Generalized cluster complexes and Coxeter combinatorics

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].

Reflectable bases for affine reflection systems

Parabolic subgroup orbits on finite root systems

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

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.

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

Let G k be the de Bruijn graph with 2k vertices. When reproduction labels from a 4-letter alphabet A 4 are assigned linearly to the edges of G k using a three row binary generating matrix, a 2k-state trellis code results which is suitable for encoding a source with alphabet A 4 at a rate of one bit per source sample with respect to the Hamming fidelity criterion. There is a special type of code which results in this way called systematic code. A systematic code belongs to three sets of codes, each set isomorphic to a residue class ring of polynomials over GF(2). A systematic code sweeps out a cycle in each of its three residue class rings via repeated multiplication by the residue class of polynomial x. The local coordinates of a systematic code are non-negative integer parameters which describe how the code is positioned in each of its three cycles. Our main result is that a systematic code is uniquely determined by its local coordinates, except in rare instances. The paper concludes with some preliminary work indicating how a code with good distortion performance in coding a memoryless source with alphabet A 4 can be found in relatively small sets of systematic codes whose local coordinates are of a certain type.

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.

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

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.

The structure of Lie algebras, Lie superalgebras and Leibniz algebras graded by finite root systems has been studied by several researchers since 1992. In this paper, we study the structure of Leibniz superalgebras graded by finite root systems, which gives an approach to study various classes of Leibniz superalgebras. Key Words: ∆-Graded; dialgebras; Steinberg Leibniz superalgebras. 1

In a previous paper, we introduced functions $\zeta_r(s,a,\Delta;q)$, which are $q$-analogues of zeta functions associated with root systems $\Delta$, and their $p$-deformations $\zeta_r(s,a,\beta,\Delta;p,q)$. It is known that zeta functions of root systems satisfy certain functional relations including Witten's volume formula. In this paper, we consider $q$-extensions of these functional relations for $\Delta=\Delta(A_2), \ \Delta(A_3), \ \Delta(B_2), \ \Delta(G_2)$. We also generalize expressions of ``Weyl group symmetric'' linear combination of $\zeta_r(s,a,\beta,\Delta;p,q)$ for $\Delta=\Delta(A_1), \ \Delta(A_2), \ \Delta(A_3)$, which were obtained in the previous paper, to arbitrary root systems $\Delta$ by using an identity due to Macdonald.

We generalize the notion of a root system by relaxing the conditions that ensure that it is invariant under reflections and study the resulting structures, which we call generalized root systems (GRSs). Since both Kostant root systems and root systems of Lie superalgebras are examples of GRSs, studying GRSs provides a uniform axiomatic approach to studying both of them. GRSs inherit many of the properties of root systems. In particular, every GRS defines a crystallographic hyperplane arrangement. We believe that GRSs provide an intrinsic counterpart to finite Weyl groupoids and crystallographic hyperplane arrangements, extending the relationship between finite Weyl groupoids and crystallographic hyperplane arrangements established by Cuntz. An important difference between GRSs and root systems is that GRSs may lack a (large enough) Weyl group. In order to compensate for this, we introduce the notion of a virtual reflection, building on a construction of Penkov and Serganova in the context of root systems of Lie superalgebras. The most significant new feature of GRSs is that, along with subsystems, one can define quotient GRSs. Both Kostant root systems and root systems of Lie superalgebras are equivalent to quotients of root systems and all root systems are isomorphic to quotients of simply-laced root systems. We classify all rank 2 GRSs and show that they are equivalent to quotients of root systems. Finally, we discuss in detail quotients of root systems. In particular we provide all isomorphisms and equivalences among them. Our results on quotient of root systems provide a different point of view on flag manifolds, reproving results of Alekseevsky and Graev.

Jim Propp recently introduced a variant of chip-firing on a line where the chips are given distinct integer labels. Hopkins, McConville, and Propp showed that this process is confluent from some (but not all) initial configurations of chips. We recast their set-up in terms of root systems: labeled chip-firing can be seen as a root-firing process which allows the moves $\lambda \to \lambda + \alpha$ for $\alpha\in \Phi^{+}$ whenever $\langle\lambda,\alpha^\vee\rangle = 0$, where $\Phi^{+}$ is the set of positive roots of a root system of Type A and $\lambda$ is a weight of this root system. We are thus motivated to study the exact same root-firing process for an arbitrary root system. Actually, this central root-firing process is the subject of a sequel to this paper. In the present paper, we instead study the interval root-firing processes determined by $\lambda \to \lambda + \alpha$ for $\alpha\in \Phi^{+}$ whenever $\langle\lambda,\alpha^\vee\rangle \in [-k-1,k-1]$ or $\langle\lambda,\alpha^\vee\rangle \in [-k,k-1]$, for any $k \geq 0$. We prove that these interval-firing processes are always confluent, from any initial weight. We also show that there is a natural way to consistently label the stable points of these interval-firing processes across all values of $k$ so that the number of weights with given stabilization is a polynomial in $k$. We conjecture that these Ehrhart-like polynomials have nonnegative integer coefficients.