Article

A class of standard bases of polynomial algebras and its applications

January 2009

DOI:10.3770/j.issn:1000-341X.2009.02.001

Authors:

Jingcheng Dong

Nanjing Agricultural University

Libin Li

Colorado State University

Chen Huixiang

Southwest Jiaotong University

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ResearchGate has not been able to resolve any references for this publication.

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The Hecke algebras for the Fricke groups

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Aloys Krieg

We discuss the Hecke algebra

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G(\sqrt 2)

⋌e and

G(\sqrt 3)

⋌e are included. Finally we give a description of modular forms for the Fricke groups in terms of new-forms in the sense of Atkin-Lehner.

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V. N. Latyshev

We give the definition of a standard basis of a T-ideal of the free associative algebra over a field of zero characteristic and indicate some basis called canonical in the linear space of n-linear forms. Using this basis, we construct a standard basis in the T-ideal of identities satisfied by the algebra of upper triangular (n × n)-matrices.

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September 1991 · Communications in Theoretical Physics

In this paper, both trigonometric and rational solutions to the Yang–Baxter equation associated with the spinor representations of the quantum universal enveloping algebras are obtained. The quantum Clebsch–Gordan matrices, the quantum projectors and the solutions are the block matrices with the dimensions of the submatrices to be 1 and , where , if is even, , if is odd. The explicit forms of the ... [Show full abstract] submatrices with the same dimensions are independent of . As examples, we discuss the solutions for the spinor representations of the quantum to , and present the explicit forms of those submatrices with the dimensions 1, 2, 8 and 32. The corresponding representations of the braid group and the link polynomials are also computed through a standard method.

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In this paper, we present a complete method for solving all polynomials of the form

\phi(z)=c_n z^n+c_{n-1} z^{n-1}+\dots+c_1 z+c_0

(``standard polynomials") over any given octonion division algebra. When

\phi(z)

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\phi(z)

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Let R=k[x,y,z] be a standard graded 3-variable polynomial ring, where k denotes any field. We study grade 3 homogeneous ideals

I \subseteq R

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, where

s \geq3

is some integer. We prove that all such ideals are obtained by a trimming process introduced by Christensen, Veliche, and Weyman. We also construct a general resolution ... [Show full abstract] for all such ideals which is minimal in sufficiently generic cases. Using this resolution, we can give bounds on the minimal number of generators

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s\geq 3

. Finally, we study the Tor-algebra structure of R/I. It is shown that these rings have Tor algebra class G(r) for

s \leq r \leq 2s-1

. Furthermore, we produce ideals I for all

s \geq 3

and all r with

s \leq r \leq 2s-1

such that

\textrm{Soc} (R/I ) = k(-s) \oplus k(-2s+1)

and R/I has Tor-algebra class G(r), partially answering a question of realizability posed by Avramov.

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The minimal degree standard identity on MnE2 and MnE3

June 2020 · Israel Journal of Mathematics

We prove an Amitsur–Levitzki-type theorem for Grassmann algebras, stating that the minimal degree of a standard identity that is a polynomial identity of the ring of n × n matrices over the m-generated Grassmann algebra is at least

2\lfloor\frac{m}{2}\rfloor+4n-4

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We introduce a novel construction which allows us to identify the elements of the skeletons of a CW-complex P(m) and the monomials in m variables. From this, we infer that there is a bijection between finite CW-subcomplexes of P(m), which are quotients of finite simplicial complexes, and some bigraded standard Artinian Gorenstein algebras, generalizing previous constructions in \cite{F:S}, ... [Show full abstract] \cite{CGIM} and \cite{G:Z}. We apply this to a generalization of Nagata idealization for level algebras. These algebras are standard graded Artinian algebras whose Macaulay dual generator is given explicitly as a bigraded polynomial of bidegree (1,d). We consider the algebra associated to polynomials of the same type of bidegree

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. We study the algebra of uniformly continuous holomorphic symmetric functions on the ball of # p , investigating in particular the spectrum of such algebras. To do so, we examine the algebra of symmetric polynomials on # p - spaces as well as finitely generated symmetric algebras of holomorphic functions. Such symmetric polynomials determine the points in # p up to a permutation. In recent ... [Show full abstract] years, algebras of holomorphic functions on the unit ball of standard complex Banach spaces have been considered by a number of authors and the spectrum of such algebras was studied in [1],[2], [7]. For example, properties of A u (BX ), the algebra of uniformly continuous holomorphic functions on the ball of a complex Banach space X have been studied by Gamelin, et al. Unfortunately, this analogue of the classical disc algebra A(D) has a very complicated, not well understood, spectrum. If X # has the approximation property, the spectrum of A u (BX ) coincides with the closed unit ball of the bidua...

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Homogeneous polynomial ideals: Standard bases and liaison

June 1990 · Journal of Pure and Applied Algebra

Giulio Campanella

Let I be a perfect height 2 homogeneous ideal in a graded polynomial algebra over a field. By a ‘liaison’ method we construct new examples of such ideals I with prescribed Hilbert function and number of generators in each degree.

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Quasi-associativity and flatness criteria for quadratic algebra deformations

December 1997 · Israel Journal of Mathematics

LetR h be the quantumR-matrix corresponding to a Drinfeld-Jimbo quantum groupU h (G). Suppose a finite dimensional representationM h ofU h (G) is given. TheR h induces an operator onM h ⊗2 andS h , its composition with the standard transposition, is the Yang-Baxter operator. It turns out that the spaceM h ⊗2 admits the decompositionM h =⊕ i n J ih whereJ ih are the eigensubspaces ... [Show full abstract] ofS h . Consider the quadratic algebras (M h , E h k ) whereE h k =⊕ i≠k J ih . We prove that all (M h ,E h k ) are flat deformations of the quadratic algebras (V 0,E 0 k ). Let End(M h ;J 1h , …,J nh ) be the quantum semigroup corresponding to this decomposition. Our second result is that this gives a flat deformation of the quantum semigroup End(M 0;J 1,0, …,J n,0).

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In this article, two types of fractional local error bounds for quadratic complementarity problems are established, one is based on the natural residual function and the other on the standard violation measure of the polynomial equalities and inequalities. These fractional local error bounds are given with explicit exponents. A fractional local error bound with an explicit exponent via the ... [Show full abstract] natural residual function is new in the tensor/polynomial complementarity problems literature. The other fractional local error bounds take into account the sparsity structures, from both the algebraic and the geometric perspectives, of the third-order tensor in a quadratic complementarity problem. They also have explicit exponents, which improve the literature significantly.

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A class of standard bases of polynomial algebras and its applications

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