Generalized Cartan Matrix Research Articles

Geometric Realizations of Lusztig’s Symmetries of Symmetrizable Quantum Groups

Let U be the quantum group and f be the Lusztig’s algebra associated with a symmetrizable generalized Cartan matrix. The algebra f can be viewed as the positive part of U. Lusztig introduced some symmetries T i on U for all i ∈ I. Since T i (f) is not contained in f, Lusztig considered two subalgebras i f and i f of f for any i ∈ I, where i f={x ∈ f | T i (x) ∈ f} and ${^{i}\mathbf {f}}=\{x\in \mathbf {f}\,\,|\,\,T^{-1}_{i}(x)\in \mathbf {f}\}$ . The restriction of T i on i f is also denoted by $T_{i}:{_{i}\mathbf {f}}\rightarrow {^{i}\mathbf {f}}$ . The geometric realization of f and its canonical basis are introduced by Lusztig via some semisimple complexes on the variety consisting of representations of the corresponding quiver. When the generalized Cartan matrix is symmetric, Xiao and Zhao gave geometric realizations of Lusztig’s symmetries in the sense of Lusztig. In this paper, we shall generalize this result and give geometric realizations of i f, i f and $T_{i}:{_{i}\mathbf {f}}\rightarrow {^{i}\mathbf {f}}$ by using the language ’quiver with automorphism’ introduced by Lusztig.

Property (T) for Kac–Moody groups over rings

Let R be a finitely generated commutative ring with 1, let A be an indecomposable 2-spherical generalized Cartan matrix of size at least 2 and M=M(A) the largest absolute value of a non-diagonal entry of A. We prove that there exists an integer n=n(A) such that the Kac–Moody group GA(R) has property (T) whenever R has no proper ideals of index less than n and all positive integers less than or equal to M are invertible in R.

Mixmaster model associated to a Borcherds algebra

The problem of integrability of the mixmaster model as a dynamical system with finite degrees of freedom is studied. The model belongs to the class of pseudo-Euclidean generalized Toda chains. It is presented as a quasi-homogeneous system after transformations of phase variables. Application of the method of getting Kovalevskaya exponents to the model leads to the generalized Adler–van Moerbeke formula for root vectors. A generalized Cartan matrix is constructed using simple root vectors inMinkowski space. The mixmaster model is associated to a Borcherds algebra. The known hyperbolic Kac–Moody algebra of the Chitre´ billiard model is obtained by using three spacelike root vectors.

Dynkin Diagrams and Root Systems of Indefinite Quasi-Hyperbolic Kac-Moody Algebra QHA<sub>4</sub><sup>(2)</sup>

Objectives: To obtain the complete classification of a particular class of indefinite type of quasi hyperbolic Kac-Moody algebra QHA4 (2) and to study the properties of imaginary roots. Methods: Pure theoretical approach for the classification of Dynkin diagrams and an analytical approach for the root system are applied. Findings: The complete classification of Dynkin diagrams associated to the Generalized Cartan Matrix of quasi hyperbolic indefinite type of Kac-Moody algebra QHA4 (2) is obtained. Here, the number of connected, non isomorphic Dynkin diagrams associated with QHA4 (2) is 858. The properties of strictly imaginary and purely imaginary roots are also studied for QHA4 (2). Applications: Kac-Moody algebra has applications in various fields of mathematics and mathematical physics such as combinatorics, number theory, partial differential equations, quantum physics etc.Keywords: Dynkin Diagram, Indefinite Quasi Hyperbolic, Kac-Moody Algebras, Purely Imaginary Roots, Strictly Imaginary Roots

On realization of fusion rings from generalized Cartan matrices

The Casimir element of a fusion ring (R,B) gives rise to the so called Casimir matrix C of (R,B). This enables us to construct a generalized Cartan matrix D − C in the sense of Kac for a suitable diagonal matrix D. In this paper, we study some elementary properties of the Casimir matrix C and use them to realize certain fusion rings from the generalized Cartan matrix D−C of finite (resp. affine) type. It turns out that there exists a fusion ring with D−C being of finite (resp. affine) type if and only if D−C has only the form A 2 (resp. A 1 (1) ). We also realize all fusion rings with D−C being a particular generalized Cartan matrix of indefinite type.

Quasi Affine Generalized Kac Moody Algebras QAGGD<sub>3</sub><sup>(2)</sup>: Dynkin diagrams and root multiplicities for a class of QAGGD<sub>3</sub><sup>(2)</sup>

Objectives: To define the class of quasi affine generalized Kac-Moody algebras QAGGD3 (2), completely classify the non isomorphic, connected Dynkin diagrams associated with QAGGD3 (2) and compute some root multiplicities for this family. Methods: The representation theory of Kac-Moody algebras is applied to compute the multiplicities of roots for a quasi affine family in QAGGD3 (2). Findings: The quasi affine generalized Kac-Moody algebras associated with symmetrizable Generalized Generalized Cartan Matrices (GGCM) of quasi affine type, obtained from the affine family D3 (2), are defined; The connected, non-isomorphic Dynkin diagrams associated with this particular family are completely classified’; Multiplicities of roots of a class GKM algebras QAGGD3 (2), with one simple imaginary root are then determined using the representation theory of Kac Moody algebras; Application: Generalized Kac-Moody algebras find interesting applications in bosonic string theory, classifications in vertex operator theory , monstrous moonshine theory etc.Keywords: Dynkin Diagram, Generalized Kac Moody Algebras (GKM), Imaginary Roots, Quasi Affine, Root Multiplicity

Symmetric bilinear form on a Lie algebra

Let \(\mathfrak {g}\) be the finite dimensional simple Lie algebra associated to an indecomposable and symmetrizable generalized Cartan matrix \(C=(a_{ij})_{n\times n}\) of finite type and let \(\mathfrak {d}\) be a finite dimensional Lie algebra related to a quantum group \(D_{q,p^{-1}}(\mathfrak {g})\) obtained by Hodges et al. (Adv Math 126:52–92, 1997) by deforming the quantum group \(U_q(\mathfrak {g})\). Here we see that \(\mathfrak {d}\) is a generalization of \(\mathfrak {g}\) and give a \(\mathfrak {d}\)-invariant symmetric bilinear form on \(\mathfrak {d}\).

Reflection group relations arising from cluster algebras

There is a well-known analogy between cluster algebras and Kac-moody algebras: roughly speaking, Kac-Moody algebras are associated with symmetrizable generalized Cartan matrices while cluster algebras correspond to skew-symmetrizable matrices. In this paper, we study an interplay between these two classes of matrices. We obtain relations in the Weyl groups of Kac-Moody algebras that come from mutation classes of skew-symmetrizable matrices. More precisely, we establish a set of relations satisfied by the reflections of the so-called companion bases; these include c-vectors, which parametrize coefficients in a cluster algebra with principal coefficients. These relations generalize the relations obtained by Barot and Marsh for finite type. For affine type, we also show that the reflections of the companion bases satisfy the relations obtained by Felikson and Tumarkin. As an application, we obtain some combinatorial properties of the mutation classes of skew-symmetrizable matrices.

Twisted Generalized Weyl Algebras and Primitive Quotients of Enveloping Algebras

To each multiquiver $\Gamma$ we attach a solution to the consistency equations associated to twisted generalized Weyl (TGW) algebras. This generalizes several previously obtained solutions in the literature. We show that the corresponding algebras $\mathcal{A}(\Gamma)$ carry a canonical representation by differential operators and that $\mathcal{A}(\Gamma)$ is universal among all TGW algebras with such a representation. We also find explicit conditions in terms of $\Gamma$ for when this representation is faithful or locally surjective. By forgetting some of the structure of $\Gamma$ one obtains a Dynkin diagram, $D(\Gamma)$. We show that the generalized Cartan matrix of $\mathcal{A}(\Gamma)$ coincides with the one corresponding to $D(\Gamma)$ and that $\mathcal{A}(\Gamma)$ contains graded homomorphic images of the enveloping algebra of the positive and negative part of the corresponding Kac-Moody algebra. Finally, we show that a primitive quotient $U/J$ of the enveloping algebra of a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic zero is graded isomorphic to a TGW algebra if and only if $J$ is the annihilator of a completely pointed (multiplicity-free) simple weight module. The infinite-dimensional primitive quotients in types $A$ and $C$ are closely related to $\mathcal{A}(\Gamma)$ for specific $\Gamma$. We also prove one result in the affine case.

On conjugacy of Cartan subalgebras in extended affine Lie algebras

That finite-dimensional simple Lie algebras over the complex numbers can be classified by means of purely combinatorial and geometric objects such as Coxeter–Dynkin diagrams and indecomposable irreducible root systems, is arguably one of the most elegant results in mathematics. The definition of the root system is done by fixing a Cartan subalgebra of the given Lie algebra. The remarkable fact is that (up to isomorphism) this construction is independent of the choice of the Cartan subalgebra. The modern way of establishing this fact is by showing that all Cartan subalgebras are conjugate.For symmetrizable Kac–Moody Lie algebras, with the appropriate definition of Cartan subalgebra, conjugacy has been established by Peterson and Kac. An immediate consequence of this result is that the root systems and generalized Cartan matrices are invariants of the Kac–Moody Lie algebras. The purpose of this paper is to establish conjugacy of Cartan subalgebras for extended affine Lie algebras; a natural class of Lie algebras that generalizes the finite-dimensional simple Lie algebra and affine Kac–Moody Lie algebras.

A fundamental representation of quantum generalized Kac-Moody algebras with one imaginary simple root

We consider the Borcherds-Cartan matrix obtained from a symmetrizable generalized Cartan matrix by adding one imaginary simple root. We extend the result of Gebert and Teschner (Lett. Math. Phys., 1994, 31: 327-334) to the quantum case. Moreover, we give a connection between the irreducible dominant representations of quantum Kac-Moody algebras and those of quantum generalized Kac-Moody algebras. As the result, a large class of irreducible dominant representations of quantum generalized Kac-Moody algebras were obtained from representations of quantum Kac-Moody algebras through tensor algebras.

Non-symmetrizable Quantum Groups: Defining Ideals and Specialization

Two generating sets of the defining ideal of a Nichols algebra of diagonal type were proposed, which are then applied to study the bar involution and the specialization problem for quantum groups associated to non-symmetrizable generalized Cartan matrices.

Weyl group orbits on Kac–Moody root systems*

Let be a Dynkin diagram and let be the simple roots of the corresponding Kac–Moody root system. Let denote the Cartan subalgebra, let W denote the Weyl group and let Δ denote the set of all roots. The action of W on , and hence on Δ, is the discretization of the action of the Kac–Moody algebra. Understanding the orbit structure of W on Δ is crucial for many physical applications. We show that for , the simple roots αi and αj are in the same W–orbit if and only if vertices i and j in the Dynkin diagram corresponding to αi and αj are connected by a path consisting only of single edges. We introduce the notion of ‘the Cayley graph of the Weyl group action on real roots’ whose connected components are in one-to-one correspondence with the disjoint orbits of W. For a symmetric hyperbolic generalized Cartan matrix A of rank we prove that any two real roots of the same length lie in the same W–orbit. We show that if the generalized Cartan matrix A contains zeros, then there are simple roots that are stabilized by simple root reflections in W, that is, W does not act simply transitively on real roots. We give sufficient conditions in terms of the generalized Cartan matrix A (equivalently ) for W to stabilize a real root. Using symmetry properties of the imaginary light cone in the hyperbolic case, we deduce that the number of W–orbits on imaginary roots on a hyperboloid of fixed radius is bounded above by the number of root lattice points on the hyperboloid that intersect the closure of the fundamental region for W.

Quantizations of Kac–Moody algebras

We analyze the extent to which a quantum universal enveloping algebra of a Kac–Moody algebra g is defined by multidegrees of its defining relations. To this end, we consider a class of character Hopf algebras defined by the same number of defining relations of the same degrees as the Kac–Moody algebra g. We demonstrate that if the generalized Cartan matrix A of g is connected then the algebraic structure, up to a finite number of exceptional cases, is defined by just one “continuous” parameter q related to a symmetrization of A, and one “discrete” parameter m related to the modular symmetrizations of A. The Hopf algebra structure is defined by n(n−1)/2 additional “continuous” parameters. We also consider the exceptional cases for Cartan matrices of finite or affine types in more detail, establishing the number of exceptional parameter values in terms of the Fibonacci sequence.

Root multiplicities of hyperbolic Kac–Moody algebras and Fourier coefficients of modular forms

In this paper we consider the hyperbolic Kac–Moody algebra \(\mathcal {F}\) associated with the generalized Cartan matrix Open image in new window. Its connection to Siegel modular forms of genus 2 was first studied by A. Feingold and I. Frenkel. The denominator function of \(\mathcal{F}\) is not an automorphic form. However, Gritsenko and Nikulin extended \(\mathcal{F}\) to a generalized Kac–Moody algebra whose denominator function is a Siegel modular form. Using the Borcherds denominator identity, the denominator function can be written as an infinite product. The exponents that appear in the product are given by Fourier coefficients of a weak Jacobi form. P. Niemann also constructed a generalized Kac–Moody algebra which contains \(\mathcal {F}\) and whose denominator function is related to a product of Dedekind η-functions. In particular, root multiplicities of the generalized Kac–Moody algebra are determined by Fourier coefficients of a modular form. As the main results of this paper, we compute asymptotic formulas for these Fourier coefficients using the method of Hardy–Ramanujan–Rademacher, and obtain an asymptotic bound for root multiplicities of the algebra \(\mathcal{F}\). Our method can be applied to other hyperbolic Kac–Moody algebras and to other modular forms as demonstrated in the later part of the paper.

Generalized McKay quivers of rank three

For each finite subgroup G of SL(n, C), we introduce the generalized Cartan matrix C_{G} in view of McKay correspondence from the fusion rule of its natural representation. Using group theory, we show that the generalized Cartan matrices have similar favorable properties such as positive semi-definiteness as in the classical case of affine Cartan matrices (the case of SL(2,C)). The complete McKay quivers for SL(3,C) are explicitly described and classified based on representation theory.

On the Lifting of Nichols Algebras

Nichols algebras are a fundamental building block of pointed Hopf algebras. Part of the classification program of finite-dimensional pointed Hopf algebras with the lifting method of Andruskiewitsch and Schneider is the determination of the liftings, i.e., all possible deformations of a given Nichols algebra. Based on recent work of Heckenberger about Nichols algebras of diagonal type we compute explicitly the liftings of all Nichols algebras with generalized Cartan matrix of type A 2, some Nichols algebras with generalized Cartan matrix of type B 2, and some Nichols algebras of two Weyl equivalence classes of non-standard type giving new classes of finite-dimensional pointed Hopf algebras.

Symmetrizable Intersection Matrix Lie Algebras

Associated to every symmetrizable generalized intersection matrix A, we define a Lie algebra, called an SIM-Lie algebra. We prove that SIM-Lie algebras keep unchange under braid-equivalences. Two special cases are considered. In the case when A is a symmetrizable generalized Cartan matrix, we show that the corresponding SIM-Lie algebra is just the Kac–Moody Lie algebra. In another case when A is an intersection matrix, we prove that the corresponding SIM-Lie algebra is just the intersection matrix Lie algebra in the sense of Slodowy.

INFINITE DESCENDING CHAINS OF COCOMPACT LATTICES IN KAC–MOODY GROUPS

Let A be a symmetrizable affine or hyperbolic generalized Cartan matrix. Let G be a locally compact Kac–Moody group associated to A over a finite field 𝔽q. We suppose that G has type ∞, that is, the Weyl group W of G is a free product of ℤ/2ℤ's. This includes all locally compact Kac–Moody groups of rank 2 and three possible locally compact rank 3 Kac–Moody groups of noncompact hyperbolic type. For every prime power q, we give a sufficient condition for the rank 2 Kac–Moody group G to contain a cocompact lattice [Formula: see text] with quotient a simplex, and we show that this condition is satisfied when q = 2s. If further Mqand [Formula: see text] are abelian, we give a method for constructing an infinite descending chain of cocompact lattices … Γ3≤ Γ2≤ Γ1≤ Γ. This allows us to characterize each of the quotient graphs of groups Γi\\X, the presentations of the Γiand their covolumes, where X is the Tits building of G, a homogeneous tree. Our approach is to extend coverings of edge-indexed graphs to covering morphisms of graphs of groups with abelian groupings. This method is not specific to cocompact lattices in Kac–Moody groups and may be used to produce chains of subgroups acting on trees in a general setting. It follows that the lattices constructed in the rank 2 Kac–Moody group have the Haagerup property. When q = 2 and rank (G) = 3 we show that G contains a cocompact lattice Γ′1that acts discretely and cocompactly on a simplicial tree [Formula: see text]. The tree [Formula: see text] is naturally embedded in the Tits building X of G, a rank 3 hyperbolic building. Moreover Γ′1≤ Λ′ for a non-discrete subgroup Λ′ ≤ G whose quotient Λ′ \ X is equal to G\X. Using the action of Γ′1on [Formula: see text] we construct an infinite descending chain of cocompact lattices …Γ′3≤ Γ′2≤ Γ′1in G. We also determine the quotient graphs of groups [Formula: see text], the presentations of the Γ′iand their covolumes.

Twisted Generalized Weyl Algebras, Polynomial Cartan Matrices and Serre-Type Relations

A twisted generalized Weyl algebra (TGWA) is defined as the quotient of a certain graded algebra by the maximal graded ideal I with trivial zero component, analogous to how Kac–Moody algebras can be defined. In this article we introduce the class of locally finite TGWAs and show that one can associate to such an algebra a polynomial Cartan matrix (a notion extending the usual generalized Cartan matrices appearing in Kac–Moody algebra theory) and that the corresponding generalized Serre relations hold in the TGWA. We also give an explicit construction of a family of locally finite TGWAs depending on a symmetric generalized Cartan matrix C and some scalars. The polynomial Cartan matrix of an algebra in this family may be regarded as a deformation of the original matrix C and gives rise to quantum Serre relations in the TGWA. We conjecture that these relations generate the graded ideal I for these algebras, and prove it in type A 2.