Graded Algebra

2015 ◽  
pp. 33-80
Author(s):  
Jean-Pierre Tignol ◽  
Adrian R. Wadsworth
Keyword(s):  
Graded Algebra

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

scholarly journals A symmetric Bloch–Okounkov theorem

2021 ◽  
Vol 8 (2) ◽  
Author(s):  
Jan-Willem M. van Ittersum
Keyword(s):  
Symmetric Functions ◽  
Graded Algebra ◽  
Symmetric Polynomials ◽  
Generating Series

AbstractThe algebra of so-called shifted symmetric functions on partitions has the property that for all elements a certain generating series, called the q-bracket, is a quasimodular form. More generally, if a graded algebra A of functions on partitions has the property that the q-bracket of every element is a quasimodular form of the same weight, we call A a quasimodular algebra. We introduce a new quasimodular algebra $$\mathcal {T}$$ T consisting of symmetric polynomials in the part sizes and multiplicities.

scholarly journals GRADED TWISTED CALABI–YAU ALGEBRAS ARE GENERALIZED ARTIN–SCHELTER REGULAR

2021 ◽  
pp. 1-54
Author(s):  
MANUEL L. REYES ◽  
DANIEL ROGALSKI
Keyword(s):  
Regularity Property ◽  
Graded Algebra ◽  
General Study ◽  
Locally Finite ◽  
Tensor Algebras ◽  
Finite Dimensional ◽  
Dimension 2 ◽  
Separable Algebras ◽  
Gk Dimension

Abstract This is a general study of twisted Calabi–Yau algebras that are $\mathbb {N}$ -graded and locally finite-dimensional, with the following major results. We prove that a locally finite graded algebra is twisted Calabi–Yau if and only if it is separable modulo its graded radical and satisfies one of several suitable generalizations of the Artin–Schelter regularity property, adapted from the work of Martinez-Villa as well as Minamoto and Mori. We characterize twisted Calabi–Yau algebras of dimension 0 as separable k-algebras, and we similarly characterize graded twisted Calabi–Yau algebras of dimension 1 as tensor algebras of certain invertible bimodules over separable algebras. Finally, we prove that a graded twisted Calabi–Yau algebra of dimension 2 is noetherian if and only if it has finite GK dimension.

scholarly journals A note on the subadditivity of Syzygies

2016 ◽  
Vol 16 (09) ◽  
pp. 1750177 ◽  
Author(s):  
Sabine El Khoury ◽  
Hema Srinivasan
Keyword(s):  
Free Resolution ◽  
Graded Algebra ◽  
Gorenstein Algebras

Let [Formula: see text] be a graded algebra with [Formula: see text] and [Formula: see text] being the minimal and maximal shifts in the minimal graded free resolution of [Formula: see text] at degree [Formula: see text]. We prove that [Formula: see text] for all [Formula: see text]. As a consequence, we show that for Gorenstein algebras of codimension [Formula: see text], the subadditivity of maximal shifts [Formula: see text] in the minimal graded free resolution holds for [Formula: see text], i.e. we show that [Formula: see text] for [Formula: see text].

PERTURBATIVE EXPANSION FOR THE t-J MODEL CONSTRUCTED FROM THE GENERATORS OF THE SUPERSYMMETRIC HUBBARD ALGEBRA

2009 ◽  
Vol 23 (14) ◽  
pp. 3159-3177
Author(s):  
CARLOS E. REPETTO ◽  
OSCAR P. ZANDRON
Keyword(s):  
Perturbative Expansion ◽  
Lagrangian Formalism ◽  
Graded Algebra ◽  
Integral Formalism ◽  
Generating Functional ◽  
First Order ◽  
Effective Lagrangian ◽  
Supersymmetric Version ◽  
Present Formalism ◽  
Ghost Field

By using the Hubbard [Formula: see text]-operators as field variables along with the supersymmetric version of the Faddeev–Jackiw symplectic formalism, a family of first-order constrained Lagrangians for the t-J model is found. In order to satisfy the Hubbard [Formula: see text]-operator commutation rules satisfying the graded algebra spl(2,1), the number and kind of constraints that must be included in a classical first-order Lagrangian formalism for this model are presented. The model is also analyzed via path-integral formalism, where the correlation-generating functional and the effective Lagrangian are constructed. In this context, the introduction of a proper ghost field is needed to render the model renormalizable. The perturbative Lagrangian formalism is developed and it is shown how propagators and vertices can be renormalized to each order. In particular, the renormalized ferromagnetic magnon propagator arising in the present formalism is discussed. As an example, the thermal softening of the magnon frequency is computed.

On the Multiplicity and the Cohen–Macaulayness of Fiber Cones of Graded Modules

2021 ◽  
Vol 28 (01) ◽  
pp. 13-32
Author(s):  
Nguyen Tien Manh
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Primary Ideal ◽  
Graded Algebra ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Noetherian Local Ring ◽  
Graded Modules ◽  
Fiber Cone

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text], [Formula: see text] an ideal of [Formula: see text], [Formula: see text] an [Formula: see text]-primary ideal of [Formula: see text], [Formula: see text] a finitely generated [Formula: see text]-module, [Formula: see text] a finitely generated standard graded algebra over [Formula: see text] and [Formula: see text] a finitely generated graded [Formula: see text]-module. We characterize the multiplicity and the Cohen–Macaulayness of the fiber cone [Formula: see text]. As an application, we obtain some results on the multiplicity and the Cohen–Macaulayness of the fiber cone[Formula: see text].

scholarly journals Schubert Class and Cyclotomic NilHecke Algebras

2021 ◽  
Vol 28 (03) ◽  
pp. 379-398
Author(s):  
Kai Zhou ◽  
Jun Hu
Keyword(s):  
Algebraic Method ◽  
Basic Algebra ◽  
Graded Algebra ◽  
Algebra Isomorphism ◽  
Type Formula ◽  
Basis Element ◽  
Schubert Classes ◽  
Positive Integers ◽  
Schubert Class

Let [Formula: see text] and [Formula: see text] be positive integers such that [Formula: see text], and let [Formula: see text] be the Grassmannian which consists of the set of [Formula: see text]-dimensional subspaces of [Formula: see text]. There is a [Formula: see text]-graded algebra isomorphism between the cohomology [Formula: see text] of [Formula: see text] and a natural [Formula: see text]-form [Formula: see text] of the [Formula: see text]-graded basic algebra of the type [Formula: see text] cyclotomic nilHecke algebra [Formula: see text]. We show that the isomorphism can be chosen such that the image of each (geometrically defined) Schubert class [Formula: see text] coincides with the basis element [Formula: see text] constructed by Hu and Liang by purely algebraic method, where [Formula: see text] with [Formula: see text] for each [Formula: see text], and [Formula: see text] is the [Formula: see text]-multipartition of [Formula: see text] associated to [Formula: see text]. A similar correspondence between the Schubert class basis of the cohomology of the Grassmannian [Formula: see text] and the [Formula: see text]'s basis ([Formula: see text] is an [Formula: see text]-multipartition of [Formula: see text] with each component being either [Formula: see text] or empty) of the natural [Formula: see text]-form [Formula: see text] of the [Formula: see text]-graded basic algebra of [Formula: see text] is also obtained. As an application, we obtain a second version of the Giambelli formula for Schubert classes.

Circle Actions

2020 ◽  
pp. 161-166
Author(s):  
Loring W. Tu
Keyword(s):  
Differential Form ◽  
Circle Action ◽  
Graded Algebra ◽  
Graded Algebras ◽  
Circle Actions ◽  
Algebra Isomorphism ◽  
Weil Algebra ◽  
Differential Graded Algebras ◽  
Cartan Model

This chapter focuses on circle actions. Specifically, it specializes the Weil algebra and the Weil model to a circle action. In this case, all the formulas simplify. The chapter derives a simpler complex, called the Cartan model, which is isomorphic to the Weil model as differential graded algebras. It considers the theorem that for a circle action, there is a graded-algebra isomorphism. Under the isomorphism F, the Weil differential δ‎ corresponds to a differential called the Cartan differential. An element of the Cartan model is called an equivariant differential form or equivariant form for a circle action on the manifold M.

Hopf Algebras from Branching Rules

1983 ◽  
Vol 35 (1) ◽  
pp. 177-192 ◽  
Author(s):  
P. Hoffman
Keyword(s):  
Abelian Group ◽  
Symmetric Group ◽  
Hopf Algebras ◽  
Homogeneous Element ◽  
Algebra Structure ◽  
Graded Algebra ◽  
Finitely Generated ◽  
Dimension Zero ◽  
Branching Rules

Below we work out the algebra structure of some Hopf algebras which arise concretely in restricting representations of the symmetric group to certain subgroups. The basic idea generalizes that used by Adams [1] for H*(BSU). The question arose in discussions with H. K. Farahat. I would like to thank him for his interest in the work and to acknowledge the usefulness of several stimulating conversations with him.1. Review and statement of results. A homogeneous element of a graded abelian group will have its gradation referred to as its dimension. In all such groups below there will be no non-zero elements with negative or odd dimension. A graded algebra (resp. coalgebra) will be associative (resp. coassociative), strictly commutative (resp. co-commutative) and in dimension zero will be isomorphic to the ground ring F, providing the unit (resp. counit). We shall deal amost entirely with F = Z or F = Z/p for a prime p; the cases F = 0 or a localization of Z will occur briefly. In every case, the component in each dimension will be a finitely generated free F-module, so dualization works simply.

scholarly journals On the Groups of Cobordism Ωk

1958 ◽  
Vol 13 ◽  
pp. 135-156 ◽  
Author(s):  
Masahisa Adachi
Keyword(s):  
Equivalence Relation ◽  
Differentiable Manifold ◽  
Equivalence Classes ◽  
Graded Algebra ◽  
Differentiable Manifolds ◽  
Free Part ◽  
Natural Way

In the papers [11] and [18] Rohlin and Thom have introduced an equivalence relation into the set of compact orientable (not necessarily connected) differentiable manifolds, which, roughly speaking, is described in the following manner: two differentiable manifolds are equivalent (cobordantes), when they together form the boundary of a bounded differentiable manifold. The equivalence classes can be added and multiplied in a natural way and form a graded algebra Ω relative to the addition, the multiplication and the dimension of manifolds. The precise structures of the groups of cobordism Ωk of dimension k are not known thoroughly. Thom [18] has determined the free part of Ω and also calculated explicitly Ωk for 0 ≦ k ≦ 7.

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