of the nilradical, these algebras contain a maximal non-abelian quasi-classical Lie algebra of dimension $2n-1$, indicating that gauge theories (with ghosts) are possible on these subalgebras.

Harish-Chandra modules for the q-analog Virasoro-like algebra

Let L ˆ be the q-analog Virasoro-like algebra. In this paper we prove that any Harish-Chandra module of L ˆ with nontrivial center is isomorphic to a generalized highest weight module. Moreover, we show that a generalized highest weight Harish-Chandra module of L ˆ is induced from an irreducible module of a Heisenberg subalgebra of L ˆ .

Nonzero level Harish-Chandra modules over the Virasoro-like algebra

In the present paper, we study the nonzero level Harish-Chandra modules for the Virasoro-like algebra. We prove that a nonzero level Harish-Chandra module of the Virasoro-like algebra is a generalized highest weight (GHW for short) module. Then we prove that a GHW module of the Virasoro-like algebra is induced from an irreducible module of a Heisenberg subalgebra.

Similarity reduction of the modified Yajima–Oikawa equation

We study a similarity reduction of the modified Yajima-Oikawa hierarchy. The hierarchy is associated with a non-standard Heisenberg subalgebra in the affine Lie algebra of type A_2^{(1)}. The system of equations for self-similar solutions is presented as a Hamiltonian system of degree of freedom two, and admits a group of B\"acklund transformations isomorphic to the affine Weyl group of type A_2^{(1)}. We show that the system is equivalent to a two-parameter family of the fifth Painlev\'e equation.

Dualities and vertex operator algebras of affine type

We notice that for any positive integer k , the set of (1,2)-specialized characters of level k standard A 1 (1) -modules is the same as the set of rescaled graded dimensions of the subspaces of level 2 k +1 standard A 2 (2) -modules that are vacuum spaces for the action of the principal Heisenberg subalgebra of A 2 (2) . We conjecture the existence of a semisimple category induced by the “equal level” representations of some algebraic structure which would naturally explain this duality-like property, and we study potential such structures in the context of generalized vertex operator algebras.

Intertwining operators and soliton equations

The fermionic approach to the Kadomtsev-Petviashvili hierarchy, suggested by the Kyoto school (Sato, Date, Jimbo, Kashiwara, and Miwa) in 1981–4, is generalized on the basis of the idea that, in a sense, the components of intertwining operators are a generalization of free fermions forgl ∞. Integrable hierarchies related to symmetries of Kac-Moody algebras are described in terms of intertwining operators. The bosonization of these operators for various choices of the Heisenberg subalgebra is explicity written out. These various realizations result in distinct hierarchies of soliton equations. For example, forsl N -symmetries this gives the hierarchies obtained by the (n 1,...,n s )-reduction from thes-component KP hierarchy introduced by Kac and van de Leur.

Vertex operator representation of the soliton tau functions in the An(1) Toda models by dressing transformations

We study the relation between the group-algebraic approach and the dressing symmetry one to the soliton solutions of the An(1) Toda field theory in 1+1 dimensions. Originally, solitons in the affine Toda models were found by Olive, Turok, and Underwood. Single solitons are created by exponentials of elements which ad-diagonalize the principal Heisenberg subalgebra. Alternatively, Babelon and Bernard exploited the dressing symmetry to reproduce the known expressions for the fundamental tau functions in the sine-Gordon model. In this paper we show the equivalence between these two methods to construct solitons in the An(1) Toda models.