Dynkin Diagrams Of Type Research Articles

Hirota Quadratic Equations for the Gromov–Witten invariants of [formula omitted

Fano orbifold lines are classified by the Dynkin diagrams of type A,D, and E. It is known that the corresponding total descendant potential is a tau-function of an appropriate Kac–Wakimoto hierarchy. It is also known that in the A-case the Kac–Wakimoto hierarchies admit an extension and that the total descendant potential is a tau-function of an extended Kac–Wakimoto hierarchy. The goal of this paper is to prove that in the D-case the total descendent potential is also a tau-function of an extended Kac–Wakimoto hierarchy.

Tetrahedron and 3D Reflection Equation from PBW Bases of the Nilpotent Subalgebra of Quantum Superalgebras

In this paper, we study transition matrices of PBW bases of the nilpotent subalgebra of quantum superalgebras associated with all possible Dynkin diagrams of type A and B in the case of rank 2 and 3, and examine relationships with three-dimensional (3D) integrability. We obtain new solutions to the Zamolodchikov tetrahedron equation via type A and the 3D reflection equation via type B, where the latter equation was proposed by Isaev and Kulish as a 3D analog of the reflection equation of Cherednik. As a by-product of our approach, the Bazhanov–Sergeev solution to the Zamolodchikov tetrahedron equation is characterized as the transition matrix for a particular case of type A, which clarifies an algebraic origin of it. Our work is inspired by the recent developments connecting transition matrices for quantum non-super algebras with intertwiners of irreducible representations of quantum coordinate rings. We also discuss the crystal limit of transition matrices, which gives a super analog of transition maps of Lusztig’s parametrizations of the canonical basis.

New large-rank Nichols algebras over nonabelian groups with commutator subgroup [formula omitted

In this article, we explicitly construct new finite-dimensional, indecomposable Nichols algebras with Dynkin diagrams of type An,Cn,Dn,E6,7,8,F4 over any group G of order 2ℓ with commutator subgroup isomorphic to Z2. The construction is generic in the sense that the type just depends on the rank and center of G, and thus positively answers for all groups of this class a question raised by Susan Montgomery in 1995 [22,4].Our construction uses the new notion of a covering Nichols algebra as a special case of a covering Hopf algebra [18] and produces non-faithful Nichols algebras. We give faithful examples of Doi twists for type A3,C3,D4,C4,F4 over several nonabelian groups of order 16 and 32. These are hence the first known examples of nondiagonal, finite-dimensional, indecomposable Nichols algebras of rank >2 over nonabelian groups.

EXT ALGEBRA OF NICHOLS ALGEBRAS OF CARTAN TYPE A2

We give the full structure of the Ext algebra of any Nichols algebra of Cartan type A2by using the Hochschild–Serre spectral sequence. As an application, we show that the pointed Hopf algebras [Formula: see text] with Dynkin diagrams of type A, D, or E, except for A1and A1× A1with the order NJ> 2 for at least one component J, are wild.

-constraints for the total descendant potential of a simple singularity

AbstractSimple, or Kleinian, singularities are classified by Dynkin diagrams of type $ADE$. Let $\mathfrak {g}$ be the corresponding finite-dimensional Lie algebra, and $W$ its Weyl group. The set of $\mathfrak {g}$-invariants in the basic representation of the affine Kac–Moody algebra $\hat {\mathfrak {g}}$ is known as a $\mathcal {W}$-algebra and is a subalgebra of the Heisenberg vertex algebra $\mathcal {F}$. Using period integrals, we construct an analytic continuation of the twisted representation of $\mathcal {F}$. Our construction yields a global object, which may be called a $W$-twisted representation of $\mathcal {F}$. Our main result is that the total descendant potential of the singularity, introduced by Givental, is a highest-weight vector for the $\mathcal {W}$-algebra.

Minuscule Heaps over Dynkin Diagrams of Type $\tilde{A}$

A minuscule heap is a partially ordered set, together with a labeling of its elements by the nodes of a Dynkin diagram, satisfying certain conditions derived by J. Stembridge. This paper classifies the minuscule heaps over the Dynkin diagram of type $\tilde{A}$.

On singular fibres of Lagrangian fibrations over holomorphic symplectic manifolds

We classify singular fibres over general points of the discriminant locus of projective Lagrangian fibrations over 4-dimensional holomorphic symplectic manifolds. The singular fibre F is the following either one: F is isomorphic to the product of an elliptic curve and a Kodaira singular fibre up to finite unramified covering or F is a normal crossing variety consisting of several copies of a minimal elliptic ruled surface of which the dual graph is Dynkin diagram of type $A_n, \tilde{A_n}$ or $\tilde{D_n}$ . Moreover, we show all types of the above singular fibres actually occur.

On Hyper-Kähler Manifolds of Type A ∞ and D ∞

We shall use an infinite dimensional hyper-Kahler quotient method to obtain hyper-Kahler 4 manifolds of type A ∞ and D ∞. Hyper-Kahler manifolds of type A ∞ and D ∞ are constructed in terms of Dynkin diagrams of type A ∞ and D ∞ respectively. A hyper-Kahler manifold of type D ∞ is the minimal resolution of the quotient space of a hyper-Kahler manifold of type A ∞ by an involution. Finally we shall show that a hyper-Kahler manifold of type A ∞ can be considered as the universal cover of elliptic fibre space of type I b .

Nahm’s conjecture and $Y$-systems

Nahm's conjecture relates $q$-hypergeometric modular functions to torsion elements in the Bloch group. An interesting class of such functions can be (conjecturally) obtained from a pair $(X,X')$ of diagrams, each of which is either a Dynkin diagram of type $ADE$ or a diagram of type $T$. Using properties of Y-systems, we prove that for a matrix of the form $A=\mathcal{C}(X)\otimes \mathcal{C}(X')^{-1}$ where $\mathcal{C}(X)$ and $\mathcal{C}(X')$ are the corresponding Cartan matrices, every solution of the equation $\mathbf{x}=(1-\mathbf{x})^A$ gives rise to a torsion element of the Bloch group.