郝曉斌,原新鳳

(河南工程學院數(shù)理科學系,河南鄭州 451191)

B2型的基及其基變換

郝曉斌,原新鳳

(河南工程學院數(shù)理科學系,河南鄭州 451191)

令v是未定元,A=Z[v,v-1],A=Q(v)是A的分式域.UA′是A′上相伴于B2型 Cartan矩陣(aij)的量子代數(shù).分析了UA′的子代數(shù)U+A′的兩組包含無限個元素的典范基的結(jié)構(gòu),對于一組基中任一元素,都可以在這組基中找到一個包含該元素的有限集合,同時在另一組基中可以找到一個對應的有限集合,這兩個集合元素個數(shù)相等,兩者元素可互相表出.

量子代數(shù);子代數(shù);基變換

量子群作為經(jīng)典李群、李代數(shù)的基本對稱概念的推廣,有著豐富的代數(shù)、幾何及物理性質(zhì).近二十年來,量子群理論引起了許多數(shù)學家和數(shù)學物理學家的注意,目前這一理論已取得了很大的發(fā)展.例如,Lambe和 Radford等系統(tǒng)地研究了量子群和一般 Hopf代數(shù)與量子 Yang-Baxter方程的解的關系[1];Luszting、Rosso和 Anderson等研究了任意有限維半單李代數(shù)g的量子包絡代數(shù)Uq(g)的表示[2-6].文獻[7]給出了A′上B2型量子代數(shù)的兩組典范基.本文進一步分析這兩組基的結(jié)構(gòu),給出了它們之間基變換.

以上兩組基都有無限個元素.本文的結(jié)論是在一組基中任取一元素,都可以在這組基中找到一個包含該元素的有限集合,對應地在另一組基中找到另一個有限集合,這兩個有限集元素個數(shù)相等,兩者元素可互相表出.

1 B2型量子代 的子代數(shù) 典范基

2 B2型量子代數(shù) 子代數(shù) 基變換

[1]LAMBE L A,RADFORD D E.Algebraic aspects of the quantum Yang-Baxter equation[J].J.Algebra,1993(154):228-288.

[2]ANDERSEN H H,POLP P,WEN K X.Representations of quantum algebras[J].Invent Math,1991(104):1-59.

[3]LUST ING G.Finite d imensional hopf algebras from quantized universal enveloping algebras[J].Amer.Math Soc,1990(3):257-296.

[4]LUST ING G.Canonical bases arising from quantized enveloping algebras[J].Amer.Math Soc,1990(3):447-498.

[5]LUST ING G.Modular representations and quantum groups[J].Contemp.Math,1989(82):59-77.

[6]LUST ING G.Quantum groups at roots 1[J].Geom.Dedicata,1990(35):89-113.

[7]XINanhua.Canonical basis for type B2[J].Journal of Algebra,1999(214):8-21.

Basis and Their Transformation offor TypeB2

HAO Xiaobin,YUAN Xinfeng
(Departm ent of M athem atical and Physical Sciences,Henan Institute of Engineering,Zhengzhou451191,China)

Letvbe an indeterminate,A=Z[v,v-1],A=Q(v)is the fraction field ofis the quantum algebra overA′associated to Cartan matrix(aij)for typeB2.The constructions of two infinite canonical bases of the subalgebraofare discussed.For any element in one basis,there is a finite setwhich includes this element.At the same time,a corresponding finite set can be found in another basis,the two sets above-mentioned have the same cardinality and can be expressed each other.

quantum algebra;subalgebra;the transfo rmation of basis

O153

A

1674-330X(2010)03-0065-03

2010-07-01

河南工程學院博士基金項目(D09008)

郝曉斌 (1974-),男,河南平頂山人,講師,博士,主要從事數(shù)學教學和有限元方法、量子群研究.