scholarly journals The Spectrum of the Graded Ring of Differential Operators of a Scored Semigroup Algebra

2010 ◽  
Vol 38 (3) ◽  
pp. 829-847
Author(s):  
Mutsumi Saito
Keyword(s):  
Differential Operators ◽  
Semigroup Algebra ◽  
Graded Ring

On Valuations, Places and Graded Rings Associated to ∗-Orderings

2007 ◽  
Vol 50 (1) ◽  
pp. 105-112 ◽  
Author(s):  
Igor Klep
Keyword(s):  
Quantum Groups ◽  
Semigroup Algebra ◽  
Graded Rings ◽  
Graded Ring ◽  
Imaginary Unit

AbstractWe study natural ∗-valuations, ∗-places and graded ∗-rings associated with ∗-ordered rings. We prove that the natural ∗-valuation is always quasi-Ore and is even quasi-commutative (i.e., the corresponding graded ∗-ring is commutative), provided the ring contains an imaginary unit. Furthermore, it is proved that the graded ∗-ring is isomorphic to a twisted semigroup algebra. Our results are applied to answer a question of Cimprič regarding ∗-orderability of quantum groups.

SIMPLE GRADED RINGS OF SIEGEL MODULAR FORMS, DIFFERENTIAL OPERATORS AND BORCHERDS PRODUCTS

2005 ◽  
Vol 16 (03) ◽  
pp. 249-279 ◽  
Author(s):  
HIROKI AOKI ◽  
TOMOYOSHI IBUKIYAMA
Keyword(s):  
Modular Forms ◽  
Differential Operators ◽  
Siegel Modular Forms ◽  
The Other ◽  
Graded Rings ◽  
Graded Ring

In this paper, we show that the graded ring of Siegel modular forms of Γ0(N) ⊂ Sp(2,ℤ) has a very simple unified structure for N = 1, 2, 3, 4, taking Neben-type case (the case with character) for N = 3 and 4. All are generated by 5 generators, and all the fifth generators are obtained by using the other four by means of differential operators, and it is also obtained as Borcherds products. As an appendix, examples of Euler factors of L-functions of Siegel modular forms of Sp(2,ℤ) of odd weight are given.

The Calder´on-Zygmund Theory I: Ellipticity

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Classical Theory ◽  
Singular Integral ◽  
Singular Integrals ◽  
Differential Operators ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Single Parameter ◽  
Partial Differential ◽  
Translation Invariant ◽  
Partial Differential Operators

This chapter discusses a case for single-parameter singular integral operators, where ρ‎ is the usual distance on ℝn. There, we obtain the most classical theory of singular integrals, which is useful for studying elliptic partial differential operators. The chapter defines singular integral operators in three equivalent ways. This trichotomy can be seen three times, in increasing generality: Theorems 1.1.23, 1.1.26, and 1.2.10. This trichotomy is developed even when the operators are not translation invariant (many authors discuss such ideas only for translation invariant, or nearly translation invariant operators). It also presents these ideas in a slightly different way than is usual, which helps to motivate later results and definitions.

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