Invariant elements in a modular lattice associated with an extended dynkin diagram $$\tilde E_6$$

1992 ◽  
Vol 26 (1) ◽  
pp. 33-42
Author(s):  
R. B. Stekol'shchik
Keyword(s):  
Modular Lattice ◽  
Dynkin Diagram ◽  
Extended Dynkin Diagram

scholarly journals RECURRENCE RELATIONS, DYNKIN DIAGRAMS AND PLÜCKER FORMULAE

2007 ◽  
Vol 49 (1) ◽  
pp. 53-59 ◽  
Author(s):  
JOHN M. BURNS ◽  
MICHAEL J. CLANCY
Keyword(s):  
Lie Algebra ◽  
Finite Groups ◽  
Dynkin Diagram ◽  
Recurrence Relations ◽  
Riemann Sphere ◽  
Simple Lie Algebra ◽  
Dynkin Diagrams ◽  
Extended Dynkin Diagram

Abstract.We prove a generalisation of an observation of N. Iwahori concerning the coefficients of the extended Dynkin diagram of a complex simple Lie algebra. We relate the combinatorics of these coefficients to the orders of finite groups that act discontinuously on the Riemann sphere and to the Plücker formulae.

scholarly journals Infinite dimensional representations of

1990 ◽  
Vol 32 (1) ◽  
pp. 25-33 ◽  
Author(s):  
A. Dean ◽  
F. Zorzitto
Keyword(s):  
Dynkin Diagram ◽  
Finite Dimension ◽  
Vector Spaces ◽  
Closed Field ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Linear Maps ◽  
Image Position ◽  
Subspace Problem ◽  
Extended Dynkin Diagram

By a representation of the extended Dynkin diagram we shall mean a list of 5 vector spaces P, E1, E2, E3, E4 over an algebraically closed field K, and 4 linear maps a1, a2, a3, a4 as shown.The spaces need not be of finite dimension.In their solution of the 4-subspace problem [6], Gelfand and Ponomarev have classified such representations when the spaces are finite dimensional. A representation like (1) can also be viewed as a module over the K-algebra R4 consisting of all 5 × 5 matrices having zeros off the first row and off the main diagonal.

scholarly journals On the complete integrability of the periodic quantum Toda lattice

2017 ◽  
Vol 29 (6) ◽  
Author(s):  
Augustin-Liviu Mare
Keyword(s):  
Dynkin Diagram ◽  
Toda Lattice ◽  
Complete Integrability ◽  
Completely Integrable ◽  
Quantum Toda Lattice ◽  
Extended Dynkin Diagram

AbstractWe prove that the periodic quantum Toda lattice corresponding to any extended Dynkin diagram is completely integrable. This has been conjectured and proved in all classical cases and

BRST EXACTNESS OF STRESS-ENERGY TENSORS

1994 ◽  
Vol 09 (12) ◽  
pp. 2033-2065 ◽  
Author(s):  
HIDEO MIYATA ◽  
HIROSHI SUGIMOTO
Keyword(s):  
Lie Algebras ◽  
Simple Root ◽  
Dynkin Diagram ◽  
Field Theories ◽  
General Level ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Stress Energy ◽  
Multiple Integrals ◽  
Extended Dynkin Diagram

BRST commutators in the topological conformal field theories obtained by twisting N=2 theories are evaluated explicitly. By our systematic calculations of the multiple integrals which contain screening operators, the BRST exactness of the twisted stress-energy tensors is deduced for classical simple Lie algebras and general level k. We can see that the paths of integrations do not affect the result, and further, the N=2 coset theories are obtained by deleting two simple roots with Kac-label 1 from the extended Dynkin diagram; in other words, by not performing the integrations over the variables corresponding to the two simple roots of Kac-Moody algebras. It is also shown that a series of N=1 theories are generated in the same way by deleting one simple root with Kac-label 2.

Perfect elements in the modular structure associated with the extended Dynkin diagram E6

1990 ◽  
Vol 23 (3) ◽  
pp. 251-254
Author(s):  
R. B. Stekol'shchik
Keyword(s):  
Dynkin Diagram ◽  
Modular Structure ◽  
Extended Dynkin Diagram

scholarly journals Tensor Product Structure of Affine Demazure Modules and Limit Constructions

2006 ◽  
Vol 182 ◽  
pp. 171-198 ◽  
Author(s):  
G. Fourier ◽  
P. Littelmann
Keyword(s):  
Tensor Product ◽  
Dynkin Diagram ◽  
Classical Case ◽  
Natural Generalization ◽  
Classical Type ◽  
Module Structure ◽  
Product Decomposition ◽  
Finite Dimensional ◽  
Demazure Modules ◽  
Extended Dynkin Diagram

AbstractLet g be a simple complex Lie algebra, we denote by ĝ the affine Kac-Moody algebra associated to the extended Dynkin diagram of g. Let Λ0 be the fundamental weight of ĝ corresponding to the additional node of the extended Dynkin diagram. For a dominant integral g-coweight λ∨, the Demazure submodule V_λ∨ (mΛ0) is a g-module. We provide a description of the g-module structure as a tensor product of “smaller” Demazure modules. More precisely, for any partition of λ∨ = λ∑j as a sum of dominant integral g-coweights, the Demazure module is (as g-module) isomorphic to ⊗jV_ (mΛ0). For the “smallest” case, λ∨ = ω∨ a fundamental coweight, we provide for g of classical type a decomposition of V_ω∨(mΛ0) into irreducible g-modules, so this can be viewed as a natural generalization of the decomposition formulas in [13] and [16]. A comparison with the Uq(g)-characters of certain finite dimensional -modules (Kirillov-Reshetikhin-modules) suggests furthermore that all quantized Demazure modules V_λ∨,q(mΛ0) can be naturally endowed with the structure of a -module. We prove, in the classical case (and for a lot of non-classical cases), a conjecture by Kashiwara [10], that the “smallest” Demazure modules are, when viewed as g-modules, isomorphic to some KR-modules. For an integral dominant ĝ-weight Λ let V(Λ) be the corresponding irreducible ĝ-representation. Using the tensor product decomposition for Demazure modules, we give a description of the g-module structure of V(Λ) as a semi-infinite tensor product of finite dimensional g-modules. The case of twisted affine Kac-Moody algebras can be treated in the same way, some details are worked out in the last section.

scholarly journals Exploring the landscape of CHL strings on Td

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Anamaría Font ◽  
Bernardo Fraiman ◽  
Mariana Graña ◽  
Carmen A. Núñez ◽  
Héctor Parra De Freitas
Keyword(s):  
Gauge Group ◽  
Moduli Space ◽  
Dynkin Diagram ◽  
Fundamental Groups ◽  
Computer Algorithms ◽  
Abelian Gauge ◽  
Reduced Rank ◽  
Systematic Exploration ◽  
String Models ◽  
Simply Connected

Abstract Compactifications of the heterotic string on special Td/ℤ2 orbifolds realize a landscape of string models with 16 supercharges and a gauge group on the left-moving sector of reduced rank d + 8. The momenta of untwisted and twisted states span a lattice known as the Mikhailov lattice II(d), which is not self-dual for d > 1. By using computer algorithms which exploit the properties of lattice embeddings, we perform a systematic exploration of the moduli space for d ≤ 2, and give a list of maximally enhanced points where the U(1)d+8 enhances to a rank d + 8 non-Abelian gauge group. For d = 1, these groups are simply-laced and simply-connected, and in fact can be obtained from the Dynkin diagram of E10. For d = 2 there are also symplectic and doubly-connected groups. For the latter we find the precise form of their fundamental groups from embeddings of lattices into the dual of II(2). Our results easily generalize to d > 2.

scholarly journals The free modular lattice, ${\rm FM}(2+2+2)$, is infinite

1966 ◽  
Vol 17 (4) ◽  
pp. 960-960
Author(s):  
Howard L. Rolf
Keyword(s):  
Modular Lattice ◽  
Free Modular Lattice

scholarly journals On Generalized Minors and Quiver Representations

2018 ◽  
Vol 2020 (3) ◽  
pp. 914-956 ◽  
Author(s):  
Dylan Rupel ◽  
Salvatore Stella ◽  
Harold Williams
Keyword(s):  
Finite Type ◽  
Dynkin Diagram ◽  
Cluster Algebra ◽  
Coordinate Ring ◽  
Coxeter Element ◽  
Quiver Representations ◽  
Group Representations ◽  
Weight Group ◽  
Highest Weight ◽  
Bruhat Cell

Abstract The cluster algebra of any acyclic quiver can be realized as the coordinate ring of a subvariety of a Kac–Moody group—the quiver is an orientation of its Dynkin diagram, defining a Coxeter element and thereby a double Bruhat cell. We use this realization to connect representations of the quiver with those of the group. We show that cluster variables of preprojective (resp. postinjective) quiver representations are realized by generalized minors of highest-weight (resp. lowest-weight) group representations, generalizing results of Yang–Zelevinsky in finite type. In type $A_{n}^{\!(1)}$ and finitely many other affine types, we show that cluster variables of regular quiver representations are realized by generalized minors of group representations that are neither highest- nor lowest-weight; we conjecture this holds more generally.

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