scholarly journals Rings of differential operators on toric varieties

1994 ◽  
Vol 37 (1) ◽  
pp. 143-160 ◽  
Author(s):  
A. G. Jones
Keyword(s):  
Line Bundle ◽  
Toric Variety ◽  
Characteristic Zero ◽  
Differential Operators ◽  
Toric Varieties ◽  
Closed Field ◽  
Finitely Generated ◽  
Algebraically Closed Field ◽  
Finite Dimensional ◽  
Rings Of Differential Operators

Let be a finite dimensional toric variety over an algebraically closed field of characteristic zero, k. Let be the sheaf of differential operators on . We show that the ring of global sections, is a finitely generated Noetherian k-algebra and that its generators can be explicitly found. We prove a similar result for the sheaf of differential operators with coefficients in a line bundle.

scholarly journals Embedding the Picard group inside the class group: the case of $$\mathbb {Q}$$-factorial complete toric varieties

2021 ◽  
Author(s):  
Michele Rossi ◽  
Lea Terracini
Keyword(s):  
Free Group ◽  
Toric Variety ◽  
Class Group ◽  
Abelian Groups ◽  
Toric Varieties ◽  
Picard Group ◽  
Closed Field ◽  
Finitely Generated ◽  
Free Part ◽  
Canonical Injection

AbstractLet X be a $$\mathbb {Q}$$ Q -factorial complete toric variety over an algebraic closed field of characteristic 0. There is a canonical injection of the Picard group $$\mathrm{Pic}(X)$$ Pic ( X ) in the group $$\mathrm{Cl}(X)$$ Cl ( X ) of classes of Weil divisors. These two groups are finitely generated abelian groups; while the first one is a free group, the second one may have torsion. We investigate algebraic and geometrical conditions under which the image of $$\mathrm{Pic}(X)$$ Pic ( X ) in $$\mathrm{Cl}(X)$$ Cl ( X ) is contained in a free part of the latter group.

scholarly journals Involutions on finite-dimensional algebras over real closed fields

2004 ◽  
Vol 77 (1) ◽  
pp. 123-128 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Characteristic Zero ◽  
Real Closed Field ◽  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Algebraically Closed Field ◽  
Real Closed Fields ◽  
Finite Dimensional ◽  
Closed Fields ◽  
Finite Dimensional Algebras

AbstractIt is shown that the following conditions on a finite-dimensional algebra A over a real closed field or an algebraically closed field of characteristic zero are equivalent: (i) A admits a special involution, in the sense of Easdown and Munn, (ii) A admits a proper involution, (iii) A is semisimple.

HOPF ALGEBRAS OF DIMENSION 30

2010 ◽  
Vol 09 (01) ◽  
pp. 11-15 ◽  
Author(s):  
DAIJIRO FUKUDA
Keyword(s):  
Hopf Algebra ◽  
Group Algebra ◽  
Hopf Algebras ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Finite Dimensional

This paper contributes to the classification of finite dimensional Hopf algebras. It is shown that every Hopf algebra of dimension 30 over an algebraically closed field of characteristic zero is semisimple and thus isomorphic to a group algebra or the dual of a group algebra.

On the Existence of the Graded Exponent for Finite Dimensional ℤp-graded Algebras

2012 ◽  
Vol 55 (2) ◽  
pp. 271-284 ◽  
Author(s):  
Onofrio M. Di Vincenzo ◽  
Vincenzo Nardozza
Keyword(s):  
Characteristic Zero ◽  
Prime Order ◽  
Closed Field ◽  
Graded Algebras ◽  
Algebraically Closed Field ◽  
Finite Dimensional

AbstractLet F be an algebraically closed field of characteristic zero, and let A be an associative unitary F-algebra graded by a group of prime order. We prove that if A is finite dimensional then the graded exponent of A exists and is an integer.

scholarly journals Group Gradings on Finite Dimensional Lie Algebras

2013 ◽  
Vol 20 (04) ◽  
pp. 573-578 ◽  
Author(s):  
Dušan Pagon ◽  
Dušan Repovš ◽  
Mikhail Zaicev
Keyword(s):  
Finite Group ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Solvable Radical ◽  
Finite Dimensional ◽  
Levi Subalgebra ◽  
Group Gradings ◽  
Finite Dimensional Lie Algebras

We study gradings by non-commutative groups on finite dimensional Lie algebras over an algebraically closed field of characteristic zero. It is shown that if L is graded by a non-abelian finite group G, then the solvable radical R of L is G-graded and there exists a Levi subalgebra B=H1⊕ ⋯ ⊕ Hm homogeneous in G-grading with graded simple summands H1,…,Hm. All Supp Hi (i=1,…,m) are commutative subsets of G.

scholarly journals Commutative algebraic monoid structures on affine spaces

2019 ◽  
Vol 22 (08) ◽  
pp. 1950064
Author(s):  
Ivan Arzhantsev ◽  
Sergey Bragin ◽  
Yulia Zaitseva
Keyword(s):  
Characteristic Zero ◽  
Affine Space ◽  
Arbitrary Dimension ◽  
Toric Varieties ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Affine Spaces ◽  
Algebraic Monoid ◽  
Algebraic Monoids

We study commutative associative polynomial operations [Formula: see text] with unit on the affine space [Formula: see text] over an algebraically closed field of characteristic zero. A classification of such operations is obtained up to dimension 3. Several series of operations are constructed in arbitrary dimension. Also we explore a connection between commutative algebraic monoids on affine spaces and additive actions on toric varieties.

scholarly journals GOLDIE RANK OF PRIMITIVE QUOTIENTS VIA LATTICE POINT ENUMERATION

2013 ◽  
Vol 55 (A) ◽  
pp. 149-168
Author(s):  
JOANNA MEINEL ◽  
CATHARINA STROPPEL
Keyword(s):  
Lattice Point ◽  
Characteristic Zero ◽  
Differential Operators ◽  
Convex Geometry ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Weyl Algebras ◽  
Primitive Ideals ◽  
Invariant Differential ◽  
Lattice Point Enumeration

AbstractLet k be an algebraically closed field of characteristic zero. I. M. Musson and M. Van den Bergh (Mem. Amer. Math. Soc., vol. 136, 1998, p. 650) classify primitive ideals for rings of torus invariant differential operators. This classification applies in particular to subquotients of localized extended Weyl algebras $\mathcal{A}_{r,n-r}=k[x_1,\ldots,x_r,x_{r+1}^{\pm1}, \ldots, x_{n}^{\pm1},\partial_1,\ldots,\partial_n],$ where it can be made explicit in terms of convex geometry. We recall these results and then turn to the corresponding primitive quotients and study their Goldie ranks. We prove that the primitive quotients fall into finitely many families whose Goldie ranks are given by a common quasi-polynomial and then realize these quasi-polynomials as Ehrhart quasi-polynomials arising from convex geometry.

scholarly journals Tilting modules and a theorem of Hoshino

1993 ◽  
Vol 35 (1) ◽  
pp. 69-77 ◽  
Author(s):  
Ibrahim Assem ◽  
Otto Kerner
Keyword(s):  
Grothendieck Group ◽  
Closed Field ◽  
Finitely Generated ◽  
Tilting Modules ◽  
Algebraically Closed Field ◽  
Finite Dimensional ◽  
Image Position

Let k be an algebraically closed field, and A a finite dimensional k-algebra, which we shall assume, without loss of generality, to be basic and connected. By module is meant throughout a finitely generated right A-module. Following Happel and Ringel [10], we shall say that a module Tλ is a tilting (respectively, cotilting) module if it satisfies the following three conditions:(1)(2)(3) the number of non-isomorphic indecomposable summands of T equals the rank of the Grothendieck group K0(A) of A.

H-Simple module algebras for eight-dimensional non-semisimple Hopf algebras

2016 ◽  
Vol 15 (07) ◽  
pp. 1650134
Author(s):  
Fengxia Gao ◽  
Shilin Yang
Keyword(s):  
Hopf Algebras ◽  
Characteristic Zero ◽  
Jacobson Radical ◽  
Simple Module ◽  
Simple Algebra ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Finite Dimensional ◽  
Module Algebras

Let [Formula: see text] be an algebraically closed field of characteristic zero. For all eight-dimensional non-semisimple Hopf algebras [Formula: see text] which are either pointed or unimodular, we characterrize all finite-dimensional [Formula: see text]-simple module algebras. As a bonus of our approach, it is shown that for any [Formula: see text]-simple algebra, the nilpotent index of the Jacobson radical is at most three.

scholarly journals Classification of Nilpotent Lie Superalgebras of Multiplier-Rank Less than or Equal to 6

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Shuang Lang ◽  
Jizhu Nan ◽  
Wende Liu
Keyword(s):  
Characteristic Zero ◽  
Lie Superalgebras ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Finite Dimensional

In this paper, we classify all the finite-dimensional nilpotent Lie superalgebras of multiplier-rank less than or equal to 6 over an algebraically closed field of characteristic zero. We also determine the covers of all the nilpotent Lie superalgebras mentioned above.

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