奈良教育大学学術リポジトリNEAR
On the Action of A(p) on the Polynomial Algebra Z3[X4, X8, X12]
著者 OCHIAI Shoji
journal or
publication title
奈良教育大学紀要. 自然科学
volume 23
number 2
page range 1‑2
year 1974‑11‑15
URL http://hdl.handle.net/10105/2614
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Bull. Nara Univ. Educ, Vol.23, No.2 (Nat.), 1974
On the Action of A(p)
on the Polynomial Algebra Zlfrl,xi,xll')
Shoji Ochiai
(Department of Mathematics, Nara University of Education, Nara, Japan) (Received April 30, 1974)
1. Let A(p) be the free associative algebra generated by the elements {P'}i=l,
2... where dim Pl=2i(p-1) and p is an odd prime. To simplify the notations, we
denote the polynomial algebra Z^Xj, x2,... xn} by ZpCdim Xj, dim x2,... dim xB}
and the generator Xi by (dim X,).
In his paper (_5), Professor N. E. Steenrod defines the action of 13, Pl, P2,... on ZpC.4, 2(p (-1)} and shows, by this action that ZPC4, 2(p fl)} is an unstable polynomial algebra over the Steenrod algebra A(p). Furthermore in the last part of his paper,
he says that any unstable polynomial algebra over A(p) with two generators (4),
(2(p |-1)) is isomorphic to the one constructed by him in C5}- That is, the unstable Hopf action CC2}> Definition p 37} on ZjCA, 2(pfl)} is unique up to equivalent CC2},
Definition p 38}. John Ewing shows in Theorem 7.2 C2} that any unstable Hopf
action on Zp(_4, 4} is equivalent to the split action. This result also means that the unstable Hopf action on ZPC4, 4} is equivalent to the split action. This result also means that the unstable Hopf action ZPC4, 4^ is unique up to equivalent.
In contrast to these results, we show that there are at least two non-equivalent unstable Hopf actions on Z3f^4, 8, 12>
2. As is well known, H*(BSP(3); Z3)=Z3C4, 8, 12} and the action of reduced
power operations on this cohomology algebra is determined by A. Borel and J. P. Serre in Cl> Especially, they obtained Pi(4)=2(4)2 )-2(8). in the next example, Pl(4)=
(4)2 holds. Cleary these two are non-equivalent actions on Z3C4, 8, 12}. This example also shows that the obvious generalization of E. Thomas' theorem CC3}, Theorem 2.1}, CC4}, Theorem 1.4} to />>2 does not hold.
Example.
On Z3C4, 8, 12}, we give the actions P1, P*... by
pi(4)=(4)2 P!(12) =(8)2
iJ2(4)= (4)8 P2(12) =0
Pi(8) =0 P»(12)=2(12)2
P2(8)=0 P*(12)=(8)2 •E (12)
2 Shoji OCHIAl
P3(8) = (12) •E(8) />s(l2)=2(8)4
F*(8) = (8)» P8(12) = (12)3
By these actions, it is clear that Z3C4, 8, 12} is the unstable algebra over the free
algebra A(p). We are left to show that Z3C4, 8, 12} is actually the unstable
polynomial algebre over A(p). We need verify only that the following equations hold.
J?i,,(4)=0 i?,,,(8)=0 i?,,2(8)=0 i?,,3(8)=0 /?3)2(8) =0
#3,3(8) =0
(a/3]
where Rz,b=P*Pb- 2 (-l)"+t 20-0-1-
a-3t
i?1,s(12)=0 fi,,8(12) =0
*,,*(12)=0 i?3>2 (12) =0
#3-3(12) =0
This is done as follows.
(pipi_2P2)(4) =Pi(4)2-2(4)s=2(4)3-2(4)3=0
(pi/u_2P*)(8) =pipi(8) -2P2(8) =0
pij>2(8)=0
piP3(8)-P*(8)=Pi((12))-(8)3=(8)s-(8)3=0 psP2(8) +PB(8) -/3*i:>1(8) =0
P»Pa(8) -fPe(8) -/^pi (8) =ps((12)(8)) =PS((12)) (8) + (12)P»((8))
=2(12)2 •E (8)+(12)2. (8)=0
(pi/»i_2P2)(12) =pi/'1(12) =i31((8)2) =0
PiP2(12) =0
(pips_p4)(i2)=pi(2(12)2)-(8)2(12)=2(Pi((12)2),)- (8)2 •E (12)
=2(2(12)(8)2)-(8)2 - (12)=0
(Pip*+P5)(l2)=pi((8)2. (12))+PS(12)=(8)2. (8)2+2(8)^=0
Therefore, by using the theory of section 4 and Theorem 6.2 in (Jf), we know
that by the operations as above, Z3C4, 8, 12^1 is the unstable polynomial algebra over
A(p).
References
£13 A.Borel and J.P.Serre, Groupes de Lie et puissances reduites de Steenrod, Amer. J. Math., 75(1953) 409-448.
(T} J°nn Ewing, On the type of associative H-space, Preprint Series 1970/71 No. 15. Aarhus Universitet.
(T) Emery Thomas, Steenrod squares and H-spaces, Ann. of Math., vol 77 (1963) 306-317.
£4} , Steenrod squares and H-spaces II, Ann. of Math., 81 (1965) 473-495.
(T] Norman Steenrod, Polynomial algebra over the algebra of cohomology operation, H-spaces, Neuchatel (Suisse) Aout 1970, Lecture notes in Math., vol 196, Springer-Verlag, 1970.
