Showing papers on "Cartan matrix published in 2010"

MacDowell-Mansouri gravity and Cartan geometry

Abstract: The geometric content of the MacDowell–Mansouri formulation of general relativity is best understood in terms of Cartan geometry. In particular, Cartan geometry gives clear geometric meaning to the MacDowell–Mansouri trick of combining the Levi-Civita connection and coframe field, or soldering form, into a single physical field. The Cartan perspective allows us to view physical spacetime as tangentially approximated by an arbitrary homogeneous 'model spacetime', including not only the flat Minkowski model, as is implicitly used in standard general relativity, but also de Sitter, anti-de Sitter or other models. A 'Cartan connection' gives a prescription for parallel transport from one 'tangent model spacetime' to another, along any path, giving a natural interpretation of the MacDowell–Mansouri connection as 'rolling' the model spacetime along physical spacetime. I explain Cartan geometry, and 'Cartan gauge theory', in which the gauge field is replaced by a Cartan connection. In particular, I discuss MacDowell–Mansouri gravity, as well as its more recent reformulation in terms of BF theory, in the context of Cartan connections.

On a class of II1 factors with at most one Cartan subalgebra, II

Abstract: This is a continuation of our previous paper studying the structure of Cartan subalgebras of von Neumann factors of type ${\rm II}_1$. We provide more examples of ${\rm II}_1$ factors having either zero, one, or several Cartan subalgebras. We also prove a rigidity result for some group measure space ${\rm II}_1$ factors.

The existence of soliton metrics for nilpotent Lie groups

Abstract: We show that a left-invariant metric g on a nilpotent Lie group N is a soliton metric if and only if a matrix U and vector v associated the manifold (N, g) satisfy the matrix equation Uv = [1], where [1] is a vector with every entry a one. We associate a generalized Cartan matrix to the matrix U and use the theory of Kac–Moody algebras to analyze the solution spaces for such linear systems. An application to the existence of soliton metrics on certain filiform Lie groups is given.

The CARTAN Form and its Generalizations in the Calculus of Variations

Abstract: In this paper, we discuss possible extensions of the concept of the Cartan form of classical mechanics to higher-order mechanics on manifolds, higher-order field theory on jet bundles and to parametric variational problems on slit tangent bundles and on bundles of nondegenerate velocities. We present a generalization of the Cartan form, known as a Lepage form, and basic properties of the Lepage forms. Both earlier and recent examples of differential forms generalizing the Cartan form are reviewed.

Classification of hyperbolic Dynkin diagrams, root lengths and Weyl group orbits

Abstract: We give a criterion for a Dynkin diagram, equivalently a generalized Cartan matrix, to be symmetrizable. This criterion is easily checked on the Dynkin diagram. We obtain a simple proof that the maximal rank of a Dynkin diagram of compact hyperbolic type is 5, while the maximal rank of a symmetrizable Dynkin diagram of compact hyperbolic type is 4. Building on earlier classification results of Kac, Kobayashi-Morita, Li and Sacliolu, we present the 238 hyperbolic Dynkin diagrams in ranks 3–10, 142 of which are symmetrizable. For each symmetrizable hyperbolic generalized Cartan matrix, we give a symmetrization and hence the distinct lengths of real roots in the corresponding root system. For each such hyperbolic root system we determine the disjoint orbits of the action of the Weyl group on real roots. It follows that the maximal number of disjoint Weyl group orbits on real roots in a hyperbolic root system is 4.

Divided power (co)homology. Presentations of simple finite dimensional modular Lie superalgebras with Cartan matrix

Abstract: For modular Lie superalgebras, new notions are introduced: Divided power homology and divided power cohomology. For illustration, we explicitly give presentations (in terms of analogs of Chevalley generators) of finite dimensional Lie (super) algebras with indecomposable Cartan matrix in characteristic 2 (and - in the arXiv version of the paper - in other characteristics for completeness of the picture). In the modular and super cases, we define notions of Chevalley generators and Cartan matrix, and an auxiliary notion of the Dynkin diagram. The relations of simple Lie algebras of the A, D, E types are not only Serre ones. These non-Serre relations are same for Lie superalgebras with the same Cartan matrix and any distribution of parities of the generators. Presentations of simple orthogonal Lie algebras having no Cartan matrix (indigenous for characteristic 2) are also given.

$E_{11}$, 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 $\mf{V}_D$ that is characterized concisely by a Cartan matrix which has been constructed explicitly for each spacetime dimension $11 \geq D \geq 3.$ 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 $\mf{V}$-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 $D \geq 1.$ This explains not only why both structures lead to the same propagating $p$-forms and their duals for $p\leq (D - 2),$ but also why one obtains the same $(D - 1)$-forms and "top" $D$-forms. The Borcherds symmetries $\mf{V}_2$ and $\mf{V}_1$ are new too. We also introduce and use the concept of a presentation of a Lie algebra that is covariant under a given subalgebra.

The classification of almost affine (hyperbolic) lie superalgebras

Abstract: We say that an indecomposable Cartan matrix A with entries in the ground field is almost affine if the Lie (super)algebra determined by it is not finite dimensional or affine (Kac–Moody) but the Lie sub(super)algebra determined by any submatrix of A, obtained by striking out any row and any column intersecting on the main diagonal, is the sum of finite dimensional or affine Lie (super)algebras. A Lie (super)algebra with Cartan matrix is said to be almost affine if it is not finite dimensional or affine (Kac–Moody), and all of its Cartan matrices are almost affine. We list all almost affine Lie superalgebras over complex numbers with indecomposable Cartan matrix correcting two earlier claims of classification.

Classification of hyperbolic Dynkin diagrams, root lengths and Weyl group orbits

Abstract: We give a criterion for a Dynkin diagram, equivalently a generalized Cartan matrix, to be symmetrizable. This criterion is easily checked on the Dynkin diagram. We obtain a simple proof that the maximal rank of a Dynkin diagram of compact hyperbolic type is 5, while the maximal rank of a symmetrizable Dynkin diagram of compact hyperbolic type is 4. Building on earlier classification results of Kac, Kobayashi-Morita, Li and Sa\c{c}lio\~{g}lu, we present the 238 hyperbolic Dynkin diagrams in ranks 3-10, 142 of which are symmetrizable. For each symmetrizable hyperbolic generalized Cartan matrix, we give a symmetrization and hence the distinct lengths of real roots in the corresponding root system. For each such hyperbolic root system we determine the disjoint orbits of the action of the Weyl group on real roots. It follows that the maximal number of disjoint Weyl group orbits on real roots in a hyperbolic root system is 4.

Stratifications and Mackey Functors II: Globally Defined Mackey Functors

Abstract: We describe structural properties of globally defined Mackey functors related to the stratification theory of algebras. We show that over a field of characteristic zero they form a highest weight category and we also determine precisely when this category is semisimple. This approach is used to show that the Cartan matrix is often symmetric and non-singular, and we are able to compute finite parts of it in some instances. We also develop a theory of vertices of globally defined Mackey functors in the spirit of group representation theory, as well as giving information about extensions between simple functors.

Killing fields of holomorphic Cartan geometries

Abstract: We study local automorphisms of holomorphic Cartan geometries This leads to classification results for compact complex manifolds admitting holomorphic Cartan geometries We prove that a compact Kahler Calabi–Yau manifold bearing a holomorphic Cartan geometry of algebraic type admits a finite unramified cover which is a complex torus

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

Abstract: 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.

Holomorphic Cartan geometries and rational curves

Abstract: We prove that any compact Kahler manifold bearing a holomorphic Cartan geometry contains a rational curve just when the Cartan geometry is inherited from a holomorphic Cartan geometry on a lower dimensional compact Kahler manifold.

Holomorphic Cartan geometries, Calabi--Yau manifolds and rational curves

Abstract: We prove that if a Calabi--Yau manifold $M$ admits a holomorphic Cartan geometry, then $M$ is covered by a complex torus. This is done by establishing the Bogomolov inequality for semistable sheaves on compact K\"ahler manifolds. We also classify all holomorphic Cartan geometries on rationally connected complex projective manifolds.

Generators of simple Lie algebras II

Abstract: This paper is a continuation of earlier work on generators of simple Lie algebras in arbitrary characteristic [3]. We show that, in contrast to classical Lie algebras, simple graded Lie algebras of Cartan type S, H or K never enjoy the “one-and-a-half generation” property. The methods rely on a study of centralisers in Cartan type Lie algebras.

Cartan invariants of symmetric groups and Iwahori–Hecke algebras

Abstract: K¨ulshammer, Olsson and Robinson conjectured that a certain set of numbers determined the invariant factors of the -Cartan matrix for Sn (equivalently, the invariant factors of the Cartan matrix for the Iwahori�Hecke algebra Hn(q), where q is a primitive th root of unity). We call these invariant factors Cartan invariants. In a previous paper, the second author calculated these Cartan invariants when  = pr, p is prime and r  p, and went on to conjecture that the formulae should hold for all r. Another result was obtained, which is surprising and counterintuitive from a block theoretic point of view. Namely, given the prime decomposition  = pr1 1 · . . . · prk k , the Cartan matrix of an -block of Sn is a product of Cartan matrices associated to pri i -blocks of Sn. In particular, the invariant factors of the Cartan matrix associated to an -block of Sn can be recovered from the Cartan matrices associated to the pri i -blocks. In this paper, we formulate an explicit combinatorial determination of the Cartan invariants of Sn, not only for the full Cartan matrix, but also for an individual block. We collect evidence for this conjecture by showing that the formulae predict the correct determinant of the -Cartan matrix. We then go on to show that Hill�s conjecture implies the conjecture of K¨ulshammer, Olsson and Robinson.

Cartan-Weyl 3-algebras and the BLG Theory II: Strong-Semisimplicity and Generalized Cartan-Weyl 3-algebras

Abstract: One of the most important questions in the Bagger-Lambert-Gustavsson (BLG) theory of multiple M2-branes is the choice of the Lie 3-algebra. The Lie 3-algebra should be chosen such that the corresponding BLG model is unitary and admits fuzzy 3-sphere as a solution. In this paper we propose another new condition: the Lie 3-algebras of use must be connected to the semisimple Lie algebras describing the gauge symmetry of D-branes via a certain reduction condition. We show that this reduction condition leads to a natural generalization of the Cartan-Weyl 3-algebras introduced in arXiv:1004.1397. Similar to a Cartan-Weyl 3-algebra, a generalized Cartan-Weyl 3-algebra processes a set of step generators characterized by non-degenerate roots. However, its Cartan subalgebra is non-abelian in general. We give reasons why having a non-abelian Cartan subalgebra may be just right to allow for fuzzy 3-sphere solution in the corresponding BLG models. We propose that generalized Cartan-Weyl 3-algebras is the right class of metric Lie 3-algebras to be used in the BLG theory.

Prolongs of (ortho-)orthogonal lie (super)algebras in characteristic 2

Abstract: Cartan described some of the finite dimensional simple Lie algebras and three of the four series of simple infinite dimensional vectorial Lie algebras with polynomial coefficients as prolongs, which now bear his name. The rest of the simple Lie algebras of these two types (finite dimensional and vectorial) are, if the depth of their grading is greater than 1, results of generalized Cartan–Tanaka–Shchepochkina (CTS) prolongs. Here we are looking for new examples of simple finite dimensional modular Lie (super)algebras in characteristic 2 obtained as Cartan prolongs. We consider pairs (an (ortho-)orthogonal Lie (super)algebra or its derived algebra, its irreducible module) and compute the Cartan prolongs of such pairs. The derived algebras of these prolongs are simple Lie (super)algebras. We point out several amazing phenomena in characteristic 2: a supersymmetry of representations of certain Lie algebras, latent or hidden over complex numbers, becomes manifest; the adjoint representation of some simple Lie...

On the Cartan Map for Crossed Products and Hopf-Galois Extensions

Abstract: We study certain aspects of the algebraic K-theory of Hopf–Galois extensions. We show that the Cartan map from K-theory to G-theory of such an extension is a rational isomorphism, provided the ring of coinvariants is regular, the Hopf algebra is finite dimensional and its Cartan map is injective in degree zero. This covers the case of a crossed product of a regular ring with a finite group and has an application to the study of Iwasawa modules.

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 Cartan matrix of type A_2, some Nichols algebras with 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.

On the Cartan matrix of Mackey algebras

Abstract: Let k be a field of characteristic p>0, and G be a finite group. The first result of this paper is an explicit formula for the determinant of the Cartan matrix of the Mackey algebra mu_k(G) of G over k. The second one is a formula for the rank of the Cartan matrix of the cohomological Mackey algebra comu_k(G) of G over k, and a characterization of the groups G for which this matrix is non singular. The third result is a generalization of this rank formula and characterization to blocks of comu_k(G) : in particular, if b is a block of kG, the Cartan matrix of the corresponding block comu_k(b) of comu_k(G) is non singular if and only if b is nilpotent with cyclic defect groups.

On the representation theory of finite J-trivial monoids

Abstract: In 1979, Norton showed that the representation theory of the 0-Hecke algebra admits a rich combinatorial description. Her constructions rely heavily on some triangularity property of the product, but do not use explicitly that the 0-Hecke algebra is a monoid algebra. The thesis of this paper is that considering the general setting of monoids admitting such a triangularity, namely J-trivial monoids, sheds further light on the topic. This is a step to use representation theory to automatically extract combinatorial structures from (monoid) algebras, often in the form of posets and lattices, both from a theoretical and computational point of view, and with an implementation in Sage. Motivated by ongoing work on related monoids associated to Coxeter systems, and building on well-known results in the semi-group community (such as the description of the simple modules or the radical), we describe how most of the data associated to the representation theory (Cartan matrix, quiver) of the algebra of any J-trivial monoid M can be expressed combinatorially by counting appropriate elements in M itself. As a consequence, this data does not depend on the ground field and can be calculated in O(n^2), if not O(nm), where n=|M| and m is the number of generators. Along the way, we construct a triangular decomposition of the identity into orthogonal idempotents, using the usual M\"obius inversion formula in the semi-simple quotient (a lattice), followed by an algorithmic lifting step. Applying our results to the 0-Hecke algebra (in all finite types), we recover previously known results and additionally provide an explicit labeling of the edges of the quiver. We further explore special classes of J-trivial monoids, and in particular monoids of order preserving regressive functions on a poset, generalizing known results on the monoids of nondecreasing parking functions.

Multiloop algebras, iterated loop algebras and extended affine Lie algebras of nullity 2

Abstract: Let M_n be the class of all multiloop algebras of finite dimensional simple Lie algebras relative to n-tuples of commuting finite order automorphisms. It is a classical result that M_1 is the class of all derived algebras modulo their centres of affine Kac-Moody Lie algebras. This combined with the Peterson-Kac conjugacy theorem for affine algebras results in a classification of the algebras in M_1. In this paper, we classify the algebras in M_2, and further determine the relationship between M_2 and two other classes of Lie algebras: the class of all loop algebras of affine Lie algebras and the class of all extended affine Lie algebras of nullity 2.

Root systems and diagram calculus. II. Quadratic forms for the Carter diagrams

Abstract: For any Carter diagram $\Gamma$ containing 4-cycle, we introduce the partial Cartan matrix $B_L$, which is similar to the Cartan matrix associated with a Dynkin diagram. A linkage diagram is obtained from $\Gamma$ by adding one root together with its bonds such that the resulting subset of roots is linearly independent. The linkage diagrams connected under the action of dual partial Weyl group (associated with $B_L$) constitute the linkage system, which is similar to the weight system arising in the representation theory of the semisimple Lie algebras. For Carter diagrams $E_6(a_i)$ and $E_6$ (resp. $E_7(a_i)$ and $E_7$; resp. $D_n(a_i)$ and $D_n$), the linkage system has, respectively, 2, 1, 1 components, each of which contains, respectively, 27, 56, $2n$ elements. Numbers 27, 56 and $2n$ are well-known dimensions of the smallest fundamental representations of semisimple Lie algebras, respectively, for $E_6$, $E_7$ and $D_n$. The 8-cell "spindle-like" linkage subsystems called loctets play the essential role in describing the linkage systems. It turns that weight systems also can be described by means of loctets.

Representations of four-derivation Lie algebras of Block type

Abstract: Block introduced certain analogues of the Zassenhaus algebras over a field of characteristic 0. The nongraded infinite-dimensional simple Lie algebras of Block type constructed by Xu can be viewed as generalizations of the Block algebras. In this paper, we construct a family of irreducible modules in terms of multiplication and differentiation operators on “polynomials” for four-devivation nongraded Lie algebras of Block type based on the finite-dimensional irreducible weight modules with multiplicity one of general linear Lie algebras. We also find a new series of submodules from which some irreducible quotient modules are obtained.

Representation theory of the higher-order peak algebras

Abstract: The representation theory (idempotents, quivers, Cartan invariants, and Loewy series) of the higher-order unital peak algebras is investigated. On the way, we obtain new interpretations and generating functions for the idempotents of descent algebras introduced in Saliola (J. Algebra 320:3866, 2008).

The quantum Cartan algebra associated to a bicovariant differential calculus

Abstract: We associate to any (suitable) bicovariant differential calculus on a quantum group a Cartan Hopf algebra which has a left, respectively right, representation in terms of left, respectively right, Cartan calculus operators. The example of the Hopf algebra associated to the $4D_+$ differential calculus on $SU_q(2)$ is described.

Cocompact lattices of minimal covolume in rank 2 kac-moody groups, part i: edge-transitive lattices

Abstract: Let G be a topological Kac-Moody group of rank 2 with symmetric Cartan matrix, defined over a finite field. An example is G = SL2(K), where K is the field of formal Laurent series over Fq. The group G acts on its Bruhat-Tits building X, a regular tree, with quotient a single edge. We classify the cocompact lattices in G which act transitively on the edges of X. Using this, for many such G we find the minimum covolume among cocompact lattices in G, by proving that the lattice which realises this minimum is edge-transitive. Our proofs use covering theory for graphs of groups, the dynamics of the G-action on X, the Levi decomposition for the parabolic subgroups of G, and finite group theory. where P1 and P2 are the standard parabolic/parahoric subgroups of G, and B = P1 \ P2 is the standard Borel/Iwahori subgroup. Now let m;nbe integers � 2. An (m;n)-amalgam is a free product with amalga- mation A1 �A0 A2, where the group A0 has index m in A1 and index n in A2. The amalgam is faithful if A0, A1 and A2 have no common normal subgroup. In Bass-Serre theory (see Section 1.2), an (m;n)-amalgam is the fundamental group of an edge of groups A = A1 A2