Showing papers on "Free algebra published in 2006"
TL;DR: The Grobner S -bases of the ideal corresponding to the universal enveloping algebra of the free nilpotent of class 2 Lie algebra and of the T-ideal generated by the polynomial identity [ x, y, z ] = 0 are calculated.
16 citations
TL;DR: A criterion for a system of polynomials to constitute a Grobner basis can be seen as a non-associative version of the Buchberger criterion and a formula is obtained for the generating series of a reducedGrobner bases for closed ideals in tree power series algebras K{{X}}.
15 citations
23 Jul 2006
TL;DR: It is shown that the locally free class group of an order in a semisimple algebra over a number field is isomorphic to a certain ray class group, and an algorithm is presented that computes this group.
Abstract: We show that the locally free class group of an order in a semisimple algebra over a number field is isomorphic to a certain ray class group. This description is then used to present an algorithm that computes the locally free class group. The algorithm is implemented in MAGMA for the case where the algebra is a group ring over the rational numbers.
14 citations
TL;DR: In this article, the authors classify all multiplicity free representations of G-invariant symplectic representations and show that all of them are commutative, i.e., the ring O(V ) G of invariants is a sub-Poisson algebra.
12 citations
TL;DR: In this paper, it was shown that q-rook monoid algebras are iterated inflations of Iwahori-Hecke algesbras, and in particular are cellular.
Abstract: We show that, over an arbitrary field, q-rook monoid algebras are iterated inflations of Iwahori-Hecke algebras, and, in particular, are cellular. Furthermore we give an algebra decomposition which shows a q-rook monoid algebra is Morita equivalent to a direct sum of Iwahori-Hecke algebras. We state some of the consequences for the representation theory of q-rook monoid algebras.
12 citations
TL;DR: In this article, it was shown that the division ring of fractions of a Lie algebra over an algebraically closed field of characteristic p > 0 is isomorphic to the ring of fraction of a Weyl algebra in the following cases: for g = gl n or sl n if p ∤ n, for the Witt algebra W 1 and for some tensor product W 1 ⊗ A of W 1 with a truncated polynomial ring.
11 citations
8 citations
TL;DR: In this article, the authors describe the set of projectors for which there exist quadruples of projected projectors, for a fixed collection of numbers αi ℝ+, � + àà= \overline {1,4} $$¯¯, such that α1� P� 1 + α2� 2 + α3� 4 = γI.
Abstract: We describe the set of γ ∈ ℝ for which there exist quadruples of projectors P
i
for a fixed collection of numbers αi ℝ+,
$$i = \overline {1,4} $$
, such that α1
P
1 + α2
P
2 + α3
P
3 + α3
P
4 = γI.
4 citations
13 Jan 2006
4 citations
TL;DR: This work uses a computer procedure to determine a basis of the elements of degree 5 in the nucleus of the free alternative algebra and makes calculations over the field Z 103 with multilinear identities.
Abstract: We use a computer procedure to determine a basis of the elements of degree 5 in the nucleus of the free alternative algebra. In order to save computer memory, we do our calculations over the field Z103. All calculations are made with multilinear identities. Our procedure is also valid for other characteristics and for determining nuclear elements of higher degree.
4 citations
TL;DR: In this paper, it was shown that if ρ1 and ρ2 have the same number of generators with the same degrees, then in some cases the Rankin-Selberg integrals which represent the corresponding two L functions, use the same Eisenstein series.
Abstract: The notion of a tower of Rankin-Selberg integrals was introduced in [G-R]. To recall this notion, let G be a reductive group defined over a global field F . Let G denote the L group of G. Let ρ denote a finite dimensional irreducible representation of G. Given an irreducible generic cuspidal representation of G(A), we let L(π, ρ, s) denote the partial L function associated with π and ρ. Here s is a complex variable and A denotes the adele ring associated with F . If ρ acts on the vector space V , we denote by C[V ] the symmetric algebra attached to the vector space V . Let C[V ] G denote the G invariant polynomials inside the symmetric algebra. As far as we know all examples of L functions represented by a Rankin-Selberg integral are associated with representations ρ such that C[V ] G is a free algebra. A list of all such groups, representations and the degrees of the generators of the invariant polynomials are given in [K]. The basic observation in [G-R] is that there is some relation between the Eisenstein series one uses to construct the Rankin-Selberg integral and the number of generators of the invariant polynomials and their degrees. This relation is far from being clear and it is mainly based on observation of all known constructions of such integrals. To summarize in an unprecise manner, the relations are: 1) If ρ1 and ρ2 have the same number of generators with the same degrees, then in some cases the Rankin-Selberg integrals which represent the corresponding two L functions, use the same Eisenstein series.
Journal Article•
TL;DR: In this paper, the complexity of representation theory of free products of finite-dimensional $C^*$-algebras is studied. But the complexity is not studied in this paper.
Abstract: In this paper we study the complexity of representation theory of free products of finite-dimensional $C^*$-algebras.
Posted Content•
TL;DR: For ungraded quotients of an arbitrary $\mathbb{Z}$-graded ring, the general PBW property is defined in this article, which covers both the classical PBW and the $N$-type PBW properties.
Abstract: For ungraded quotients of an arbitrary $\mathbb{Z}$-graded ring, we define the general PBW property, that covers the classical PBW property and the $N$-type PBW property studied via the $N$-Koszulity by several authors ([BG1], BG2], [FV]). In view of the noncommutative Grobner basis theory, we conclude that every ungraded quotient of a path algebra (or a free algebra) has the general PBW property. We remark that an earlier result of Golod [Gol] concerning Grobner bases can be used to give a homological characterization of the general PBW property in terms of Shafarevich complex. Examples of application are given.
TL;DR: In this article, it was shown that the complex semigroup algebra of a free monoid of rank at least two is -primitive, where denotes the involution on the algebra induced by word-reversal on the monoid.
Abstract: It is shown that the complex semigroup algebra of a free monoid of rank at least two is -primitive, where denotes the involution on the algebra induced by word-reversal on the monoid.
TL;DR: In this paper, the Cayley-Dickson algebra over the ring of rational integers is shown to be an RA loop, that is, a loop whose loop ring in any characteristic is an alternative, but not associative ring.
Abstract: Let L be an RA loop, that is, a loop whose loop ring in any characteristic is an alternative, but not associative, ring. Suppose that L is finite and that any noncommutative division algebra appearing as a simple component in the Wedderburn decomposition of Q
L is the classical Cayley–Dickson algebra over Q. Then the unit loop of the alternative loop ring Z
L of L over the ring of rational integers is finitely generated.
Journal Article•
TL;DR: In this article, a framework for Kleene algebras with embedded structures is proposed to guarantee the existence of free algebra if the embedded structures satisfy certain conditions, i.e., they satisfy certain properties.
Abstract: This paper proposes a framework for Kleene algebras with embedded structures that enables different kinds of Kleene algebras such as a Kleene algebra with tests and a Kleene algebra with relations to be handled uniformly. This framework guarantees the existence of free algebra if the embedded structures satisfy certain conditions.
01 Jan 2006
TL;DR: In this paper, a family of mappings (x,y) is defined on a ring A and subscripted with elements of a multiplicative monoid G, and a monoid algebra of G over A is constructed explicitly, and the universality property of it is shown.
Abstract: We define on an arbitrary ring A a family of mappings (�x,y) subscripted with elements of a multiplicative monoid G. The assigned properties allow to call these mappings derivations of the ring A. A monoid algebra of G over A is constructed explicitly, and the universality property of it is shown. In this note we consider monoid algebras over non-commutative rings. First, we introduce axiomatically a family of mappings � = (�x,y) defined on a ring A and subscripted with elements of a multiplicative monoid G. Due to their assigned properties these mappings can be called derivations of A. Next, we construct a monoid algebra Ah Gi by means of the family �, and the universality of it is shown. 1. Let A be a ring (in general non-commutative) and G a multiplicative monoid. Throughout the paper we consider 1 6 0 (where 0 is the null element of A, and 1 is the unit element for multiplication), the unit element of G is denoted by e. We introduce a family of mappings of A into itself by the following assumption.