Showing papers on "Cartan matrix published in 2012"

Quivers of monoids with basic algebras

Abstract: We compute the quiver of any nite monoid that has a basic algebra over an algebraically closed eld of characteristic zero. More generally, we reduce the computation of the quiver over a splitting eld of a class of monoids that we term rectangular monoids (in the semigroup theory literature the class is known as DO) to representation-theoretic computations for group algebras of maximal subgroups. Hence in good characteristic for the maximal subgroups, this gives an essentially complete computation. Since groups are examples of rectangular monoids, we cannot hope to do better than this. For the subclass of R-trivial monoids, we also provide a semigroup-theoretic description of the projective indecomposable modules and compute the Cartan matrix.

Weighted Khovanov-Lauda-Rouquier algebras

Abstract: In this paper, we define a generalization of Khovanov-Lauda-Rouquier algebras which we call weighted Khovanov-Lauda-Rouquier algebras. We show that these algebras carry many of the same structures as the original Khovanov-Lauda-Rouquier algebras, including induction and restriction functors which induce a twisted biaglebra structure on their Grothendieck groups. We also define natural quotients of these algebras, which in an important special case carry a categorical action of an associated Lie algebra. Special cases of these include the algebras categorifying tensor products and Fock spaces defined by the author and Stroppel in past work. For symmetric Cartan matrices, weighted KLR algebras also have a natural gometric interpretation as convolution algebras, generalizing that for the original KLR algebras by Varagnolo and Vasserot; this result has positivity consequences important in the theory of crystal bases. In this case, we can also relate the Grothendieck group and its bialgebra structure to the Hall algebra of the associated quiver.

E 11, Borcherds algebras and maximal supergravity

Abstract: The dynamical p-forms of torus reductions of maximal supergravity theory have been shown some time ago to possess remarkable algebraic structures. The set (“dynamical spectrum”) of propagating p-forms has been described as a (truncation of a) real Borcherds superalgebra D that is characterized concisely by a Cartan matrix which has been constructed explicitly for each spacetime dimension 11 ≥ D ≥ 3. In the equations of motion, each differential form of degree p is the coefficient of a (super-) group generator, which is itself of degree p for a specific gradation (the -gradation). A slightly milder truncation of the Borcherds superalgebra enables one to predict also the “spectrum” of the non-dynamical (D − 1) and D-forms. The maximal supergravity p-form spectra were reanalyzed more recently by truncation of the field spectrum of E 11 to the p-forms that are relevant after reduction from 11 to D dimensions. We show in this paper how the Borcherds description can be systematically derived from the split (“maximally non compact”) real form of E 11 for D ≥ 1. This explains not only why both structures lead to the same propagating p-forms and their duals for p ≤ (D − 2), but also why one obtains the same (D−1)-forms and “top” D-forms. The Borcherds symmetries 2 and 1 are new too. We also introduce and use the concept of a presentation of a Lie algebra that is covariant under a given subalgebra.

Amalgamated free product type III factors with at most one Cartan subalgebra

Abstract: We investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras $M_1 \ast_B M_2$ over an amenable von Neumann subalgebra $B$. First, we settle the problem of the absence of Cartan subalgebra in arbitrary free product von Neumann algebras. Namely, we show that any nonamenable free product von Neumann algebra $(M_1, \varphi_1) \ast (M_2, \varphi_2)$ with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established in \cite{Io12a}. Next, we prove that any countable nonsingular ergodic equivalence relation $\mathcal R$ defined on a standard measure space and which splits as the free product $\mathcal R = \mathcal R_1 \ast \mathcal R_2$ of recurrent subequivalence relations gives rise to a nonamenable factor $\rL(\mathcal R)$ with a unique Cartan subalgebra, up to unitary conjugacy. Finally, we prove unique Cartan decomposition for a class of group measure space factors $\rL^\infty(X) \rtimes \Gamma$ arising from nonsingular free ergodic actions $\Gamma \curvearrowright (X, \mu)$ on standard measure spaces of amalgamated groups $\Gamma = \Gamma_1 \ast_{\Sigma} \Gamma_2$ over a finite subgroup $\Sigma$.

Explicit expression of Cartan’s connection for Levi-nondegenerate 3-manifolds in complex surfaces, and identification of the Heisenberg sphere

Abstract: We study effectively the Cartan geometry of Levi-nondegenerate C6-smooth hypersurfaces M3 in ℂ2. Notably, we present the so-called curvature function of a related Tanaka-type normal connection explicitly in terms of a graphing function for M, which is the initial, single available datum. Vanishing of this curvature function then characterizes explicitly the local biholomorphic equivalence of such M3 ⊂ ℂ2 to the Heisenberg sphere ℍ3, such M’s being necessarily real analytic.

Affine and Finite Lie Algebras and Integrable Toda Field Equations on Discrete Space-Time ?

Abstract: Difference-difference systems are suggested corresponding to the Cartan matri- ces of any simple or affine Lie algebra. In the cases of the algebras AN , BN , CN , G2, D3, A (1) , A (2) , D (2) these systems are proved to be integrable. For the systems corresponding to the algebras A2, A (1) , A (2) generalized symmetries are found. For the systems A2, B2, C2, G2, D3 complete sets of independent integrals are found. The Lax representation for the difference-difference systems corresponding to AN , BN , CN , A (1) , D (2) N are presented.

Lie algebroids and Cartan’s method of equivalence

Abstract: Elie Cartan's general equivalence problem is recast in the language of Lie algebroids. The resulting formalism, being coordinate and model-free, allows for a full geometric interpretation of Cartan's method of equivalence via reduction and prolongation. We show how to construct certain normal forms (Cartan algebroids) for objects of finite-type, and are able to interpret these directly as infinitesimal symmetries deformed by curvature.' Details are developed for transitive structures but rudiments of the theory include intransitive structures (intransitive symmetry deformations). Detailed illustrations include subriemannian contact structures and conformal geometry.

The Cardy-Cartan modular invariant

Abstract: Using factorizable Hopf algebras, we construct modular invariant partition functions of charge conjugation, or Cardy, type as characters of coends in categories that share essential features with the ones appearing in logarithmic CFT. The coefficients of such a partition function are given by the Cartan matrix of the theory.

On the Lifting of Nichols Algebras

Abstract: 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.

Bimodules over Cartan MASAs in von Neumann Algebras, Norming Algebras, and Mercer's Theorem

Abstract: In a 1991 paper, R. Mercer asserted that a Cartan bimodule isomorphism between Cartan bimodule algebras A_1 and A_2 extends uniquely to a normal *-isomorphism of the von Neumann algebras generated by A_1 and A_2 [13, Corollary 4.3]. Mercer's argument relied upon the Spectral Theorem for Bimodules of Muhly, Saito and Solel [15, Theorem 2.5]. Unfortunately, the arguments in the literature supporting [15, Theorem 2.5] contain gaps, and hence Mercer's proof is incomplete. In this paper, we use the outline in [16, Remark 2.17] to give a proof of Mercer's Theorem under the additional hypothesis that the given Cartan bimodule isomorphism is weak-* continuous. Unlike the arguments contained in [13, 15], we avoid the use of the Feldman-Moore machinery from [8]; as a consequence, our proof does not require the von Neumann algebras generated by the algebras A_i to have separable preduals. This point of view also yields some insights on the von Neumann subalgebras of a Cartan pair (M,D), for instance, a strengthening of a result of Aoi [1]. We also examine the relationship between various topologies on a von Neumann algebra M with a Cartan MASA D. This provides the necessary tools to parametrize the family of Bures-closed bimodules over a Cartan MASA in terms of projections in a certain abelian von Neumann algebra; this result may be viewed as a weaker form of the Spectral Theorem for Bimodules, and is a key ingredient in the proof of our version of Mercer's theorem. Our results lead to a notion of spectral synthesis for weak-* closed bimodules appropriate to our context, and we show that any von Neumann subalgebra of M which contains D is synthetic. We observe that a result of Sinclair and Smith shows that any Cartan MASA in a von Neumann algebra is norming in the sense of Pop, Sinclair and Smith.

Cartan-Lepage forms in geometric mechanics

Abstract: We present variational principles for mechanics on the basis of the Cartan 1-form. We extend the theory to the Cartan–Lepage 2-forms related with dynamical forms and equations of motion. Explicit coordinate formulae for Lepage equivalents of dynamical forms are given.

Consequences of the Littelmann Path Theory for the Structure of the Kashiwara B(∞) Crystal

Abstract: These lectures give a review of results on the Kashiwara B(∞) crystal defined for a Kac–Moody Lie algebra, which can be obtained using the Littelmann path model. In this we do not need to assume, as does Kashiwara, that the Cartan matrix is symmetrizable.

Bi-null Cartan curves in semi-Euclidean spaces of index 2

Abstract: We give the notion of bi-null Cartan curves in semi-Euclidean spaces of index 2, together with the unique Frenet frame and the Cartan curvatures. We also discuss some properties of bi-null Cartan curves in terms of the Cartan curvatures.

Notes on parameters of quiver Hecke algebras

Abstract: Varagnolo-Vasserot and Rouquier proved that, in a symmetric generalized Cartan matrix case, the simple modules over the quiver Hecke algebra with a special parameter correspond to the upper global basis. In this note we show that the simple modules over the quiver Hecke algebras with a generic parameter also correspond to the upper global basis in a symmetric generalized Cartan matrix case.

Structure and Representation Theory of Kac–Moody Superalgebras

Abstract: The aim of these lecture notes is to give an introduction to the structure and representation theory of Lie superalgebras.

On Fundamental Domains and Volumes of Hyperbolic Coxeter–Weyl Groups

Abstract: We present a simple method for determining the shape of fundamental domains of generalized modular groups related to Weyl groups of hyperbolic Kac-Moody algebras. These domains are given as subsets of certain generalized upper half planes, on which the Weyl groups act via generalized modular transformations. Our construction only requires the Cartan matrix of the underlying finite-dimensional Lie algebra and the associated Coxeter labels as input information. We present a simple formula for determining the vol- ume of these fundamental domains. This allows us to re-produce in a simple manner the known values for these volumes previously obtained by other methods.

Counting fine gradings on matrix algebras and on classical simple Lie algebras

Abstract: Known classification results allow us to find the number of (equivalence classes of) fine gradings on matrix algebras and on classical simple Lie algebras over an algebraically closed field $\mathbb{F}$ (assuming $\mathrm{char} \mathbb{F} e 2$ in the Lie case). The computation is easy for matrix algebras and especially for simple Lie algebras of type $B_r$ (the answer is just $r+1$), but involves counting orbits of certain finite groups in the case of Series $A$, $C$ and $D$. For $X\in\{A,C,D\}$, we determine the exact number of fine gradings, $N_X(r)$, on the simple Lie algebras of type $X_r$ with $r\le 100$ as well as the asymptotic behaviour of the average, $\hat N_X(r)$, for large $r$. In particular, we prove that there exist positive constants $b$ and $c$ such that $\exp(br^{2/3})\le\hat N_X(r)\le\exp(cr^{2/3})$. The analogous average for matrix algebras $M_n(\mathbb{F})$ is proved to be $a\ln n+O(1)$ where $a$ is an explicit constant depending on $\mathrm{char} \mathbb{F}$.

Cartan invariant matrices for finite monoids

Abstract: Let $M$ be a finite monoid. In this paper we describe how the Cartan invariant matrix of the monoid algebra of $M$ over a field $\mathbb{K}$ of characteristic zero can be expressed using characters and some simple combinatorial statistic. In particular, it can be computed efficiently from the composition factors of the left and right class modules of $M$. When $M$ is aperiodic, this approach works in any characteristic, and generalizes to $\mathbb{K}$ a principal ideal domain like $\mathbb{Z}$. When $M$ is $\mathcal{R}$-trivial, we retrieve the formerly known purely combinatorial description of the Cartan matrix.

Generalized McKay quivers of rank three

Abstract: 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.

Lie Groups and Lie Algebras - A Physicist's Perspective

Abstract: 1 Generalities 2 Lie Groups and Lie Algebras 3 Rotations: SO(3) an SU(2) 4 Representations of SU(2) 5 The so(n) Algebra and Clifford Numbers 6 Reality Properties of Spinors 7 Clebsch-Gordan Series for Spinors 8 The Center and Outer Automorphisms of Spin(n) 9 Composition Algebras 10 The Exceptional Group G2 11 Casimir Operators for Orthogonal Groups 12 Classical Groups 13 Unitary Groups 14 The Symmetric Group Sr and Young Tableaux 15 Reduction of SU(n) Tensors 16 Cartan Basis, Simple Roots and Fundamental Weights 17 Cartan Classification of Semisimple Algebras 18 Dynkin Diagrams 19 The Lorentz Group 20 The Poincare and Liouville Groups 21 The Coulomb Problem in n Space Dimensions

Mutation Classes of 3×3 Generalized Cartan Matrices

Abstract: One of the recent developments in representation theory has been the introduction of cluster algebras by Fomin and Zelevinsky. It is now well known that these algebras are closely related with different areas of mathematics. A particular analogy exists between combinatorial aspects of cluster algebras and Kac–Moody algebras: roughly speaking, cluster algebras are associated with skew-symmetrizable matrices, while Kac–Moody algebras correspond to (symmetrizable) generalized Cartan matrices. In this paper, we describe an interplay between these two classes of matrices in size 3. In particular, we give a characterization of the mutation classes associated with the generalized Cartan matrices of size 3, generalizing results of Beineke–Bruestle–Hille.

Highlights in Lie algebraic methods

Abstract: Preface.- Part I: The Courses.- 1 Spherical Varieties.- 2 Consequences of the Littelmann Path Model for the Structure of the Kashiwara B(infinity) Crystal.- 3 Structure and Representation Theory of Kac-Moody Superalgebras.- 4 Categories of Harish-Chandra Modules.- 5 Generalized Harish-Chandra Modules.- Part II: The Papers.- 6 B-Orbits of 2-Nilpotent Matrices.- 7 The Weyl Denominator Identity for Finite-Dimensional Lie Superalgebras.- 8 Hopf Algebras and Frobenius Algebras in Finite Tensor Categories.- 9 Mutation Classes of 3 x 3 Generalized Cartan Matrices.- 10 Contractions and Polynomial Lie Algebras.

Discovering Real Lie Subalgebras of e6 using Cartan Decompositions

Abstract: The process of complexification is used to classify a Lie algebra and identify its Cartan subalgebra. However, this method does not distinguish between real forms of a complex Lie algebra, which can differ in signature. In this paper, we show how Cartan decompositions of a complexified Lie algebra can be combined with information from the Killing form to identify real forms of a given Lie algebra. We apply this technique to sl(3,O), a real form of e6 with signature (52,26), thereby identifying chains of real subalgebras and their corresponding Cartan subalgebras within e6. Motivated by an explicit construction of sl(3,O), we then construct an abelian group of order 8 which acts on the real forms of e6, leading to the identification of 8 particular copies of the 5 real forms of e6, which can be distinguished by their relationship to the original copy of sl(3,O).

Notes on parameters of quiver Hecke algebras

Abstract: Varagnolo-Vasserot and Rouquier proved that, in a symmetric generalized Cartan matrix case, the simple modules over the quiver Hecke algebra with a special parameter correspond to the upper global basis. In this note we show that the simple modules over the quiver Hecke algebras with a generic parameter also correspond to the upper global basis in a symmetric generalized Cartan matrix case.

Nottingham Lie algebras with diamonds of finite and infinite type

Abstract: We consider a class of infinite-dimensional, modular, graded Lie algebras, which includes the graded Lie algebra associated to the Nottingham group with respect to its lower central series. We identify two subclasses of Nottingham Lie algebras as loop algebras of finite-dimensional simple Lie algebras of Hamiltonian Cartan type. A property of Laguerre polynomials of derivations, which is related to toral switching, plays a crucial role in our constructions.

II$_1$ factors and equivalence relations with distinct fundamental groups

Abstract: We construct a group measure space II$_1$ factor that has two non-conjugate Cartan subalgebras. We show that the fundamental group of the II$_1$ factor is trivial, while the fundamental group of the equivalence relation associated with the second Cartan subalgebra is non-trivial. This is not absurd as the second Cartan inclusion is twisted by a 2-cocycle.

Dirac and Generalized Complex Structures on Cartan Geometries

Abstract: We introduce linear Dirac and generalized complex structures on Cartan geometries and give criteria for Dirac subalgebras of $\frkg\ltimes\frkg^*$ representing Dirac structures on a Cartan geometry We prove that there is a bijection between the linear generalized structures on a torsion free Cartan geometry and the equivariant generalized structures on its corresponding homogeneous space