Leibniz Superalgebras Graded by Finite Root Systems

Dialgebras are generalizations of associative algebras which give rise to Leibniz algebras instead of Lie algebras. In this paper we study super dialgebras and Leibniz superalgebras, which are $\z_2$-graded dialgebras and Leibniz algebras. We also study universal central extesions of Leibniz superalgebras and obtain some results as in the case of Lie superalgebras. We determine universal central extensions of basic classical Lie superalgebras in the category of Leibniz superalgebras. They play a key role in studying Leibniz superalgebras graded by finite root systems.

We introduce and study a family of simplicial complexes associated to an arbitrary finite root system and a nonnegative integer parameter m. For m=1, our construction specializes to the (simplicial) generalized associahedra or, equivalently, to the cluster complexes for the cluster algebras of finite type. Our computation of the face numbers and h-vectors of these complexes produces the enumerative invariants defined in other contexts by C.A.Athanasiadis, suggesting links to a host of well studied problems in algebraic combinatorics of finite Coxeter groups, root systems, and hyperplane arrangements. Recurrences satisfied by the face numbers of our complexes lead to combinatorial algorithms for determining Coxeter-theoretic invariants. That is, starting with a Coxeter diagram of a finite Coxeter group, one can compute the Coxeter number, the exponents, and other classical invariants by a recursive procedure that only uses most basic graph-theoretic concepts applied to the input diagram. In types A and B, we rediscover the constructions and results obtained by E.Tzanaki .

Parabolic subgroup orbits on finite root systems

Lie algebras graded by finite irreducible reduced root systems have been classified up to central extensions by Berman and Moody, Benkart and Zelmanov, and Neher. In this paper we determine the central extensions of these Lie algebras and hence describe them completely up to isomorphism.

Nichols algebras are fundamental objects in the construction of quantized enveloping algebras and in the classification of pointed Hopf algebras by the lifting method of Andruskiewitsch and Schneider. The structure of Cartan graphs can be attached to any Nichols algebras of diagonal type and plays an important role in the classification of Nichols algebras of diagonal type with a finite root system. In this paper, the main properties of all simply connected Cartan graphs attached to rank 6 Nichols algebras of diagonal type are determined. As an application, we obtain a subclass of rank 6 finite dimensional Nichols algebras of diagonal type.

Lusztig isomorphisms for Drinfel'd doubles of bosonizations of Nichols algebras of diagonal type

Reflectable bases for affine reflection systems

Finite Fields, Root Systems, and Orbit Numbers of Chevalley Groups

Rank 4 finite-dimensional Nichols algebras of diagonal type in positive characteristic

We continue our study of Cartan schemes and their Weyl groupoids. The results in this paper provide an algorithm to determine connected simply connected Cartan schemes of rank three, where the real roots form a finite irreducible root system. The algorithm terminates: Up to equivalence there are exactly 55 such Cartan schemes, and the number of corresponding real roots varies between 6 and 37. We identify those Weyl groupoids which appear in the classification of Nichols algebras of diagonal type.

The goals of this paper are two-fold. First, we prove, for an arbitrary finite root system D, the periodicity conjecture of Al. B. Zamolodchikov [24] that concerns Y-systems, a particular class of functional relations playing an important role in the theory of thermodynamic Bethe ansatz. Algebraically, Y-systems can be viewed as families of rational functions defined by certain birational recurrences formulated in terms of the root system (. We obtain explicit formulas for these rational functions, which always turn out to be Laurent polynomials, and prove that they exhibit the periodicity property conjectured by Zamolodchikov. In a closely related development, we introduce and study a simplicial complex A(b), which can be viewed as a generalization of the Stasheff polytope (also known as associahedron) for an arbitrary root system (D. In type A, this complex is the face complex of the ordinary associahedron, whereas in type B, our construction produces the Bott-Taubes polytope, or cyclohedron. We enumerate the faces of the complex A(@), prove that its geometric realization is always a sphere, and describe it in concrete combinatorial terms for the classical types ABCD. The primary motivation for this investigation came from the theory of cluster algebras, introduced in [9] as a device for studying dual canonical bases and total positivity in semisimple Lie groups. This connection remains behind the scenes in the text of this paper, and will be brought to light in a forthcoming sequel1 to [9].

A uniform proof is given that the entries of the hvector of the cluster complex ∆(Φ), associated by S. Fomin and A. Zelevinsky to a finite root system Φ, count elements of the lattice L of noncrossing partitions of corresponding type by rank. Similar interpretations for the h-vector of the positive part of ∆(Φ) are provided. The proof utilizes the appearance of the complex ∆(Φ) in the context of the lattice L in recent work of two of the authors, as well as an explicit shelling of ∆(Φ). This article has appeared in International Mathematical Research Notices, (2007), article ID 69705. Department of Mathematics (Division of Algebra-Geometry), University of Athens, Panepistimioupolis, 15784 Athens, Greece E-mail address: caath@math.uoa.gr School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland E-mail address: tom.brady@dcu.ie Mathematics Department, University of California, Santa Barbara, Santa Barbara, CA 93106, U.S.A E-mail address: jon.mccammond@math.ucsb.edu School of Mathematical Sciences, Dublin Institute of Technology, Dublin 8, Ireland E-mail address: colum.watt@dit.ie Date: June 20, 2006, Final Version 26/6/06. 2000 Mathematics Subject Classification. Primary 20F55; Secondary 05E99. 1

Using the vanishing cycles of simple singularities, we study the eigenvectors of Cartan matrices of finite root systems, and of q-deformations of these matrices.

In this paper, we study the sharp Jackson inequality for the best approximation of f ∈ L2,κ(ℝd) by a subspace Eκ2(σ) (SEκ2(σ)), which is a subspace of entire functions of exponential type (spherical exponential type) at most σ. Here L2,κ(ℝd) denotes the space of all d-variate functions f endowed with the L2-norm with the weight \(v_\kappa (x) = \prod\nolimits_{\xi \in R_ + } {|(\xi ,x)|^{2\kappa (\xi )} } \), which is defined by a positive subsystem R+ of a finite root system R ⊂ ℝd and a function κ(ξ): R → ℝ+ invariant under the reflection group G(R) generated by R. In the case G(R) = ℤ2d, we get some exact results. Moreover, the deviation of best approximation by the subspace Eκ2(σ) (SEκ2(σ)) of some class of the smooth functions in the space L2,κ(ℝd) is obtained.

For the weight \(v_k \left( x \right) = \prod _{\alpha \in \mathbb{R}_ + } \left| {\left( {\alpha ,x} \right)} \right|^{2k\left( \alpha \right)}\) defined by a positive subsystem R+ of a finite root system R ⊂ ℝd and by a function k(α): R → ℝ+ invariant under the reflection group generated by R, a sharp Jackson inequality in L2(ℝd) is proved.