Group Gradings in Incidence Algebras

If X is a bounded countable locally finite partially ordered set, R an integral domain, and G a group having the property that no non-identity element has order a unit of R. Then it is shown that any G-grading of the incidence algebra I(X, R) is equivalent to a good grading. Further, an example is given showing that not all group gradings of incidence algebras are equivalent to good gradings.     

Torsion Elements in Incidence Algebras

Abstract

Let I(X, F) denote the incidence algebra of the locally finite partially ordered set, X, defined over the field, F. This paper considers when the set of torsion elements of I(X, F) forms a group. If F is of finite characteristic, it is shown that the torsion elements of I(X, F) form a group if and only if X is bounded, while if F is of characteristic 0, the torsion elements of the incidence algebra form a group if and only if X is an antichain.     

An $ \mathfrak{X} $-crown of a finite soluble group

Let G be a finite soluble group and F_\mathfrakX(G) {\Phi_\mathfrak{X}}(G) an intersection of all those maximal subgroups M of G for which G

A Theorem About Elementary Cuts and Flow Polynomials

It is known that the functions F_G(k) and I_G(k) evaluating the numbers of nowhere-zero _k- and k-flows in a graph G, respectively, are polynomials of k. If X is a totally cyclic orientation of G, then the number of integral flows having values 1,,k–1 on the arcs of X can be evaluated by a polynomial I_X(k). F_G(k) and I_G(k) can be expressed as sums of I_X(k). In this paper we show that the value I_X(k) is positive for every totally cyclic orientation X of G if and only if k is greater than or equal to the maximum cardinality of an elementary edge-cut of G.Acknowledgments.This paper was finished during visiting School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia.Mathematics Subject Classification (1991): 05C15, 05C20     

Nilpotent orbits over ground fields of good characteristic

Let X be an F-rational nilpotent element in the Lie algebra of a connected and reductive group G defined over the ground field F. Suppose that the Lie algebra has a non-degenerate invariant bilinear form. We show that the unipotent radical of the centralizer of X is F-split. This property has several consequences. When F is complete with respect to a discrete valuation with either finite or algebraically closed residue field, we deduce a uniform proof that G(F) has finitely many nilpotent orbits in (F). When the residue field is finite, we obtain a proof that nilpotent orbital integrals converge. Under some further (fairly mild) assumptions on G, we prove convergence for arbitrary orbital integrals on the Lie algebra and on the group. The convergence of orbital integrals in the case where F has characteristic 0 was obtained by Deligne and Ranga Rao (1972).     

Graded polynomial identities for matrices with the transpose involution over an infinite field

Let F be an infinite field and let M_n(F) be the algebra of n×n matrices over F endowed with an elementary grading whose neutral component coincides with the main diagonal. In this paper, we find a basis for the graded polynomial identities of M_n(F) with the transpose involution. Our results generalize for infinite fields of arbitrary characteristic previous results in the literature, which were obtained for the field of complex numbers and for a particular class of elementary G-gradings.     

Group Gradings on Polynomial Algebras

Let k be a field. We consider gradings on a polynomial algebra k[X₁,…, X_n] by an arbitrary abelian group G, such that the indeterminates are homogeneous elements of nontrivial degree. We classify the isomorphism types of such gradings, and we count them in the case where G is finite. We present some examples of good gradings and find a minimal set of generators of the subalgebra of elements of trivial degree.     

Generalized bisymmetry on a restricted domain

Let X ⊂ ℝ be an interval of positive length and define the set Δ = {(x, y) ∈ X × X | x ≧ y}. We give the solution of the equation

Some results on generalized exponents

A digraph G = (V, E) is primitive if, for some positive integer k, there is a u → v walk of length k for every pair u, v of vertices of V. The minimum such k is called the exponent of G, denoted exp(G). The exponent of a vertex u ∈ V, denoted exp(u), is the least integer k such that there is a u → v walk of length k for each v ∈ V. For a set X ⊆ V, exp(X) is the least integer k such that for each v ∈ V there is a X → v walk of length k, i.e., a u → v walk of length k for some u ∈ X. Let F(G, k) : = max{exp(X) : |X| = k} and F(n, k) : = max{F(G, k) : |V| = n}, where |X| and |V| denote the number of vertices in X and V, respectively. Recently, B. Liu and Q. Li proved F(n, k) = (n − k)(n − 1) + 1 for all 1 ≤ k ≤ n − 1. In this article, for each k, 1 ≤ k ≤ n − 1, we characterize the digraphs G such that F(G, k) = F(n, k), thereby answering a question of R. Brualdi and B. Liu. We also find some new upper bounds on the (ordinary) exponent of G in terms of the maximum outdegree of G, Δ⁺(G) = max{d⁺(u) : u ∈ V}, and thus obtain a new refinement of the Wielandt bound (n − 1)² + 1. © 1998 John Wiley & Sons, Inc. J. Graph Theory 28: 215–225, 1998     

Automorphisms of Free Left Nilpotent Leibniz Algebras and Fixed Points

ABSTRACT

Let F _m (N) be the free left nilpotent (of class two) Leibniz algebra of finite rank m, with m ≥ 2. We show that F _m (N) has non-tame automorphisms and, for m ≥ 3, the automorphism group of F _m (N) is generated by the tame automorphisms and one more non-tame IA-automorphism. Let F(N) be the free left nilpotent Leibniz algebra of rank greater than 1 and let G be an arbitrary non-trivial finite subgroup of the automorphism group of F(N). We prove that the fixed point subalgebra F(N) ^G is not finitely generated.     

C*-index of observable algebras in <Emphasis Type="Italic">G</Emphasis>-spin model

In two-dimensional lattice spin systems in which the spins take values in a finite group G, one can define a field algebra F which carries an action of a Hopf algebra D(G), the double algebra of G and moreover, an action of D(G;H), which is a subalgebra of D(G) determined by a subgroup H of G, so that F becomes a modular algebra. The concrete construction of D(G;H)-invariant subspace A _H in F is given. By constructing the quasi-basis of conditional expectation γ _G of A _H onto A _G , the C*-index of γ _G is exactly the index of H in G.     

Positive circuits and maximal number of fixed points in discrete dynamical systems

We consider a product X of n finite intervals of integers, a map F from X to itself, the asynchronous state transition graph Γ(F) on X that Thomas proposed as a model for the dynamics of a network of n genes, and the interaction graph G(F) that describes the topology of the system in terms of positive and negative interactions between its n components. Then, we establish an upper bound on the number of fixed points for F, and more generally on the number of attractors in Γ(F), which only depends on X and on the topology of the positive circuits of G(F). This result generalizes the following discrete version of Thomas’ conjecture recently proved by Richard and Comet: If G(F) has no positive circuit, then Γ(F) has a unique attractor. This result also generalizes a result on the maximal number of fixed points in Boolean networks obtained by Aracena, Demongeot and Goles. The interest of this work in the context of gene network modeling is briefly discussed.     

Reduced invariant sets

Let K be a compact Lie group and W a finite-dimensional real K-module. Let X be a K-stable real algebraic subset of W. Let I(X){\mathcal{I}(X)} denote the ideal of X in \mathbbR[W]{\mathbb{R}[W]} and let I_K(X){\mathcal{I}_{K}(X)} be the ideal generated by I(X)^K{\mathcal{I}(X)^{K}} . We find necessary conditions and sufficient conditions for I(X) = I_K(X){{\mathcal{I}(X) = \mathcal{I}_{K}(X)}} and for ?{I_K(X)} = I(X){{\sqrt{\mathcal{I}_{K}(X)} = \mathcal{I}(X)}} . We consider analogous questions for actions of complex reductive groups.     

Tension polynomials of graphs

The tension polynomial F_G(k) of a graph G, evaluating the number of nowhere‐zero ?_k‐tensions in G, is the nontrivial divisor of the chromatic polynomial χ_G(k) of G, in that χ_G(k) = k^c(G)F_G(k), where c(G) denotes the number of components of G. We introduce the integral tension polynomial I_G(k), which evaluates the number of nowhere‐zero integral tensions in G with absolute values smaller than k. We show that 2^r(G)F_G(k)≥I_G(k)≥ (r(G)+1)F_G(k), where r(G)=|V(G)|?c(G), and, for every k>1, F_G(k+1)≥ F_G(k)˙k / (k?1) and I_G(k+1)≥I_G(k)˙k/(k?1). © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 137–146, 2002     

On the action of derivations on nilpotent ideals of associative algebras

Let I be a nilpotent ideal of an associative algebra A over a field F and let D be a derivation of A. We prove that the ideal I + D(I) is nilpotent if char F = 0 or the nilpotency index I is less than char F = p in the case of the positive characteristic of the field F. In particular, the sum N(A) of all nilpotent ideals of the algebra A is a characteristic ideal if char F = 0 or N(A) is a nilpotent ideal of index < p = char F.     

A brill - noether theory for<Emphasis Type="Italic">κ</Emphasis>-gonal nodal curves

LetV(g, x, k, y) be the set of all pairs (X, F), whereX is an integral projective nodal curve withp _a(X)=g and card(Sing(X))=x andF is a rank 1 torsion free sheaf onX with deg(F)=k, card(Sing(F))=y andh ⁰(X, F)≥2. Here we study a general (X, F) εV(g, x, k, y) and in particular the Brill-Noether theory ofX and the scrollar invariants ofF.     

On the structure and representations of non-balanced quantum doubles

For a field k and two finite groups G and X, when G acts on X from the right by group automorphisms, there is a Hopf algebra structure on k-space (kX ^op )* ? kG (see Theorem 2.1), called a non-balanced quantum double and denoted by D _X (G). In this paper, some Hopf algebra properties of D _X (G) are given, the representation types of D _X (G) viewed as a k-algebra are discussed, the algebra structure and module category over D _X (G) are studied. Since the Hopf algebra structure of non-balanced quantum double D _X (G) generalizes the usual quantum double D(G) for a finite group G, all results about D _X (G) in this paper can also be used to describe D(G) as a special case and the universal R-matrix of D _X (G) provides more solutions of Yang-Baxter equation.     

Good Gradings on Upper Block Triangular Matrix Algebras

We investigate group gradings of upper block triangular matrix algebras over a field such that all the matrix units lying there are homogeneous elements. We describe these gradings as endomorphism algebras of graded flags and classify them as orbits of a certain biaction of a Young subgroup and the group G on the set G ⁿ , where G is the grading group and n is the size of the matrix algebra. In particular, the results apply to algebras of upper triangular matrices.