scholarly journals Semiclassical limits of quantum affine spaces

2009 ◽  
Vol 52 (2) ◽  
pp. 387-407 ◽  
Author(s):  
K. R. Goodearl ◽  
E. S. Letzter
Keyword(s):  
Semiclassical Limit ◽  
Poisson Algebra ◽  
Closed Field ◽  
Algebra Structure ◽  
Affine Spaces ◽  
Torsion Free Subgroup ◽  
Symplectic Leaves ◽  
Semiclassical Limits ◽  
Coordinate Rings ◽  
Torsion Free

AbstractSemiclassical limits of generic multi-parameter quantized coordinate rings A=$\mathcal{O}$q(kn) of affine spaces are constructed and related to A, for k an algebraically closed field of characteristic zero and q a multiplicatively antisymmetric matrix whose entries generate a torsion-free subgroup of k×. A semiclassical limit of A is a Poisson algebra structure on the corresponding classical coordinate ring R=$\mathcal{O}$(kn), and results of Oh, Park, Shin and the authors are used to construct homeomorphisms from the Poisson-prime and Poisson-primitive spectra of R onto the prime and primitive spectra of~A. The Poisson-primitive spectrum of R is then identified with the space of symplectic cores in kn in the sense of Brown and Gordon, and an example is presented (over ℂ) for which the Poisson-primitive spectrum of R is not homeomorphic to the space of symplectic leaves in kn. Finally, these results are extended from quantum affine spaces to quantum affine toric varieties.

On solvable centerless groups of Morley rank 3

1993 ◽  
Vol 58 (2) ◽  
pp. 546-556
Author(s):  
Mark Kelly Davis ◽  
Ali Nesin
Keyword(s):  
Nilpotent Group ◽  
General Structure ◽  
Nilpotent Groups ◽  
Simple Groups ◽  
Closed Field ◽  
Morley Rank ◽  
Nonnilpotent Group ◽  
Finite Morley Rank ◽  
Torsion Free Subgroup ◽  
Torsion Free

We know quite a lot about the general structure of ω-stable solvable centerless groups of finite Morley rank. Abelian groups of finite Morley rank are also well-understood. By comparison, nonabelian nilpotent groups are a mystery except for the following general results:• An ω1-categorical torsion-free nonabelian nilpotent group is an algebraic group over an algebraically closed field of characteristic 0 [Z3].• A nilpotent group of finite Morley rank is the central product of a definable subgroup of finite exponent and of a definable divisible subgroup [N3].• A divisible nilpotent group of finite Morley rank is the direct product of its torsion part (which is central) and of a torsion-free subgroup [N3].However, we do not understand nilpotent groups of bounded exponent. It seems that the classification of nilpotent (but nonabelian) p-groups of finite Morley rank is impossible. Even the nilpotent groups of Morley rank 2 contain insurmountable difficulties [C], [T] . At first glance, this may seem to be an obstacle to proving the Cherlin-Zil'ber conjecture (“simple groups of finite Morley rank are algebraic groups”). Our purpose in this article is to show that if such a group is a definable subgroup of a nonnilpotent group, then it is possible to obtain a classification within the boundaries of our present knowledge. In this respect, our article may be considered as a relief to those who are trying to classify simple groups of finite Morley rank.Before explicitly stating our result, we need the following definition.

scholarly journals Relationships between quantized algebras and their semiclassical limits

2018 ◽  
Vol 20 ◽  
pp. 01008
Author(s):  
Sei-Qwon Oh
Keyword(s):  
Mechanical System ◽  
Semiclassical Limit ◽  
Star Product ◽  
Poisson Algebra ◽  
Nonzero Element ◽  
Point Of View ◽  
Quantum Mechanical System ◽  
Algebraic Point ◽  
Semiclassical Limits ◽  
The Given

A Poisson ℂ-algebra R appears in classical mechanical system and its quantized algebra appearing in quantum mechanical system is a ℂ[[ħ]]-algebra Q = R[[ħ]] with star product * such that for any a,b Є R ⊆ Q, a*b = ab + B1(a,b)ħ + B2(a,b)ħ2 + … subject to {a,b}= ħ-1(a * b ‒ b * a)|ħ=0, … (**) where Bi : R ⨯ R → R are bilinear products. The given Poisson algebra R is recovered from its quantized algebra Q by R = Q/ħQ with Poisson bracket (**), which is called its semiclassical limit. But it seems that the star product in Q is complicate and that Q is difficult to understand at an algebraic point of view since it is too big. For instance, if λ is a nonzero element of ℂ then ħ - λ is a unit in Q and thus a so-called deformation of R, Q/(ħ - λ)Q, is trivial. Hence it seems that we need an appropriate 픽-subalgebra A of Q such that A contains all generators of Q, ħ є A and A is understandable at an algebraic point of view, where 픽 is a subring of C[[ħ]]. Here we discuss how to find nontrivial deformations from quantized algebras and the natural map in [6] from a class of infinite deformations onto its semiclassical limit. The results are illustrated by examples.

Some Results in the Connective K-Theory of Lie Groups

1988 ◽  
Vol 31 (2) ◽  
pp. 194-199
Author(s):  
L. Magalhães
Keyword(s):  
Hopf Algebra ◽  
Lie Groups ◽  
Lie Group ◽  
Algebra Structure ◽  
Hopf Algebra Structure ◽  
K Theory ◽  
Torsion Free ◽  
Connected Lie Group

AbstractIn this paper we give a description of:(1) the Hopf algebra structure of k*(G; L) when G is a compact, connected Lie group and L is a ring of type Q(P) so that H*(G; L) is torsion free;(2) the algebra structure of k*(G2; L) for L = Z2 or Z.

scholarly journals Some properties of complete intersections in “good” projective varieties

1976 ◽  
Vol 61 ◽  
pp. 103-111 ◽  
Author(s):  
Lorenzo Robbiano
Keyword(s):  
Complete Intersection ◽  
Singular Surface ◽  
Complete Intersections ◽  
Closed Field ◽  
Irreducible Curve ◽  
Projective Varieties ◽  
Algebraically Closed Field ◽  
Torsion Free

In [10] it was proved that, if X denotes a non singular surface which is a complete intersection in (k an algebraically closed field of characteristic 0) and C an irreducible curve on X, which is a set-theoretic complete intersection in X, then C is actually a complete intersection in X; the key point was to show that Pic (X) modulo the subgroup generated by the class of is torsion-free.

On a hyperbolic 3-manifold with some special properties

1993 ◽  
Vol 113 (1) ◽  
pp. 87-90 ◽  
Author(s):  
B. Zimmermann
Keyword(s):  
Coxeter Groups ◽  
Universal Covering ◽  
Finite Subgroup ◽  
Heegaard Splitting ◽  
Heegaard Genus ◽  
Normal Torsion ◽  
Covering Group ◽  
Minimal Index ◽  
Torsion Free Subgroup ◽  
Torsion Free

We present a closed hyperbolic 3-manifold M with some surprising properties. The universal covering group of M is a normal torsion-free subgroup of minimal index in one of the nine Coxeter groups G, generated by the reflections in the faces of one of the nine Lannér-tetrahedra (bounded tetrahedra in hyperbolic 3-space all of whose dihedral angles are of the form π/n with n ∈ ℕ see [1] or [3]). The corresponding Coxeter group G splits as a semidirect product G = π1M⋉A, where A is a finite subgroup of G, and G is the only one of the nine Coxeter groups associated to the Lannér-tetrahedra which admits such a splitting (this follows using results in [4]). We derive a presentation of π1M and show that the first homology group H1(M) of M is isomorphic to ℚ11. This is in sharp contrast to other torsion-free (non-normal) subgroups of finite index in Coxeter groups constructed in [1] which all have finite first homology (though it is known that they are all virtually ℚ-representable (see [5], p. 434). It follows from our computations that the Heegaard genus of M is 11, and that there exists a Heegaard splitting of M of genus 11 invariant under the action of the group I+(M) ≌ S5 ⊕ ℚ2 of orientation-preserving isometries of M (we compute this group in [4]), so that the Heegaard genus of M is equal to the equivariant Heegaard genus of the action of I+(M) on M. Moreover M is maximally symmetric in the sense of [4, 6]: the order 120 of the subgroup of index 2 in I+(M) which preserves both handle-bodies of the Heegaard splitting is the maximal possible order of a group of orientation-preserving diffeomorphisms of a handle-body of genus 11. (This maximal order is 12(g—1) for a handle-body of genus g; see [7].) By taking the coverings Mq of M corresponding to the surjections π1M→H1(M) ≌ ℚ11→(ℚq)11 for q ∈ ℕ, we obtain explicitly an infinite series of maximally symmetric hyperbolic 3-manifolds.

scholarly journals A double Poisson algebra structure on Fukaya categories

2015 ◽  
Vol 98 ◽  
pp. 57-76
Author(s):  
Xiaojun Chen ◽  
Hai-Long Her ◽  
Shanzhong Sun ◽  
Xiangdong Yang
Keyword(s):  
Poisson Algebra ◽  
Algebra Structure ◽  
Fukaya Categories

The Symplectic Representation and the Torelli Group

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Finite Index ◽  
Algebraic Structure ◽  
Symplectic Form ◽  
Intersection Number ◽  
Free Subgroup ◽  
Torelli Group ◽  
Residually Finite ◽  
Basic Properties ◽  
Torsion Free Subgroup ◽  
Torsion Free

This chapter discusses the basic properties and applications of a symplectic representation, denoted by Ψ‎, and its kernel, called the Torelli group. After describing the algebraic intersection number as a symplectic form, the chapter presents three different proofs of the surjectivity of Ψ‎, each illustrating a different theme. It also illustrates the usefulness of the symplectic representation by two applications to understanding the algebraic structure of Mod(S). First, the chapter explains how this representation is used by Serre to prove the theorem that Mod(Sɡ) has a torsion-free subgroup of finite index. It thens uses the symplectic representation to prove, following Ivanov, the following theorem of Grossman: Mod(Sɡ) is residually finite. It also considers some of the pioneering work of Dennis Johnson on the Torelli group. In particular, a Johnson homomorphism is constructed and some of its applications are given.

On Complete Intersections Over an Algebraically Non-Closed Field

1986 ◽  
Vol 29 (2) ◽  
pp. 140-145 ◽  
Author(s):  
Maria Grazia Marinari ◽  
Mario Raimondo
Keyword(s):  
Complete Intersection ◽  
Affine Space ◽  
Complete Intersections ◽  
Closed Field ◽  
Affine Variety ◽  
Affine Spaces ◽  
Real Curve

AbstractWe give a criterion in order that an affine variety defined over any field has a complete intersection (ci.) embedding into some affine space. Moreover we give an example of a smooth real curve C all of whose embeddings into affine spaces are c.i.; nevertheless it has an embedding into ℝ3 which cannot be realized as a c.i. by polynomials.

scholarly journals ON ABELIAN COSET GENERALIZED VERTEX ALGEBRAS

2001 ◽  
Vol 03 (02) ◽  
pp. 287-340 ◽  
Author(s):  
HAISHENG LI
Keyword(s):  
Vertex Operator Algebra ◽  
Free Abelian Group ◽  
Vertex Algebra ◽  
Heisenberg Algebra ◽  
Algebra Structure ◽  
Torsion Free Abelian Group ◽  
Main Motivation ◽  
Highest Weight ◽  
Torsion Free ◽  
Vacuum Space

This paper studies the algebraic aspect of a general abelian coset theory with a work of Dong and Lepowsky as our main motivation. It is proved that the vacuum space ΩV (or the space of highest weight vectors) of a Heisenberg algebra in a general vertex operator algebra V has a natural generalized vertex algebra structure in the sense of Dong and Lepowsky and that the vacuum space ΩW of a V-module W is a natural ΩV-module. The automorphism group Aut ΩVΩV of the adjoint ΩV-module is studied and it is proved to be a central extension of a certain torsion free abelian group by C×. For certain subgroups A of Aut ΩVΩV, certain quotient algebras [Formula: see text] of ΩV are constructed. Furthermore, certain functors among the category of V-modules, the category of ΩV-modules and the category of [Formula: see text]-modules are constructed and irreducible ΩV-modules and [Formula: see text]-modules are classified in terms of irreducible V-modules. If the category of V-modules is semisimple, then it is proved that the category of [Formula: see text]-modules is semisimple.

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