cliffordfilt cooper
PDF · 2 pages · 47.4 KB
Open PDF file
Two-page note by Yaim Cooper, dated May 11, 2005, apparently collected among Phil's Wedge World files. It defines algebras, tensor products, the tensor algebra, the exterior algebra as a quotient, and graded algebras. It then defines the Clifford algebra by the relation v⊗v = |v|^2, explains why it is filtered rather than graded, and states that the associated graded algebra is isomorphic to the exterior algebra.
AI-written summary; may contain errors.
Extracted text (machine-read; may contain errors)
Cli ord Algebras asFiltered Algebras
YaimCooper
May11,2005
1TheTensor Algebra
De nition. AnalgebraisavectorspaceVovera eldFwithamultiplic ation.
Themultiplic ationmustbedistributive and,forevery f2Fandx;y2Vmust
satisfy f(xy)=(fx)y=x(fy).
Ourfavoriteexample ofanalgebra isMn(F)|nbynmatrices overF.It
hasthevectorspace structure overFofFn2,andtheusual matrix multiplication
makesitanalgebra overF.
There areverygeneral waysofde ning thetensor productoftwovector
spaces VandW,buthereIwillsticktoaveryconcrete basis-dep enden tone,for
nite dimensional vectorspaces VandWwithbases fe1;:::engandff1;:::fmg,
respectively.
De nition. Thetensor productofVandWisthevectorspaceVNWis
spannedbyelements oftheformv
w,wherethefollowing rulesaresatis e d,
withaandelement ofthe eldF:
1.(v1+v2)
w=v1
w+v2
w
2.v
(w1+w2)=v
w1+v
w2
3.a(v
w)=(av)
w=v
(aw)
Aside: Forthose whoknowaboutdualspaces, WNVisthespace ofall
linear maps fromVtoW.Soourmostfamiliar example ofanalgebra |nn
matrices |isinfactthetensor productVNV.Oneofthecannonical ways
ofde ning thetensor productVNWisinfactasthespace oflinear maps from
VtoW.
Wecancertainly takethetensor productofavectorspace Vwithitself. We
canalsodothisasmanytimes aswefeellike.
De nition. Fork1,de ne
Tk(V)=VNV:::NV(kfactors)
andT0(V)=F
Now,wede ne thetensor algebra as
De nition. T(V)=L1
k=0Tk(V)
1
2TheExterior Algebra
Thetensor algebra isaring-ithasamultiplication whichisdistributiv eover
thevectorspace addition. InthisringisanidealA(V)generated byallelemen ts
oftheformv
v,v2V.
De nition. Thequotient ofT(V)bytheidealA(V)iscalledtheexterior algebra
ofV,anddenote dbyVV.
Nowwede ne theconcept ofagraded algebra.
De nition. AnalgebraAisgradedifitisthedirectsumofsubspacesA=
A0LA1L:::suchthatAiAjAi+jforalli;j0.Theelements ofAkare
saidtobehomogenous ofdegreek.
Both thetensor algebra andexterior algebras de ned thusfararegraded
algebras, thegraded kthpieceinbothcasesbeinggivenbyk-products ofvectors.
3TheCli ord Algebra
Thede nition ofaCli ord algebra isparallel tothatoftheexterior algebra.
Thistime, weconstruct theidealD(V)generated byallelemen tsoftheform
v
v jvj2.
De nition. TheCli or dalgebraofVC(V)isthequotient T(V)=D(V).
TheCli ord algebra isnolonger graded. Thisisbecause nowwhen one
multiplies twoelemen tstogether, theresulting termmaydropindegree without
becoming zero,bytherelation v
v=jvj2.Forexample, takevitself, ofdegree
1.v
vreduces tojvj2,ofdegree zero,rather thanahomogenous termofdegree
2.Still,theproductofadegree itermanddegree jtermcanstillgivearesult
ofdegree atmosti+j.Sothealgebra isnolonger graded, butitdoeshavea
weakerproperty,thatofbeinga ltered algebra.
De nition. A ltrationofanalgebraAisasequenceofsubspacesA0A1:::
suchthatAiAjAi+jforalli;j0.
Thispropertyisweakerthanbeinggraded inthatwenolonger haveadirect
sumdecomp osition, butrather adecomp osition intoasequence ofinclusiv e
subspaces.
However,wecantaketheCli ord algebra andobtain fromitagraded algebra
again. Wedosobyde ning thesubspaces gr(C(V)p)=C(V)p=C(V)p 1,and
letting gr(C(V))=C(V)0Lgr(C(V)1)Lgr(C(V)2):::
Weendwiththeremark abletheorem,
Theorem 1.gr(C(V))=VV:
2