D_4型量子包络代数的Gelfand-Kirillov维数.pdfVIP

数 学 杂 志 Vo1．35(2015) No．6 ： 垦：( 2 THE GELFAND．KIRILL0V DIM ENS10N 0F QUANTIZED ENVELOPING ALGEBRA OFTYPED6 M IAO Yue，ABDUKADIR Obul (CollegeofMathezntics0ndSyste，nScieces，XinjiangUniversitUrumgi83004~Chin口) Abstract：Inthispaper，weresearchtheproblem ofcomputingtheGelfand—Kirillov dimen— sion ofquantized envelopingalgebraoftypeD4byusingthemethodofcomputingtheGelfand— Kirillovdimensiongivenin[1]andtheGrSbner—Shirshovbasisforquantizedenvelopingalgebraof typeD4givenin[2]．ThemainresultwegetisthattheGelfand—Kirillovdimensionofquantized enveloping algebraoftypeD4is28．Wehopethisresultwillprovidesomeideasto computethe Gelfand—KirillovdimensionofquantizedenvelopingalgebraoftypeDn． Keywords： Gr6bner-·Shirshov basis； Poincar6．．Birkhoff-W itt algebra； weight vector； Gelfand—Kjrjllovdimension 2010MR SubjectClassification： 16P90；17B37 Documentcode：A ArticleID： 0255-·7797(2015)06--1329--12 1 Introduction Incontrastwiththecommutativecase，fornon—commutativealgebrastheclassicalKrull dimensionisusuallynotaveryusefultool，becauseitisdefinedbyusingchainsofprime ideals． Forfinitely generated 一algebrasR．theGelfand—Kirillov dimension isfarbetter invariantandcoincideswith theKrulldimension in thecommutativecase．TheGelfand— Kirillov dimensionmeasurestheasymptoticrateofgrowthofalgebrasandprovidesimpor— tantstructuralinformation，SOthisinvarianthasbecomeoneofthestandardtoolsin the studyoffinitelygeneratedinfinitedimensionalalgebras．Butingeneral，theGelfand—Kirillov dimensionisextremelyhardtocompute． In[1]，theauthorsgaveadetaileddiscussionoftheGelfand—Kirillovdimensionoffinitely generated 一algebrasandmodulesoverthem．andalsointroducedanalgorithm tocompute theGelfand—Kirillovdimensionofseveralclassicalandnon—classical