Showing papers on "Cartan matrix published in 1998"

The Jacobians of Non‐Split Cartan Modular Curves

Abstract: Let E be an elliptic curve defined over a number field K. The galois group Gal(K|K) acts on the K-points of E. In particular, this action leaves stable the p-torsion points of E, denoted E[p]. Hence, one obtains a representation ρE,p : Gal(K|K) → Aut(E[p]) ∼= GL2(Fp), called the mod p representation of E. Let p be an odd prime. It is known from the theory of complex multiplication that the mod p representation of an elliptic curve over Q with complex multiplication has image lying in the normaliser of a split or non-split Cartan subgroup of GL2(Fp), if it is irreducible. Such a mod p representation is called split or non-split dihedral, respectively. Conversely, one may ask whether these are the only elliptic curves over Q with dihedral mod p representation (see [25], section 4.3). There exist modular curves X split(p) and X + non-split(p) defined over Q which classify elliptic curves with split dihedral and non-split dihedral mod p representation, respectively, in the sense that the Q-rational points of X split(p) and X + non-split(p) correspond to elliptic curves over Q with split dihedral and non-split dihedral mod p representation, respectively [7]. The above question can therefore be rephrased by asking whether the non-cuspidal Q-rational points on the modular curves X split(p) and X non-split(p) arise only from elliptic curves over Q with complex multiplication. If the genus of X split(p) or X + non-split(p) is zero, then it has infinitely many Qrational points. Thus, this question has a negative answer in case of genus zero, which occurs for p = 3, 5, 7. Only X non-split(p) achieves genus one and this occurs for p = 11. It can be shown that X non-split(11) is the elliptic curve 121E in [3] and that its Mordell-Weil group has rank one. Thus, there are infinitely many elliptic curves over Q, non-isomorphic over Q, with non-split dihedral mod 11 representation. For all other values of p, X split(p) and X + non-split(p) have genus greater than one so there are only finitely many elliptic curves over Q with dihedral mod p representation by Faltings’ Theorem. Hence, in these cases it is plausible that the non-cuspidal rational points arise only from elliptic curves over Q with complex multiplication, although it may be possible for some exceptions to occur for small values of p. Indeed, in [25], it is asked whether this is the case for p ≥ 19. Because of the isomorphism X split(p) ∼= X + 0 (p ), the methods of [18] [19] can be used to tackle this problem in the split case. In [21], some progress has been made in this direction. However, Mazur states in [18] that the non-split case does not seem to be approachable by known methods. In an effort to understand X non-split(p), we prove the following theorem: Theorem 1. The jacobian of X non-split(p) is isogenous to the new part of the jacobian of X 0 (p ).

Third order ordinary differential equations and Legendre connections

Abstract: We use N. Tanaka's theory of normal Cartan connections associated with simple graded Lie algebras to study Cartan's equivalence problem of single third order ordinary differential equations under contact transformations. As a result we obtain the complete structure equation with two differential inv riants, which is applied on general Legendre Grassmann bundles of three-dimensional contact manifolds.

Quantum Kac– Algebras and Vertex Representations

Abstract: We introduce an affinization of the quantum Kac–Moody algebra associated to a symmetric generalized Cartan matrix. Based on the affinization, we construct a representation of the quantum Kac–Moody algebra by vertex operators from bosonic fields. We also obtain a combinatorial indentity about Hall–Littlewood polynomials.

Hochschild cohomology for von neumann algebras with cartan subalgebras

Abstract: The main result of this paper is that H n ( M, M ) = 0, n ≥ 1, for von Neumann algebras M with Cartan subalgebras and separable preduals.

Isomorphisms between generalized Cartan type $W$ Lie algebras in characteristic $0$

Abstract: In this paper, we determine when two simple generalized Cartan type W Lie algebras Wd(A T ) are isomorphic, and discuss the relationship between the Jacobian conjecture and the generalized Cartan type W Lie algebras.

Central extensions of the families of quasi-unitary Lie algebras

Abstract: The most general possible central extensions of two whole families of Lie algebras, which can be obtained by contracting the special pseudo-unitary algebras of the Cartan series and the pseudo-unitary algebras , are completely determined and classified for arbitrary p and q. In addition to the and algebras, whose second cohomology group is well known to be trivial, each family includes many non-semisimple algebras; their central extensions, which are explicitly given, can be classified into three types as far as their properties under contraction are involved. A closed expression for the dimension of the second cohomology group of any member of these families of algebras is given.

The structure of quantum Lie algebras for the classical series ?, ? and ?

Abstract: The structure constants of quantum Lie algebras depend on a quantum deformation parameter q and they reduce to the classical structure constants of a Lie algebra at q = 1. We explain the relationship between the structure constants of quantum Lie algebras and quantum Clebsch-Gordan coefficients for . We present a practical method for the determination of these quantum Clebsch-Gordan coefficients and are thus able to give explicit expressions for the structure constants of the quantum Lie algebras associated to the classical Lie algebras , and . In the quantum case the structure constants of the Cartan subalgebra are non-zero and we observe that they are determined in terms of the simple quantum roots. We introduce an invariant Killing form on the quantum Lie algebras and find that it takes values which are simple q-deformations of the classical ones.

Finite Dimensional Pointed Hopf Algebras with Abelian Coradical and Cartan matrices

Abstract: In a previous work \cite{AS2} we showed how to attach to a pointed Hopf algebra A with coradical $\k\Gamma$, a braided strictly graded Hopf algebra R in the category $_{\Gamma}^{\Gamma}\Cal{YD}$ of Yetter-Drinfeld modules over $\Gamma$ In this paper, we consider a further invariant of A, namely the subalgebra R' of R generated by the space V of primitive elements Algebras of this kind are known since the pioneering work of Nichols It turns out that R' is completely determined by the braiding c:V\otimes V \to V \otimes V We denote R' = B(V) We assume further that $\Gamma$ is finite abelian Then c is given by a matrix (b_{ij}) whose entries are roots of unity; we also suppose that they have odd order We introduce for these braidings the notion of "braiding of Cartan type" and we attach a generalized Cartan matrix to a braiding of Cartan type We prove that B(V) is finite dimensional if its corresponding matrix is of finite Cartan type and give sufficient conditions for the converse statement As a consequence, we obtain many new families of pointed Hopf algebras When $\Gamma$ is a direct sum of copies of a group of prime order, the conditions hold and any matrix is of Cartan type As a sample, we classify all the finite dimensional pointed Hopf algebras which are coradically graded, generated in degree one and whose coradical has odd prime dimension p We also characterize coradically graded pointed Hopf algebras of order p^4, which are generated in degree one

The family of quaternionic quasi-unitary Lie algebras and their central extensions

Abstract: The family of quaternionic quasi-unitary (or quaternionic unitary Cayley--Klein algebras) is described in a unified setting. This family includes the simple algebras sp(N+1) and sp(p,q) in the Cartan series C_{N+1}, as well as many non-semisimple real Lie algebras which can be obtained from these simple algebras by particular contractions. The algebras in this family are realized here in relation with the groups of isometries of quaternionic hermitian spaces of constant holomorphic curvature. This common framework allows to perform the study of many properties for all these Lie algebras simultaneously. In this paper the central extensions for all quasi-simple Lie algebras of the quaternionic unitary Cayley--Klein family are completely determined in arbitrary dimension. It is shown that the second cohomology group is trivial for any Lie algebra of this family no matter of its dimension.

Monoid gradings on algebras and the cartan determinant conjecture

Abstract: In this work we tackle the Cartan determinant conjecture for finite-dimensional algebras through monoid gradings. Given an adequate ∑-grading on the left Artinian ring A, where ∑ is a monoid, we construct a generalized Cartan matrix with entries in ℤ∑, which is right invertitale whenever gl.dim A < ∞. That gives a positive answer to the conjecture when A admits a strongly adequate grading by an aperiodic commutative monoid. We then show that, even though this does not give a definite answer to the conjecture, it strictly widens the class of known graded algebras for which it is true.

Central extensions of the families of quasi-unitary Lie

Abstract: The most general possible central extensions of two whole families of Lie algebras, which can be obtained by contracting the special pseudo-unitary algebras su.p;q/ of the Cartan series Al and the pseudo-unitary algebras u.p;q/, are completely determined and classified for arbitrary p and q. In addition to the su.p;q/ and u.p;q/ algebras, whose second cohomology group is well known to be trivial, each family includes many non-semisimple algebras; their central extensions, which are explicitly given, can be classified into three types as far as their properties under contraction are involved. A closed expression for the dimension of the second cohomology group of any member of these families of algebras is given.

A Theory of Lorentzian Kac-Moody algebras

Abstract: We present a variant of the Theory of Lorentzian (i. e. with a hyperbolic generalized Cartan matrix) Kac-Moody algebras recently developed by V. A. Gritsenko and the author. It is closely related with and strongly uses results of R. Borcherds. This theory should generalize well-known Theories of finite Kac-Moody algebras (i. e. classical semisimple Lie algebras corresponding to positive generalized Cartan matrices) and affine Kac-Moody algebras (corresponding to semi-positive generalized Cartan matrices). Main features of the Theory of Lorentzian Kac-Moody algebras are: One should consider generalized Kac-Moody algebras introduced by Borcherds. Denominator function should be an automorphic form on IV type Hermitian symmetric domain (first example of this type related with Leech lattice was found by Borcherds). The Kac-Moody algebra is graded by an integral hyperbolic lattice $S$. Weyl group acts in the hyperbolic space related with $S$ and has a fundamental polyhedron $\Cal M$ of finite (or almost finite) volume and a lattice Weyl vector. There are results and conjectures which permit (in principle) to get a ``finite'' list of all possible Lorentzian Kac-Moody algebras. Thus, this theory looks very similar to Theories of finite and affine Kac-Moody algebras but is much more complicated. There were obtained some classification results on Lorentzian Kac-Moody algebras and many of them were constructed.

Quasi-Hopf Superalgebras and Elliptic Quantum Supergroups

Abstract: We introduce the quasi-Hopf superalgebras which are $Z_2$ graded versions of Drinfeld's quasi-Hopf algebras. We describe the realization of elliptic quantum supergroups as quasi-triangular quasi-Hopf superalgebras obtained from twisting the normal quantum supergroups by twistors which satisfy the graded shifted cocycle condition, thus generalizing the quasi-Hopf twisting procedure to the supersymmetric case. Two types of elliptic quantum supergroups are defined, that is the face type $B_{q,\lambda}(G)$ and the vertex type $A_{q,p}[\hat{sl(n|n)}]$ (and $A_{q,p}[\hat{gl(n|n)}]$), where $G$ is any Kac-Moody superalgebra with symmetrizable generalized Cartan matrix. It appears that the vertex type twistor can be constructed only for $U_q[\hat{sl(n|n)}]$ in a non-standard system of simple roots, all of which are fermionic.

Simple generalized restricted modules for graded Lie algebras of Cartan type

Abstract: Let L = X (m:n)(2), Xe ¦W, S, H, K ¦ be a simple graded Lie algebra of Cartan type over a fieldF of characteristic p> 3. With the aid of Farnsteiner’s generalized reduced Verma module, a connection between the simple GR modules and simple graded modules of L is eastablished.

A presentation of toroidal algebras a homomorphic images of G.I.M. algebras

Abstract: Let A [1] be a 1-fold affinization of a generalized Cartan matrix A of finite or affine type so that A [1] has two identical columns and two identical rows. We consider a presentation of a toroidal Lie algebra as a homomorphic image of the generalized intersection matrix Lie algebra of A [1] and thus obtain an explicit description of the generators of the kernel from the defining relations

Abelianization of SU(N) Gauge Theory with Gauge Invariant Dynamical Variables and Magnetic Monopoles

Abstract: It is shown that SU(N) gauge theory coupled to adjoint Higgs can be explicitly re-written in terms of SU(N) gauge invariant dynamical variables with local physical interactions. The resultant theory has a novel compact abelian $U(1)^{(N - 1)}$ gauge invariance. The above abelian gauge invariance is related to the adjoint Higgs field and not to the gauge group SU(N). In this abelianized version the magnetic monopoles carrying the magnetic charges of $(N-1)$ types have a natural origin and therefore appear explicitly in the partition function as Dirac monopoles along with their strings. The gauge invariant electric and magnetic charges with respect to $U(1)^{(N-1)}$ gauge groups are shown to be vectors in root and co-root lattices of SU(N) respectively. Therefore, the Dirac quantization condition corresponds to SU(N) Cartan matrix elements being integers. We also study the effect of the $\theta$ term in the abelian version of the theory.

Lie algebras of an affinization of a generalized Cartan matrix

Abstract: Let A [1] be a 1-fold affinization of a generalized Cartan matrix A of finite or affine type so that A [1] has two identical columns and two identical rows. One can associate to A [1] a Lie algebra $ {\frak g}_{a} (A^{[1]}) $ which is isomorphic to either the u.c.a. of a loop algebra or a 2-toroidal Lie algebra. On the other hand, there is also the radical free contragredient Lie algebra $ {\frak g}_{c}(A^{[1]}) $ of A [1]. We show that there is a Lie algebra $ {\frak k} $ so that both $ {\frak g}_{c}(A^{[1]}) $ and $ {\frak g}_{a}(A^{[1]}) $ are homomorphic images of $ \frak k $ with finitely generated kernels.