Untwisted Affine Kac-Moody Algebra Research Articles

Teleparallel Geroch geometry

We construct the teleparallel dynamics for extended geometry where the structure algebra is (an extension of) an untwisted affine Kac-Moody algebra. This provides a geometrisation of the Geroch symmetry appearing on dimensional reduction of a gravitational theory to two dimensions. The formalism is adapted to the underlying tensor hierarchy algebra, and will serve as a stepping stone towards the geometrisation of other infinite-dimensional, e.g. hyperbolic, symmetries.

Characterization of simple smooth modules

In this paper, we characterize simple smooth modules over some infinite-dimensional Z-graded Lie algebras. More precisely, we prove that if one specific positive root element of a Z-graded Lie algebra g locally finitely acts on a simple g-module V, then V is a smooth g-module. These infinite-dimensional Z-graded Lie algebras include the Virasoro algebra, affine-Virasoro algebras, the (twisted, mirror) Heisenberg-Virasoro algebras, the planar Galilean conformal algebra, and many others. This result for untwisted affine Kac-Moody algebras holds unless we change the condition from “locally finitely” to “locally nilpotently”. We also show that these are not the case for the Heisenberg algebra.

3-Dimensional mixed BF theory and Hitchin’s integrable system

The affine Gaudin model, associated with an untwisted affine Kac–Moody algebra, is known to arise from a certain gauge fixing of 4-dimensional mixed topological–holomorphic Chern–Simons theory in the Hamiltonian framework. We show that the finite Gaudin model, associated with a finite-dimensional semisimple Lie algebra, or more generally the tamely ramified Hitchin system on an arbitrary Riemann surface, can likewise be obtained from a similar gauge fixing of 3-dimensional mixed BF theory in the Hamiltonian framework.

Real Hamiltonian forms of affine Toda field theories: Spectral aspects

The paper is devoted to real Hamiltonian forms of 2-dimensional Toda field theories related to exceptional simple Lie algebras, and to the spectral theory of the associated Lax operators. Real Hamiltonian forms are a special type of "reductions" of Hamiltonian systems, similar to real forms of semi-simple Lie algebras. Examples of real Hamiltonian forms of affine Toda field theories related to exceptional complex untwisted affine Kac-Moody algebras are studied. Along with the associated Lax representations, we also formulate the relevant Riemann-Hilbert problems and derive the minimal sets of scattering data that determine uniquely the scattering matrices and the potentials of the Lax operators.

Lattice of dominant weights of affine Kac–Moody algebras

The dual space of the Cartan subalgebra in a Kac–Moody algebra has a partial ordering defined by the rule that two elements are related if and only if their difference is a non-negative or non-positive integer linear combination of simple roots. In this paper, we study the subposet formed by dominant weights in affine Kac–Moody algebras. We give a more explicit description of the covering relations in this poset. We also study the structure of basic cells in this poset of dominant weights for untwisted affine Kac–Moody algebras of type A.

Borodin–Okounkov formula, string equation and topological solutions of Drinfeld–Sokolov hierarchies

We give a general method to compute the expansion of topological tau functions for Drinfeld-Sokolov hierarchies associated to an arbitrary untwisted affine Kac-Moody algebra. Our method consists of two main steps: first these tau functions are expressed as (formal) Fredholm determinants of the type appearing in the Borodin-Okounkov formula, then the kernels for these determinants are found using a reduced form of the string equation. A number of explicit examples are given.

Integrable representations for toroidal extended affine Lie algebras

Let g be any untwisted affine Kac–Moody algebra, μ any fixed complex number, and g˜(μ) the corresponding toroidal extended affine Lie algebra of nullity two. For any k-tuple λ=(λ1,⋯,λk) of weights of g, and k-tuple a=(a1,⋯,ak) of distinct non-zero complex numbers, we construct a class of modules V˜(λ,a) for the extended affine Lie algebra g˜(μ). We prove that the g˜(μ)-module V˜(λ,a) is completely reducible. We also prove that the g˜(μ)-module V˜(λ,a) is integrable when all weights λi in λ are dominant. Thus, we obtain a new class of irreducible integrable weight modules for the toroidal extended affine Lie algebra g˜(μ).

On Integrable Field Theories as Dihedral Affine Gaudin Models

Abstract We introduce the notion of a classical dihedral affine Gaudin model, associated with an untwisted affine Kac–Moody algebra $\widetilde{\mathfrak{g}}$ equipped with an action of the dihedral group $D_{2T}$, $T \geq 1$ through (anti-)linear automorphisms. We show that a very broad family of classical integrable field theories can be recast as examples of such classical dihedral affine Gaudin models. Among these are the principal chiral model on an arbitrary real Lie group $G_0$ and the $\mathbb{Z}_T$-graded coset $\sigma $-model on any coset of $G_0$ defined in terms of an order $T$ automorphism of its complexification. Most of the multi-parameter integrable deformations of these $\sigma $-models recently constructed in the literature provide further examples. The common feature shared by all these integrable field theories, which makes it possible to reformulate them as classical dihedral affine Gaudin models, is the fact that they are non-ultralocal. In particular, we also obtain affine Toda field theory in its lesser-known non-ultralocal formulation as another example of this construction. We propose that the interpretation of a given classical non-ultralocal integrable field theory as a classical dihedral affine Gaudin model provides a natural setting within which to address its quantisation. At the same time, it may also furnish a general framework for understanding the massive ordinary differential equations (ODE)/integrals of motion (IM) correspondence since the known examples of integrable field theories for which such a correspondence has been formulated can all be viewed as dihedral affine Gaudin models.

Local charges in involution and hierarchies in integrable sigma-models

Integrable σ-models, such as the principal chiral model, {mathbb{Z}}_T -coset models for Tin {mathbb{Z}}_{ge 2} and their various integrable deformations, are examples of non-ultralocal integrable field theories described by r/s-systems with twist function. In this general setting, and when the Lie algebra mathfrak{g} underlying the r/s-system is of classical type, we construct an infinite algebra of local conserved charges in involution, extending the approach of Evans, Hassan, MacKay and Mountain developed for the principal chiral model and symmetric space σ-model. In the present context, the local charges are attached to certain ‘regular’ zeros of the twist function and have increasing degrees related to the exponents of the untwisted affine Kac-Moody algebra widehat{mathfrak{g}} associated with mathfrak{g} . The Hamiltonian flows of these charges are shown to generate an infinite hierarchy of compatible integrable equations.

Soliton equations related to the affine Kac-Moody algebra D 4 (1)

We have derived the hierarchy of soliton equations associated with the untwisted affine Kac-Moody algebra D (1)4 by calculating the corresponding recursion operators. The Hamiltonian formulation of the equations from the hierarchy is also considered. As an example we have explicitly presented the first non-trivial member of the hierarchy, which is an one-parameter family of mKdV equations. We have also considered the spectral properties of the Lax operator and introduced a minimal set of scattering data.

Symmetric quiver Hecke algebras and R -matrices of quantum affine algebras, II

Let $\g$ be an untwisted affine Kac-Moody algebra of type $A^{(1)}_n$ $(n \ge 1)$ or $D^{(1)}_n$ $(n \ge 4)$ and let $\g_0$ be the underlying finite-dimensional simple Lie subalgebra of $\g$. For each Dynkin quiver $Q$ of type $\g_0$, Hernandez and Leclerc (\cite{HL11}) introduced a tensor subcategory $\CC_Q$ of the category of finite-dimensional integrable $\uqpg$-modules and proved that the Grothendieck ring of $\CC_Q$ is isomorphic to $\C [N]$, the coordinate ring of the unipotent group $N$ associated with $\g_0$. We apply the generalized quantum affine Schur-Weyl duality introduced in \cite{KKK13} to construct an exact functor $\F$ from the category of finite-dimensional graded $R$-modules to the category $\CC_Q$, where $R$ denotes the symmetric quiver Hecke algebra associated to $\g_0$. We prove that the homomorphism induced by the functor $\F$ coincides with the homomorphism of Hernandez and Leclerc and show that the functor $\F$ sends the simple modules to the simple modules.

Fusion Product Structure of Demazure Modules

Let 𝔤 be a finite–dimensional complex simple Lie algebra. Given a non–negative integer ℓ, we define $\mathcal {P}^{+}_{\ell }$ to be the set of dominant weights λ of 𝔤 such that ℓΛ0+λ is a dominant weight for the corresponding untwisted affine Kac–Moody algebra $\widehat {{\mathfrak {g}}}$ . For the current algebra 𝔤[t] associated to 𝔤, we show that the fusion product of an irreducible 𝔤–module V(λ) such that $\lambda \in \mathcal {P}^{+}_{\ell }$ and a finite number of special family of 𝔤–stable Demazure modules of level ℓ (considered in Fourier and Littelmann, Nagoya Math. J. 182, 171–198 (2006), Adv. Math. 211(2), 566–593 2007) again turns out to be a Demazure module. This fact is closely related with several important conjectures. We use this result to construct the 𝔤[t]–module structure of the irreducible ${\widehat {\mathfrak {g}}}$ –module V(ℓ Λ0 + λ) as a semi–infinite fusion product of finite dimensional 𝔤[t]–modules as conjectured in Fourier and Littelmann, Adv. Math. 211(2), 566–593 (2007). As a second application we give further evidence to the conjecture on the generalization of Schur positivity (see Chari, Fourier and Sagaki 2013).

A category of modules for the full toroidal Lie algebra

Toroidal Lie algebras are very natural multi-variable generalizations of affine Kac-Moody algebras. The theory of affine Lie algebras is rich and beautiful, having connections with diverse areas of mathematics and physics. Toroidal Lie algebras are also proving themselves to be useful for the applications. Frenkel, Jing and Wang [FJW] used representations of toroidal Lie algebras to construct a new form of the McKay correspondence. Inami et al., studied toroidal symmetry in the context of a 4-dimensional conformal field theory [IKUX], [IKU]. There are also applications of toroidal Lie algebras to soliton theory. Using representations of the toroidal algebras one can construct hierarchies of non-linear PDEs [B2], [ISW]. In particular, the toroidal extension of the Korteweg-de Vries hierarchy contains the Bogoyavlensky’s equation, which is not in the classical KdV hierarchy [IT]. One can use the vertex operator realizations to construct n-soliton solutions for the PDEs in these hierarchies. We hope that further development of the representation theory of toroidal Lie algebras will help to find new applications of this interesting class of algebras. The construction of a toroidal Lie algebra is totally parallel to the well-known construction of an (untwisted) affine Kac-Moody algebra [K1]. One starts with a finite-dimensional simple Lie algebra ġ and considers Fourier polynomial maps from an N + 1-dimensional torus into ġ. Setting tk = e ixk , we may identify the algebra of Fourier polynomials on a torus with the Laurent polynomial algebra R = C[t0 , t ± 1 , . . . , t ± N ], and the Lie algebra of the ġ-valued maps from a torus with the multi-loop algebra C[t±0 , t ± 1 , . . . , t ± N ] ⊗ ġ. When N = 0, this yields the usual loop algebra. Just as for the affine algebras, the next step is to build the universal central extension (R⊗ ġ)⊕K of R⊗ ġ. However unlike the affine case, the center K is infinite-dimensional when N ≥ 1. The infinite-dimensional center makes this Lie algebra highly degenerate. One can show, for example, that in an irreducible bounded weight module, most of the center should act trivially. To eliminate this degeneracy, we add the Lie algebra of vector fields on a torus, D = Der (R) to (R⊗ ġ)⊕K. The resulting algebra,

Verma-Type Modules for Quantum Affine Lie Algebras

Let \(\mathfrak{g}\) be an untwisted affine Kac–Moody algebra and MJ(λ) a Verma-type module for \(\mathfrak{g}\) with J-highest weight λ∈P. We construct quantum Verma-type modules MJq(λ) over the quantum group \(U_{q}(\mathfrak{g})\) , investigate their properties and show that MJq(λ) is a true quantum deformation of MJ(λ) in the sense that the weight structure is preserved under the deformation. We also analyze the submodule structure of quantum Verma-type modules.

The R-matrix action of untwisted affine quantum groups at roots of 1

Let gˆ be an untwisted affine Kac–Moody algebra. The quantum group Uq(gˆ) is known to be a quasitriangular Hopf algebra (to be precise, a braided Hopf algebra). Here we prove that its unrestricted specializations at odd roots of 1 are braided too: in particular, specializing q at 1 we have that the function algebra F[Hˆ] of the Poisson proalgebraic group Ĥ dual of Ĝ (a Kac–Moody group with Lie algebra gˆ) is braided. This in turn implies also that the action of the universal R-matrix on the tensor products of pairs of Verma modules can be specialized at odd roots of 1.

Dual affine quantum groups

Let $\hat{\mathfrak{g}}$ be an untwisted affine Kac-Moody algebra, with its Sklyanin-Drinfel'd structure of Lie bialgebra, and let $\hat{\mathfrak{h}}$ be the dual Lie bialgebra. By dualizing the quantum double construction - via formal Hopf algebras - we construct a new quantum group $U_q(\hat{\mathfrak{h}})$, dual of $U_q(\hat{\mathfrak{g}})$. Studying its restricted and unrestricted integer forms and their specializations at roots of 1 (in particular, their classical limits), we prove that $U_q(\hat{\mathfrak{h}})$ yields quantizations of $\hat{\mathfrak{h}}$ and $\hat{G}^\infty$ (the formal group attached to $\hat{\mathfrak{g}}$), and we construct new quantum Frobenius morphisms. The whole picture extends to the untwisted affine case the results known for quantum groups of finite type.

A PBW basis for lusztig’s form of untwisted affine quantum groups

Let ĝ an untwisted affine Kac-Moody algebra over the field C, and let Uq(ĝ) be the associated quantum enveloping algebra; let Uq(ĝ) be the Lusztig’s integer form of Uq(ĝ), generated by q-divided powers of Chevalley generators over a suitable sub ring R of C(q). We prove a Poincaré-Birkhoff-Witt like theorem for Uq(ĝ), yielding a basis over R made of ordered products of q-divided powers of suitable quantum root vectors.

The matrix theory of automorphisms for twisted affine Kac-Moody algebras (for ) and (for )

=-1 The application to the two series of classical twisted affine Kac-Moody algebras (for ) and (for ) of the matrix formulation of automorphisms developed earlier for untwisted affine Kac-Moody algebras is investigated. The conjugacy classes of involutive automorphisms and the corresponding real forms are deduced explicitly for the special case of .

From hydrogen atom to generalized Dynkin diagrams

We identify the “dynamical algebras" H D of the D-dimensional hydrogen atom as positive subalgebras of twisted and untwisted affine Kac-Moody algebras: For odd D≥5 we obtain H 2l+1≃D l+1 (2)+ . But for even D≥6, H 2l is a parabolic subalgebra of B l (1). H 4 is a parabolic subalgebra of C 2 (1), H 3≃D 2 (2)+≃A 1 (1)+ , while H 2 is isomorphic to the Borel subalgebra of A 1 (1). Along the way we prove a theorem on the untwisting of positive subalgebras of twisted affine algebras, and introduce generalized Dynkin diagrams which enable us to represent graphically automorphisms and parabolic subalgebras of finite and affine algebras.

Formula omitted]-Algebras from Soliton Equations and Heisenberg Subalgebras

We derive sufficient conditions under which the "second" Hamiltonian structure of a class of generalized KdV-hierarchies defines one of the classical W-algebras obtained through Drinfel′d-Sokolov Hamiltonian reduction. These integrable hierarchies are associated to the Heisenberg subalgebras of an untwisted affine Kac-Moody algebra. When the principal Heisenberg subalgebra is chosen, the well-known connection between the Hamiltonian structure of the generalized Drinfel′d-Sokolov hierarchies-the Gel′fand-Dickey algebras-and the W-algebras associated to the Casimir invariants of a Lie algebra is recovered. After carefully discussing the relations between the embeddings of A1 = sl(2, C) into a simple Lie algebra g and the elements of the Heisenberg subalgebras of g(1), we identify the class of W-algebras that can be defined in this way. For An, this class only includes those associated to the embeddings labelled by partitions of the form n + 1 = k(m) + q(1) and n + 1 = k(m + 1) + k(m) + q(1).