Affine Algebras Research Articles

Tropical geometry, quantum affine algebras, and scattering amplitudes

Tropical geometry, quantum affine algebras, and scattering amplitudes

Integral closure of an affine algebra

Let R be a commutative ring with identity and R ′ be the integral closure of R. In this paper, we show that if R is an affine algebra over a field K, then every regular ideal of R ′ is finitely generated, i.e., R ′ is an r-Noetherian ring. We also study when the integral closure of an affine algebra is Noetherian. First we show that if R is a Krull ring such that R/P is a Noetherian domain for each minimal regular prime ideal P of R, then R is an r-Noetherian ring, which is a generalization of Nishimura’s result. As an application of this result, we prove that if R is an r-Noetherian ring with reg-dim R ≤ 2 , then R ′ is an r-Noetherian ring. We finally construct a couple of r-Noetherian rings, e.g., an r-Noetherian ring R that is not Noetherian and reg-dim R = ∞ or reg-dim R = n ≤ dim R = n + m − 1 for arbitrary positive integers n, m.

A twist at infinite distance in the CHL string

We analyze a space-time algebra of BPS states that emerges in the infinite distance limit in the moduli space of the nine-dimensional CHL string as the theory decompactifies to the ten-dimensional E8 × E8 heterotic string. We find an affine algebra as expected from the heterotic case, but in a twisted version.

Algebraic Complete Integrability of the $a_4^{(2)}$ Toda Lattice

The aim of this work is focused on the investigation of the algebraic complete integrability of the Toda lattice associated with the twisted affine Lie algebra $a_4^{(2)}$. First, we prove that the generic fiber of the momentum map for this system is an affine part of an abelian surface. Second, we show that the flows of integrable vector fields on this surface are linear. Finally, using the formal Laurent solutions of the system, we provide a detailed geometric description of these abelian surfaces and the divisor at infinity.

Shifted Yangians and Polynomial $R$-Matrices

We study the category \mathcal{O}^{\mathrm{sh}} of representations over a shifted Yangian. This category has a tensor product structure and contains distinguished modules, the positive prefundamental modules and the negative prefundamental modules. Motivated by the representation theory of the Borel subalgebra of a quantum affine algebra and by the relevance of quantum integrable systems in this context, we prove that tensor products of prefundamental modules with irreducible modules are either cyclic or cocyclic. This implies the existence and uniqueness of morphisms, the R -matrices, for such tensor products. We prove the R -matrices are polynomial in the spectral parameter, and we establish functional relations for the R -matrices. As applications, we prove the Jordan–Hölder property in the category \mathcal{O}^{\mathrm{sh}} . We also obtain a proof, uniform for any finite type, that any irreducible module factorizes through a truncated shifted Yangian.

The finite presentation problem for Noetherian algebras

We prove that there exists an affine Noetherian algebra over a field of characteristic zero which is not finitely presented. Specifically, we prove that the Medvedev–Passman–Resco–Small algebra, recently shown to form a counterexample to the stability problem for Noetherian algebras, is not finitely presented. This answers a question due to Bergman and Irving in characteristic zero, a case explicitly left open by Resco and Small in 1993, who gave a counterexample to this question in positive characteristic.

Affine Standard Lyndon Words: A-Type

Abstract We generalize an algorithm of Leclerc [ 6] describing explicitly the bijection of Lalonde–Ram [ 5] from finite to affine Lie algebras. In type $A_{n}^{(1)}$, we compute all affine standard Lyndon words for any order of the simple roots and establish some properties of the induced orders on the positive affine roots.

Young walls and graded dimension formulas for finite quiver Hecke algebras of types A2n−1(2), Dn(1) and Bn(1)

In this paper, we establish the graded dimension formulas of finite quiver Hecke algebras using level-[Formula: see text] Fock space representations of quantum affine algebras of types [Formula: see text], [Formula: see text] and [Formula: see text].

Quantum Sugawara operators in type A

The quantum Sugawara operators associated with a simple Lie algebra g are elements of the center of a completion of the quantum affine algebra Uq(gˆ) at the critical level. By the foundational work of Reshetikhin and Semenov-Tian-Shansky (1990), such operators occur as coefficients of a formal Laurent series ℓV(z) associated with every finite-dimensional representation V of the quantum affine algebra. As demonstrated by Ding and Etingof (1994), the quantum Sugawara operators generate all singular vectors in generic Verma modules over Uq(gˆ) at the critical level and give rise to a commuting family of transfer matrices. Furthermore, as observed by E. Frenkel and Reshetikhin (1999), the operators are closely related with the q-characters and q-deformed W-algebras via the Harish-Chandra homomorphism.We produce explicit quantum Sugawara operators for the quantum affine algebra of type A which are associated with primitive idempotents of the Hecke algebra and parameterized by Young diagrams. This opens a way to understand all the related objects via their explicit constructions. We consider one application by calculating the Harish-Chandra images of the quantum Sugawara operators. The operators act by scalar multiplication in the q-deformed Wakimoto modules and we calculate the eigenvalues by identifying them with the Harish-Chandra images.

Young wall realizations of level 1 irreducible highest weight and Fock space crystals of quantum affine algebras in type E

We construct Young wall models for the crystal bases of level 1 irreducible highest weight representations and Fock space representations of quantum affine algebras in types E6(1), E7(1) and E8(1). In each case, Young walls consist of coloured blocks stacked inside the relevant Young wall pattern which satisfy a certain combinatorial condition. Moreover the crystal structure is described entirely in terms of adding and removing blocks.

Representations of shifted quantum affine algebras and cluster algebras I: The simply laced case

AbstractWe introduce a family of cluster algebras of infinite rank associated with root systems of type , , . We show that suitable completions of these cluster algebras are isomorphic to the Grothendieck rings of the categories of the corresponding shifted quantum affine algebras. The cluster variables of a class of distinguished initial seeds are certain formal power series defined by E. Frenkel and the second author, which satisfy a system of functional relations called ‐system. We conjecture that all cluster monomials are classes of simple objects of . In the final section, we show that these cluster algebras contain infinitely many cluster subalgebras isomorphic to the coordinate ring of the open double Bruhat cell of the corresponding simple simply connected algebraic group. This explains the similarity between ‐system relations and certain generalized minor identities discovered by Fomin and Zelevinsky.

A combinatorial model for q-characters of fundamental modules of type Dn

In this paper, we introduce a combinatorial path model of representation of the quantum affine algebra of type D n , inspired by Mukhin and Young’s combinatorial path models of representations of the quantum affine algebras of types A n and B n . In particular, we give a combinatorial formula for q-characters of fundamental modules of type D n by assigning each path to a monomial or binomial. By counting our paths, a new expression on dimensions of fundamental modules of type D n is obtained.

Lepowsky's and Wakimoto's product formulas for the affine Lie algebras [formula omitted]

In this paper, we recall Lepowsky's and Wakimoto's product character formulas formulated in a new way by using arrays of specialized weighted crystals of negative roots for affine Lie algebras of type Cl(1), Dl+1(2) and A2l(2). Lepowsky–Wakimoto's infinite periodic products appear as one side of (conjectured) Rogers–Ramanujan-type combinatorial identities for affine Lie algebras of type Cl(1).

A new class of simple smooth modules over the affine algebra [formula omitted]

In this paper, we construct a new class of simple smooth modules over the affine algebra A1(1). Specifically, we present a family of sl(2)ˆ-modules Wφ for functions φ which are not Whittaker functions, and provide a simplicity criterion as well as an isomorphism theorem for such modules. For each simple sl(2)ˆ-module Wφ, a class of simple sl(2)˜-module Wφ,γ is also constructed for any constant γ. The sl(2)ˆ-modules Wφ are weight modules, while the sl(2)˜-modules Wφ,γ are not.

On the parallelism between algebraic and analytic Picard groups of algebraic affine hypersurfaces

For any completeC-algebraic variety X and its associated compactC-analytic space X, it follows from the well known GAGA principle that the algebraic Picard group Pic(X) and the analytic Picard group Pic(X) are isomorphic. Our main purpose here is to investigate an analogous situation for non complete situation, namely C-algebraic affine hypersurfaces.

Eigenvalues of quantum Gelfand invariants

We consider the quantum Gelfand invariants which first appeared in a landmark paper by Reshetikhin et al. [Algebra Anal. 1(1), 178–206 (1989)]. We calculate the eigenvalues of the invariants acting in irreducible highest weight representations of the quantized enveloping algebra for gln. The calculation is based on Liouville-type formulas relating two families of central elements in the quantum affine algebras of type A.

Triangular solutions to the reflection equation for Uq(sln^)

We study solutions of the reflection equation related to the quantum affine algebra Uq(sln^) . First, we explain how to construct a family of stochastic integrable vertex models with fixed boundary conditions. Then, we construct upper- and lower-triangular solutions of the reflection equation related to symmetric tensor representations of Uq(sln^) with arbitrary spin. We also prove the star–star relation for the Boltzmann weights of the Ising-type model, conjectured by Bazhanov and Sergeev, and use it to verify certain properties of the solutions obtained.

Schur-Weyl dualities for shifted quantum affine algebras and Ariki-Koike algebras

We establish a Schur-Weyl duality between a shifted quantum affine algebra and an Ariki-Koike algebra. Then, we realize a cyclotomic q-Schur algebra in the context of the Schur-Weyl duality.

THE LEVI DECOMPOSITION OF THE LIE ALGEBRA M_2 (R)⋊gl_2 (R)

The idea of the Lie algebra is studied in this research. The decomposition between Levi sub-algebra and the radical can be used to define the finite dimensional Lie algebra. The Levi decomposition is the name for this type of decomposition. The goal of this study is to obtain a Levi decomposition of the Lie algebra of dimension 8. We compute its Levi sub-algebra and the radical of Lie algebra with respect to its basis to achieve this goal. We use literature studies on the Levi decomposition and Lie algebra in Dagli result to produce the radical and Levi sub-algebra. It has been shown that can be decomposed in the terms of the Levi sub-algebra and its radical. In this resulst, it has been given by direct computations and we obtained that the explicit formula of Levi decomposition of the affine Lie algebra whose basis is is written by with is is the Levi sub-algebra of .

Strong Duality Data of Type A and Extended T-Systems

AbstractThe extended T-systems are a number of relations in the Grothendieck ring of the category of finite-dimensional modules over the quantum affine algebras of types $$A_n^{(1)}$$ A n ( 1 ) and $$B_n^{(1)}$$ B n ( 1 ) , introduced by Mukhin and Young as a generalization of the T-systems. In this paper we establish the extended T-systems for more general modules, which are constructed from an arbitrary strong duality datum of type A. Our approach does not use the theory of q-characters, and so also provides a new proof to the original Mukhin–Young’s extended T-systems.