Vietnam Journal of Mathematics 33:1 (2005) 19–32
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On an Invariant -T heoret ic D escript ion
of the Lamb da A lgebra*
N guyen Sum
Department of M athematics, Uni versi ty of Quy Nhon,
170 An Duong Vuong, Quy Nhon, Binh Di nh, Vietnam
Received May 12, 2003
Revised Sept emb er 15, 2004
D edicated t o P rofessor Hu`yn h M `ui on the occasion of his sixtieth birt hday
A b st ract T he purp ose of t his paper is t o give a mod-panalogue of t he Lom onaco
invariant -t heoret ic descr ipt ion of t he lamb da algebra for pan odd prime. More pre-
cisely, using mod ular invariant s of t he general linear group GL n=GL (n , Fp)and it s
Borel subgroup Bn, we const ru ct a d ifferent ial algeb ra Q−which is isomor phic t o t he
lamb da algebra Λ= Λ
p.
Int ro d uct ion
For t he last few decades, t he modular invariant t heory has been playing an
im port ant role in st a ble hom ot opy t heory. Singer [9] gave a n int er pr et at ion for
t he dua l of t he lamb da algeb ra Λp, wh ich wa s int rod uced by t he six au t hors
[1], in t erms of mod ula r invariant t heory of t he gen era l lin ear group at t h e
prime p= 2. In [8], Hu ng and t h e a ut h or gave a mod-pa nalogu e of t he Sin ger
invariant -t heoret ic descr ipt ion of t he dua l of t h e lamb da a lgebr a for pan odd
pr ime. Lom onaco [6] also gave an int erpret a t ion for t h e lamb da a lgebra in t er ms
of m odular invar iant t heory of t he Borel su bgroup of t he gener al lin ear gr oup at
p= 2.
∗
T h is wor k was su pp or t ed in p ar t by t he Viet na m N at ion al R esea rch P r ogra m G rant 140801.
20 Nguyen Sum
T h e p urp ose of t his pap er is t o give a m od-pa nalogu e of t he Lomon aco
invariant -t heor et ic descrip t ion of t he lamb da algebr a for pan odd prime. More
precisely, using modu lar invaria nt s of t he general linear grou p GL n=G L (n, Fp)
and it s Borel sub group Bn, we const ruct a d ifferent ia l algebr a Q−which is iso-
morphic t o t he lamb da algebra Λ = Λp. Here and in what follows, Fpdenot es t he
prime field of pelem ent s. R ecall t hat , Λpis t he E1-t erm of t he Adam s sp ect ral
sequence of spheres for pan odd prime, whose E2-t erm is E xt ∗
A(p)(Fp,Fp)where
A(p) den ot es t he mod pSt eenrod algebra, a nd E∞-t erm is a grad ed a lgebr a
associat ed t o t he p-pr ima ry comp onent s of t he st ab le hom ot opy of sph eres.
It sh ould b e not ed t hat t he id ea for t he invariant -t h eoret ic descript ion of t he
lamb da algebra is du e t o Lomonaco, who realizes it for p= 2 in [6]. In t his
pap er, we develope of his work for pany odd prim e. Our m ain cont ribut ions are
t he comput at ions a t odd degrees, wher e t he b eh avior of t h e lamb da algebra is
com plet ely different from that for p= 2.
T he pap er cont ains 4 sect ions. Sec. 1 is a pr elim inary on t he m od ula r invari-
ant t heor y and it s loca lizat ion. In Sec. 2 we const ruct t he differ ent ial algeb ra Q
by using modular invariant t heory and show t hat Qcan b e present ed by a set of
gener at or s a nd some r elat ions on t hem. In Sec. 3 we recall som e result s on t he
la mb da algebr a and show t hat it is isom or phic t o a differ ent ial subalgebra Q−
of Q. F inally, in Sec. 4 we give an Fp-vect or sp ace basis for Q.
1. P relim inaries on t he Invariant T heory
For an odd prime p,let Enb e a n elem ent ary abelian p-group of rank n,andlet
H∗
(B E n)= E(x1,x2,... ,xn)⊗Fp(y1,y
2,... ,y
n)
be the mod-pcohomology ring of En. It is a t ensor produ ct of a n ext erior algebra
on generat or s xiof dimension 1 wit h a polynomial algebra on generat ors yiof
dim ension 2. Here and t h roughout t he pa per, t he coefficient s are t aken over t h e
prime field Fpof pelem ent s.
Let GL n=G L (n , Fp)andBnb e it s Borel subgroup consist ing of all invert -
ible upp er t ria ngular mat rices. T hese gr oups act nat u rally on H∗
(B E n). Let S
b e t he mult iplicat ive subset of H∗
(B E n) gen erat ed by all elem ent s of dim ension
2andlet
Φn=H∗
(B E n)S
b e t h e localiza t ion of H∗
(B E n) obt ain ed by invert ing all elem ent s of S.The
act ion of GL non H∗
(B E n) ext en ds t o a n a ct ion of it s on Φn. We recall here
some resu lt s on t h e invariant rin gs Γn= Φ
G L n
nand ∆ n= Φ
Bn
n.
Let Lk , s and Mk , s denot e t he following gr aded det erminant s (in t he sense of
Mui [3])
On an I nvar iant-T heoreti c Descri pti on of the Lambda Algebra 21
Lk , s =
y1y2... y
k
yp
1yp
2... yp
k
.
.
..
.
.... .
.
.
yps−1
1yps−1
2... yps−1
k
yps+ 1
1yps+ 1
2... yps+ 1
k
.
.
..
.
.... .
.
.
ypk
1ypk
2... ypk
k
,
Mk , s =
x1x2... xk
y1y2... y
k
yp
1yp
2... yp
k
.
.
..
.
.... .
.
.
yps−1
1yps−1
2... yps−1
k
yps+ 1
1yps+ 1
2... yps+ 1
k
.
.
..
.
.... .
.
.
ypk−1
1ypk−1
2... ypk−1
k
.
for 0 ≤s≤k≤nand Mk , k = 0. Weset Lk=Lk , k ,1≤k≤n , L 0= 1. Recall
t ha t Lkis invert ible in Φn.
As is well known Lk , s is divisible by Lk. Dickson inva riant s Qk , s and Mui
invaria nt s Rk , s ,V
k,0≤s≤k, are defined by
Qk , s =Lk , s / L k,R
k , s =Mk , s Lp−2
k,V
k=Lk/ L k−1.
Not e t hat dim Qk , s = 2(pk−ps),dim Rk , s = 2(pk−ps)−1,dim Vk= 2pk−1,
Qk , 0=Lp−1
k,L
k=VkVk−1...V
2V1.
From t h e result s in Dickson [2] a nd Mu i [3, 4.17] we observe
T heorem 1.1. (see Singer [9])
Γn=E(Rn , 0,Rn , 1,... ,Rn , n −1)⊗Fp(Q±1
n , 0,Qn , 1,... ,Qn , n −1).
Following Li–Singer [7], we set
Nk=Mk , k −1Lp−2
k,W
k=Vp−1
k,1≤k≤n.
T hen we have
T heorem 1.2. (see Li–Singer [7])
∆n=E(N1,N2,... ,Nn)⊗Fp(W±1
1,W±1
2,... ,W±1
n).
Forlatteruse,weset
tk=Nk/ Qp−1
k−1,0,w
k=Wk/ Qp−1
k−1,0,1≤k≤n.
Observe t hat d im tk= 2p−3,dim wk= 2p−2. Fr om T h eor em 1.2 we ob t ain
22 Nguyen Sum
Corollary 1.3.
∆n=E(t1,t2,... ,tn)⊗Fp(w±1
1,w±1
2,... ,w±1
n).
Moreover, from Dickson [2], Mui [3], we have
P rop o sit ion 1.4 .
(i) Qn , s =Qp
n−1, s −1+Qp−1
n−1,0Qn−1, s wn,
(ii) Rn , s =Qp−1
n−1,0(Rn−1, s wn+Qn−1, s tn).
2. T he A lge bra Q
In t h is sect ion, we const r uct t he different ial algebra Qby using modular invari-
ant t heory. In Sec. 4, we will sh ow t hat t he la mb da algeba is isom orph ic t o a
subalgebra of Q.
D efi nit io n 2.1. L e t ∆nbe a s i n S ec . 1.Set
∆ = ⊕
n≥0
∆n.
H ere, by con ven t ion , ∆0=Fp.T his is a direct sum of vector spaces over Fp.
R e m a r k . F o r I= (ε1,ε2,... ,εn,i1,i2,... ,in)withεj= 0,1,ij∈Z,set
wI=tε1
1tε2
2...tεn
nwi1+ε1
1wi2+ε2
2...win+εn
n,
even in t he case when som e of εjor ijare zero. For exam ple, t he elem ent
t1∈∆2will b e writ t en as t1t0
2w0
1w0
2, t o b e d ist inguished from t1∈∆1,since
t1=t1t0
2w0
1w0
2. For any n > 0wehaveamonomial
t0
1t0
2...t0
nw0
1w0
2...w0
n∈∆n
which is t h e ident it y of ∆ n. All t hese elem ent s are dist inct in ∆ .
Now we equip ∆ wit h a n algebra st ru ct ure as follows. For any n on-n egat ive
int egers k, ℓ , we define an isomorphism of algebras
μk , ℓ :∆k⊗∆ℓ→∆k+ℓ
by set t ing
μk , ℓ (tε1
1tε2
2...tεk
kwi1+ε1
1wi2+ε2
2...wik+εk
k⊗tσ1
1tσ2
2...tσℓ
ℓwj1+σ1
1wj2+σ2
2...wjℓ+σℓ
ℓ)
=tε1
1tε2
2...tεk
ktσ1
k+ 1 tσ2
k+ 2 ...tσℓ
k+ℓwi1+ε1
1wi2+ε2
2...wik+εk
kwj1+σ1
k+ 1 wj2+σ2
k+ 2 ...wjℓ+σℓ
k+ℓ,
for any i1,i2,... ,ik,j1,j2,... ,jℓ∈Z,ε1,ε2,... ,εk,σ
1,σ
2,... ,σ
ℓ= 0,1.
We assemble μk , ℓ ,k,ℓ ≥0, t o obt ain a mult ip lica t ion
μ:∆ ⊗∆→∆.
T his mult iplicat ion makes ∆ int o an algebra.
For simplicit y, we denot e μ(x⊗y)= x∗yfor a ny elem ent s x , y ∈∆ .
On an I nvar iant-T heoreti c Descri pti on of the Lambda Algebra 23
D efi nit io n 2 .2. L e t Γden ot e t he two-sides ideal of ∆gen erat ed by all elem en t s
of t he form s
t0
1t0
2w−1
1w0
2Qa
2,0Qb
2,1,
t0
1t0
2w−1
1w0
2R2,0Qa
2,0Qb
2,1−R2,1Qa
2,0Qb
2,1,
2t0
1t0
2w1w0
2R2,1Qa
2,0Qb
2,1−R2,0Qa
2,0Qb
2,1,
t0
1t0
2w1w0
2R2,0R2,1Qa
2,0Qb
2,1,
where a, b ∈Z,b≥0.
We define
Q= ∆/Γ
t o be t he quot ient of ∆ by t he id eal Γ.
For any non-n egat ive int eger n, we define a homomorphism
¯
δn:∆n→∆n+ 1
by set t ing
¯
δn(x)= −t1w−1
1∗x+ (−1)d i m xx∗t1w−1
1,
for any homogeneous element x∈∆n. By assembling ¯
δn,n ≥0, we obt a in an
endomorphism
¯
δ:∆ →∆.
T heorem 2.3. T he en dom orphism ¯
δ:∆ →∆in du ces an en dom orphism δ:
Q→Qwhich is a diff eren t ial.
P roof. Let u∈∆nbe a homogeneous element and suppose u∈Γ. From t he
definit ion of Γ we see t ha t uis a sum of elem ent s of t he form
ui∗si∗zi,
where ui∈∆ni,z
i∈∆n−ni−2and siis on e of t he elem ent s given in Definit ion
2.2. T hen ¯
δ(u)isasumofelementsoftheform
−t1w−1
1∗ui∗si∗zi+ (−1)d i m uui∗si∗zi∗t1w−1
1.
Since t1w−1
1∗ui∈∆ni+ 1 ,z
i∗t1w−1
1∈∆n−ni−1,we ob t a in ¯
δ(u)∈Γ. So, ¯
δ
induces an endomorphism
δ:Q→Q.
Now we p rove t hat δδ = 0. It suffices t o ch eck t h at if x∈∆nis a homogeneous
element t hen ¯
δ¯
δ(x)∈Γ. In fact , from t he d efin it ion of ¯
δwe have
¯
δ¯
δ(x)= t1t2w−1
1w−1
2∗x−x∗t1t2w−1
1w−1
2.
A d irect com put at ion using P r op osit ion 1.4 sh ows t hat
R2,0Q−1
2,0=t1t0
2w−1
1w0
2+t0
1t2w0
1w−1
2,
R2,1Q−1
2,0=t0
1t2w−1
1w−1
2.