A The proof of Theorem 3.1 and its computations
Theorem 3.1 is an essential and basic result on which we work throughout this article.
This result was stated without detailed proofs in [19]. For the reader’s convenience
and to make the exposition self-contained, we would like to form this appendix, which allows
one to become acquainted with related topics without computational difficulties. The computations
involved in the proof of this theorem are extremely tedious. It is a formidable project to demonstrate
them in more detail in the main text. Let us complete this task in this appendix.
The proof of Theorem 3.1 and its computations.
(1) This result follows immediately from the following calculations:
[y12pr,y13ps]=[x12pr,x13ps]
=[1pr+10010001][10ps+1010001]-[10ps+1010001][1pr+10010001]
=[1pr+1ps+1010001]-[1pr+1ps+1010001]=0,
=[x13pr,x23ps]
=[10pr+1010001][10001ps+1001]-[10001ps+1001][10pr+1010001]
=[10pr+101ps+1001]-[10pr+101ps+1001]=0.
(2) One can directly verify the relation
(A.1)[y12pr,y23ps]=[x12pr,x23ps]=x12pr(1-x12-prx23psx12prx23-ps)x23ps.
Thus it suffices to determine
x12-prx23psx12prx23-ps=[1-pr+10010001][10001ps+1001][1pr+10010001][10001-ps+1001]
=[1-pr+1-pr+s+201ps+1001][1pr+10010001][10001-ps+1001]
=[10-pr+s+201ps+1001][10001-ps+1001]
=[10-pr+s+2010001]
(A.2)=x13-pr+s+1.
Taking (A.2) into (A.1), we obtain
[y12pr,y23ps]=(1+y12)pr[1-(1+y13)-pr+s+1](1+y23)ps.
(3)–(4) These assertions are straightforward to check.
(5) The computational method is similar to that of (2). We present some brief calculations as follows:
[y21pr,y32ps]=[x21pr,x32ps]=x21pr(1-x21-prx32psx21prx32-ps)x32ps,
x21-prx32psx21prx32-ps=[100-pr+110001][1000100ps+11][100pr+110001][1000100-ps+11]
=[100-pr+1100ps+11][100pr+110001][1000100-ps+11]
=[100010pr+s+2ps+11][1000100-ps+11]
=[100010pr+s+201]
=x31pr+s+1,
=(1+y21)pr[1-(1+y31)pr+s+1](1+y32)ps.
(6) Let us consider the relation
(A.3)[y12pr,y1122ps]=[x12pr,x1122ps]=x12pr(1-x12-prx1122psx12prx1122-ps)x1122ps.
One can compute that
x12-prx1122psx12prx1122-ps=[1-pr+10010001][(1+p)ps000(1+p)-ps0001]
×[1pr+10010001][(1+p)-ps000(1+p)ps0001]
=[(1+p)ps-pr+1(1+p)-ps00(1+p)-ps0001][1pr+10010001][(1+p)-ps000(1+p)ps0001]
=[(1+p)pspr+1(1+p)ps-pr+1(1+p)-ps00(1+p)-ps0001][(1+p)-ps000(1+p)ps0001]
=[1pr+1(1+p)2ps-pr+10010001]
=x12pr(1+p)2ps-pr
(A.4)=(1+y12)pr(1+p)2ps-pr.
Combining (A.3) with (A.4) gives
[y12pr,y1122ps]=(1+y12)pr[1-(1+y12)pr(1+p)2ps-pr](1+y1122)ps.
(7)–(11) The computational methods of
(7)–(11) are similar to that of (6).
Below is a list of simple calculations for the reader’s convenience.
The calculation of [y13pr,y1122ps]:
[y13pr,y1122ps]=[x13pr,x1122ps]=x13pr(1-x13-prx1122psx13prx1122-ps)x1122ps,
x13-prx1122psx13prx1122-ps=[10-pr+1010001][(1+p)ps000(1+p)-ps0001]
×[10pr+1010001][(1+p)-ps000(1+p)ps0001]
=[(1+p)ps0-pr+10(1+p)-ps0001][10pr+1010001][(1+p)-ps000(1+p)ps0001]
=[(1+p)ps0pr+1(1+p)ps-pr+10(1+p)-ps0001][(1+p)-ps000(1+p)ps0001]
=[10pr+1(1+p)ps-pr+1010001]
=x13pr(1+p)ps-pr
=(1+y13)pr(1+p)ps-pr,
=(1+y13)pr[1-(1+y13)pr(1+p)ps-pr](1+y1122)ps.
The calculation of [y23pr,y1122ps]:
[y23pr,y1122ps]=[x23pr,x1122ps]=x23pr(1-x23-prx1122psx23prx1122-ps)x1122ps,
x23-prx1122psx23prx1122-ps=[10001-pr+1001][(1+p)ps000(1+p)-ps0001]
×[10001pr+1001][(1+p)-ps000(1+p)ps0001]
=[(1+p)ps000(1+p)-ps-pr+1001][10001pr+1001][(1+p)-ps000(1+p)ps0001]
=[(1+p)ps000(1+p)-pspr+1(1+p)-ps-pr+1001][(1+p)-ps000(1+p)ps0001]
=[10001pr+1(1+p)-ps-pr+1001]
=x23pr(1+p)-ps-pr
=(1+y23)pr(1+p)-ps-pr,
=(1+y23)pr[1-(1+y23)pr(1+p)-ps-pr](1+y1122)ps.
The calculation of [y12pr,y2233ps]:
[y12pr,y2233ps]=[x12pr,x2233ps]=x12pr(1-x12-prx2233psx12prx2233-ps)x2233ps,
x12-prx2233psx12prx2233-ps=[1-pr+10010001][1000(1+p)ps000(1+p)-ps]
bmatrix×[1pr+10010001][1000(1+p)-ps000(1+p)ps]
bmatrix=[1-pr+1(1+p)ps00(1+p)ps000(1+p)-ps][1pr+10010001][1000(1+p)-ps000(1+p)ps]
bmatrix=[1pr+1-pr+1(1+p)ps00(1+p)ps000(1+p)-ps][1000(1+p)-ps000(1+p)ps]
=[1pr+1(1+p)-ps-pr+10010001]
=x12pr(1+p)-ps-pr
=(1+y12)pr(1+p)-ps-pr,
=(1+y12)pr[1-(1+y12)pr(1+p)-ps-pr](1+y2233)ps.
The calculation of [y13pr,y2233ps]:
[y13pr,y2233ps]=[x13pr,x2233ps]=x13pr(1-x13-prx2233psx13prx2233-ps)x2233ps,
x13-prx2233psx13prx2233-ps=[10-pr+1010001][1000(1+p)ps000(1+p)-ps]
×[10pr+1010001][1000(1+p)-ps000(1+p)ps]
=[10-pr+1(1+p)-ps0(1+p)ps000(1+p)-ps][10pr+1010001][1000(1+p)-ps000(1+p)ps]
=[10pr+1-pr+1(1+p)-ps0(1+p)ps000(1+p)-ps][1000(1+p)-ps000(1+p)ps]
=[10pr+1(1+p)ps-pr+1010001]
=x13pr(1+p)ps-pr
=(1+y13)pr(1+p)ps-pr,
=(1+y13)pr[1-(1+y13)pr(1+p)ps-pr](1+y2233)ps.
The calculation of [y23pr,y2233ps]:
[y23pr,y2233ps]=[x23pr,x2233ps]=x23pr(1-x23-prx2233psx23prx2233-ps)x2233ps,
x23-prx2233psx23prx2233-ps=[10001-pr+1001][1000(1+p)ps000(1+p)-ps]
×[10001pr+1001][1000(1+p)-ps000(1+p)ps]
=[1000(1+p)ps-pr+1(1+p)-ps00(1+p)-ps][10001pr+1001][1000(1+p)-ps000(1+p)ps]
=[1000(1+p)pspr+1(1+p)ps-pr+1(1+p)-ps00(1+p)-ps][1000(1+p)-ps000(1+p)ps]
=[10001pr+1(1+p)2ps-pr+1001]
=x23pr(1+p)2ps-pr
=(1+y23)pr(1+p)2ps-pr,
=(1+y23)pr[1-(1+y23)pr(1+p)2ps-pr](1+y2233)ps.
(12) In light of the relation
(A.5)[y12pr,y21ps]=[x12pr,x21ps]=x12pr(1-x12-prx21psx12prx21-ps)x21ps,
it is sufficient for us to compute
x12-prx21psx12prx21-ps=[1-pr+10010001][100ps+110001][1pr+10010001][100-ps+110001]
=[1-pr+s+2-pr+10ps+110001][1pr+10010001][100-ps+110001]
=[1-pr+s+2-p2r+s+30ps+11+pr+s+20001][100-ps+110001]
(A.6)=[1-pr+s+2+p2(r+s+2)-p2r+s+30-pr+2s+31+pr+s+20001].
Applying the triangular decomposition formula to the matrix in (A.6) yields
[1-pr+s+2+p2(r+s+2)-p2r+s+30-pr+2s+31+pr+s+20001]
=[1-p2r+s+3(1+pr+s+2)-10010001]
×[1-pr+s+2+p2(r+s+2)-p3(r+s+2)(1+pr+s+2)-10001+pr+s+20001]
×[100-pr+2s+3(1+pr+s+2)-110001].
In particular, we see that
[1-pr+s+2+p2(r+s+2)-p2r+s+30-pr+2s+31+pr+s+20001]
(A.7)=(1+y12)-p2r+s+2(1+pr+s+2)-1[(1+pr+s+2)-10001+pr+s+20001](1+y21)-pr+2s+2(1+pr+s+2)-1.
One should note that
(1+pr+s+2)-1=1-pr+s+2+p2(r+s+2)-p3(r+s+2)+p4(r+s+2)+⋯.
It follows from the properties of p-adic integers that there exists one element β such that
(1+pr+s+2)-1=(1+p)β,
where β=β0+β1p+β2p2+…+βr+spr+s+βr+s+1pr+s+1+⋯, βk∈ℤ and
0≤βk≤(p-1). According to the expansion formula of (1+pr+s+2)-1, we can compute all βk. For instance,
β0=β1=…=βr+s=0,βr+s+1=p-1,βr+s+2=(pr+s+1-12-1)modp.
Thus (A.7) can be rewritten as
[1-pr+s+2+p2(r+s+2)-p2r+s+30-pr+2s+31+pr+s+20001]
=(1+y12)-p2r+s+2(1+pr+s+2)-1[(1+p)β000(1+p)-β0001](1+y21)-pr+2s+2(1+pr+s+2)-1
(A.8)=(1+y12)-p2r+s+2(1+pr+s+2)-1(1+y1122)β(1+y21)-pr+2s+2(1+pr+s+2)-1.
Relations (A.5), (A.6) and (A.8) jointly lead to
[y12pr,y21ps]=(1+y12)pr[1-(1+y12)-p2r+s+2(1+pr+s+2)-1(1+y1122)β(1+y21)-pr+2s+2(1+pr+s+2)-1](1+y21)ps.
(13) We have the Lie bracket
[y13pr,y21ps]=[x13pr,x21ps]=x13pr(1-x13-prx21psx13prx21-ps)x21ps.
Note that
x13-prx21psx13prx21-ps=[10-pr+1010001][100ps+110001][10pr+1010001][100-ps+110001]
=[10-pr+1ps+110001][10pr+1010001][100-ps+110001]
=[100ps+11pr+s+2001][100-ps+110001]
=[10001pr+s+2001]=x23pr+s+1.
We therefore have
[y13pr,y21ps]=(1+y13)pr[1-(1+y23)pr+s+1](1+y21)ps.
(14)–(18)
The proofs of these assertions are very similar to what we did in the previous ones. The calculations are briefly listed below.
The calculation of [y12pr,y31ps]:
[y12pr,y31ps]=[x12pr,x31ps]=x12pr(1-x12-prx31psx12prx31-ps)x31ps,
x12-prx31psx12prx31-ps=[1-pr+10010001][100010ps+101][1pr+10010001][100010-ps+101]
=[1-pr+10010ps+101][1pr+10010001][100010-ps+101]
=[100010ps+1pr+s+21][100010-ps+101]
=[1000100pr+s+21]=x32pr+s+1,
=(1+y12)pr[1-(1+y32)pr+s+1](1+y31)ps.
The calculation of [y13pr,y31ps]:
[y13pr,y31ps]=[x13pr,x31ps]=x13pr(1-x13-prx31psx13prx31-ps)x31ps,
x13-prx31psx13prx31-ps=[10-pr+1010001][100010ps+101][10pr+1010001][100010-ps+101]
=[1-pr+s+20-pr+1010ps+101][10pr+1010001][100010-ps+101]
=[1-pr+s+20-p2r+s+3010ps+101+pr+s+2][100010-ps+101]
=[1-pr+s+2+p2(r+s+2)0-p2r+s+3010-pr+2s+301+pr+s+2]
=[10-p2r+s+3(1+pr+s+2)-1010001]
×[1-pr+s+2+p2(r+s+2)-p3(r+s+2)(1+pr+s+2)-100010001+pr+s+2]
×[100010-pr+2s+3(1+pr+s+2)-101]
=(1+y13)-p2r+s+2(1+pr+s+2)-1[(1+pr+s+2)-100010001+pr+s+2](1+y31)-pr+2s+2(1+pr+s+2)-1
=(1+y13)-p2r+s+2(1+pr+s+2)-1[(1+pr+s+2)-10001+pr+s+20001]
×[1000(1+pr+s+2)-10001+pr+s+2](1+y31)-pr+2s+2(1+pr+s+2)-1
=(1+y13)-p2r+s+2(1+pr+s+2)-1[(1+p)β000(1+p)-β0001]
×[1000(1+p)β000(1+p)-β](1+y31)-pr+2s+2(1+pr+s+2)-1
=(1+y13)-p2r+s+2(1+pr+s+2)-1(1+y1122)β(1+y2233)β(1+y31)-pr+2s+2(1+pr+s+2)-1,
=(1+y13)pr[1-(1+y13)-p2r+s+2(1+pr+s+2)-1(1+y1122)β
×(1+y2233)β(1+y31)-pr+2s+2(1+pr+s+2)-1](1+y31)ps.
The calculation of [y23pr,y31ps]:
[y23pr,y31ps]=[x23pr,x31ps]=x23pr(1-x23-prx31psx23prx31-ps)x31ps,
x23-prx31psx23prx31-ps=[10001-pr+1001][100010ps+101][10001pr+1001][100010-ps+101]
=[100-pr+s+21-pr+1ps+101][10001pr+1001][100010-ps+101]
=[100-pr+s+210ps+101][100010-ps+101]
=[100-pr+s+210001]=x21-pr+s+1,
=(1+y23)pr[1-(1+y21)-pr+s+1](1+y31)ps.
The calculation of [y13pr,y32ps]:
[y13pr,y32ps]=[x13pr,x32ps]=x13pr(1-x13-prx32psx13prx32-ps)x32ps,
x13-prx32psx13prx32-ps=[10-pr+1010001][1000100ps+11][10pr+1010001][1000100-ps+11]
=[1-pr+s+2-pr+10100ps+11][10pr+1010001][1000100-ps+11]
=[1-pr+s+200100ps+11][1000100-ps+11]
=[1-pr+s+20010001]=x12-pr+s+1,
=(1+y13)pr[1-(1+y12)-pr+s+1](1+y32)ps.
The calculation of [y23pr,y32ps]:
[y23pr,y32ps]=[x23pr,x32ps]=x23pr(1-x23-prx32psx23prx32-ps)x32ps,
x23-prx32psx23prx32-ps=[10001-pr+1001][1000100ps+11][10001pr+1001][1000100-ps+11]
=[10001-pr+s+2-pr+10ps+11][10001pr+1001][1000100-ps+11]
=[10001-pr+s+2-p2r+s+30ps+11+pr+s+2][1000100-ps+11]
=[10001-pr+s+2+p2(r+s+2)-p2r+s+30-pr+2s+31+pr+s+2]
=[10001-p2r+s+3(1+pr+s+2)-1001]
×[10001-pr+s+2+p2(r+s+2)-p3(r+s+2)(1+pr+s+2)-10001+pr+s+2]
×[1000100-pr+2s+3(1+pr+s+2)-11]
=(1+y23)-p2r+s+2(1+pr+s+2)-1[1000(1+pr+s+2)-10001+pr+s+2](1+y32)-pr+2s+2(1+pr+s+2)-1
=(1+y23)-p2r+s+2(1+pr+s+2)-1[1000(1+p)β000(1+p)-β](1+y32)-pr+2s+2(1+pr+s+2)-1
=(1+y23)-p2r+s+2(1+pr+s+2)-1(1+y2233)β(1+y32)-pr+2s+2(1+pr+s+2)-1,
=(1+y23)pr[1-(1+y23)-p2r+s+2(1+pr+s+2)-1(1+y2233)β
×(1+y32)-pr+2s+2(1+pr+s+2)-1](1+y32)ps.
(19)–(24) Let us sketch the proof of (19),
and the remainder will be briefly listed later. Here again, the Lie bracket can be written as
[y1122pr,y21ps]=[x1122pr,x21ps]=x1122pr(1-x1122-prx21psx1122prx21-ps)x21ps.
We compute that
x1122-prx21psx1122prx21-ps=[(1+p)-pr000(1+p)pr0001][100ps+110001]
×[(1+p)pr000(1+p)-pr0001][100-ps+110001]
=[(1+p)-pr00ps+1(1+p)pr(1+p)pr0001][(1+p)pr000(1+p)-pr0001][100-ps+110001]
=[100ps+1(1+p)2pr10001][100-ps+110001]
=[100ps+1(1+p)2pr-ps+110001]
=x21ps(1+p)2pr-ps
=(1+y21)ps(1+p)2pr-ps.
This shows that
[y1122pr,y21ps]=(1+y1122)pr[1-(1+y21)ps(1+p)2pr-ps](1+y21)ps.
The calculation of [y1122pr,y31ps]:
[y1122pr,y31ps]=[x1122pr,x31ps]=x1122pr(1-x1122-prx31psx1122prx31-ps)x31ps,
x1122-prx31psx1122prx31-ps=[(1+p)-pr000(1+p)pr0001][100010ps+101]
×[(1+p)pr000(1+p)-pr0001][100010-ps+101]
=[(1+p)-pr000(1+p)pr0ps+101][(1+p)pr000(1+p)-pr0001][100010-ps+101]
=[100010ps+1(1+p)pr01][100010-ps+101]
=[100010ps+1(1+p)pr-ps+101]
=x31ps(1+p)pr-ps
=(1+y31)ps(1+p)pr-ps,
=(1+y1122)pr[1-(1+y31)ps(1+p)pr-ps](1+y31)ps.
The calculation of [y1122pr,y32ps]:
[y1122pr,y32ps]=[x1122pr,x32ps]=x1122pr(1-x1122-prx32psx1122prx32-ps)x32ps,
x1122-prx32psx1122prx32-ps=[(1+p)-pr000(1+p)pr0001][1000100ps+11]
×[(1+p)pr000(1+p)-pr0001][1000100-ps+11]
=[(1+p)-pr000(1+p)pr00ps+11][(1+p)pr000(1+p)-pr0001][1000100-ps+11]
=[1000100ps+1(1+p)-pr1][1000100-ps+11]
=[1000100ps+1(1+p)-pr-ps+11]
=x32ps(1+p)-pr-ps
=(1+y32)ps(1+p)-pr-ps,
=(1+y1122)pr[1-(1+y32)ps(1+p)-pr-ps](1+y32)ps.
The calculation of [y2233pr,y21ps]:
[y2233pr,y21ps]=[x2233pr,x21ps]=x2233pr(1-x2233-prx21psx2233prx21-ps)x21ps,
x2233-prx21psx2233prx21-ps=[1000(1+p)-pr000(1+p)pr][100ps+110001]
×[1000(1+p)pr000(1+p)-pr][100-ps+110001]
=[100ps+1(1+p)-pr(1+p)-pr000(1+p)pr][1000(1+p)pr000(1+p)-pr][100-ps+110001]
=[100ps+1(1+p)-pr10001][100-ps+110001]
=[100ps+1(1+p)-pr-ps+110001]
=x21ps(1+p)-pr-ps
=(1+y21)ps(1+p)-pr-ps,
=(1+y2233)pr[1-(1+y21)ps(1+p)-pr-ps](1+y21)ps.
The calculation of [y2233pr,y31ps]:
[y2233pr,y31ps]=[x2233pr,x31ps]=x2233pr(1-x2233-prx31psx2233prx31-ps)x31ps,
x2233-prx31psx2233prx31-ps=[1000(1+p)-pr000(1+p)pr][100010ps+101]
×[1000(1+p)pr000(1+p)-pr][100010-ps+101]
=[1000(1+p)-pr0ps+1(1+p)pr0(1+p)pr][1000(1+p)pr000(1+p)-pr][100010-ps+101]
=[100010ps+1(1+p)pr01][100010-ps+101]
=[100010ps+1(1+p)pr-ps+101]
=x31ps(1+p)pr-ps
=(1+y31)ps(1+p)pr-ps,
=(1+y2233)pr[1-(1+y31)ps(1+p)pr-ps](1+y31)ps.
The calculation of [y2233pr,y32ps]:
[y2233pr,y32ps]=[x2233pr,x32ps]=x2233pr(1-x2233-prx32psx2233prx32-ps)x32ps,
x2233-prx32psx2233prx32-ps=[1000(1+p)-pr000(1+p)pr][1000100ps+11]
×[1000(1+p)pr000(1+p)-pr][1000100-ps+11]
=[1000(1+p)-pr00ps+1(1+p)pr(1+p)pr][1000(1+p)pr000(1+p)-pr][1000100-ps+11]
=[1000100ps+1(1+p)2pr1][1000100-ps+11]
=[1000100ps+1(1+p)2pr-ps+11]
=x32ps(1+p)2pr-ps
=(1+y32)ps(1+p)2pr-ps,
=(1+y2233)pr[1-(1+y32)ps(1+p)2pr-ps](1+y32)ps.∎
B Some complicated computations involved in Theorem 4.1
This section is devoted to the proofs of the eight key relations appearing in (4.4).
These subtle and complicated computations are fairly necessary for the proof of Theorem 4.1.
The anonymous reviewer advised us to present these tedious computations for the reader to follow our arguments.
To make our exposition space-saving and self-contained, we compile them into this appendix.
Now let us consider the calculation of [y1pr,wm]∘. We begin by noticing that
[y1pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8y1pr(y1ps)i1(y2ps)i2⋯(y8ps)i8-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y1pr}∘.
Note the fact that [y1pr,y2ps]∘=0, which is due to (3.2). It follows that
[y1pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2y1pr(y3ps)i3⋯(y8ps)i8-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y1pr}∘.
Putting the relation [y1pr,y3ps]∘=y2pr+s+1 into the above identity gives
[y1pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2(y3ps)y1pr(y3ps)i3-1⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y1pr}∘
+∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2y2pr+s+1(y3ps)i3-1⋯(y8ps)i8.
Using [y1pr,y3ps]∘=y2pr+s+1 again, we further get
[y1pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2(y32ps)y1pr(y3ps)i3-2⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y1pr}∘
+∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2y2pr+s+1(y3ps)i3-1⋯(y8ps)i8
+∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2(y3ps)y2pr+s+1(y3ps)i3-2⋯(y8ps)i8.
Repeating the above process until y1pr commutes with all y3ps in the term (y3ps)i3, we arrive at
[y1pr,wm]∘={∑i1=0α1⋯∑i8=0α8ai1⋯i8(y1ps)i1(y2ps)i2(y3ps)i3y1pr⋯(y8ps)i8
-∑i1=0α1⋯∑i8=0α8ai1⋯i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y1pr}∘
+∑i1=0α1⋯∑i8=0α8ai1⋯i8(y1ps)i1(y2ps)i2y2pr+s+1(y3ps)i3-1⋯(y8ps)i8
+∑i1=0α1⋯∑i8=0α8ai1⋯i8(y1ps)i1(y2ps)i2(y3ps)y2pr+s+1(y3ps)i3-2⋯(y8ps)i8
(B.1)+⋯+∑i1=0α1⋯∑i8=0α8ai1⋯i8(y1ps)i1(y2ps)i2(y3ps)i3-1y2pr+s+1⋯(y8ps)i8.
Anyway, we must be careful of the meaning of “=” in (B.1). Here, “=” makes sense against the backdrop of [⋅,⋅]∘.
We now make the best use of (3.2) and compute
[y2pr+s+1,y3ps]=0,
=y2pr+s+1y4ps-y4psy2pr+s+1=-y2pr+2s+2+⋯,
=y2pr+s+1y5ps-y5psy2pr+s+1=-y2pr+2s+2+⋯,
=y2pr+s+1y6ps-y6psy2pr+s+1=-y3pr+2s+2+⋯,
=y2pr+s+1y7ps-y7psy2pr+s+1=y4pr+2s+2+y5pr+2s+2+⋯,
=y2pr+s+1y8ps-y8psy2pr+s+1=y1pr+2s+2+⋯.
We therefore obtain
(B.2)y2pr+s+1y3ps=y3psy2pr+s+2,
(B.3)y2pr+s+1y4ps=y4psy2pr+s+1-y2pr+2s+2+⋯,
(B.4)y2pr+s+1y5ps=y5psy2pr+s+1-y2pr+2s+2+⋯,
(B.5)y2pr+s+1y6ps=y6psy2pr+s+1-y3pr+2s+2+⋯,
(B.6)y2pr+s+1y7ps=y7psy2pr+s+1+y4pr+2s+2+y5pr+2s+2+⋯,
(B.7)y2pr+s+1y8ps=y8psy2pr+s+1+y1pr+2s+2+⋯.
Let us now put the right-hand sides of (B.2)–(B.7) into the following terms of relation (B.1):
∑i1=0α1⋯∑i8=0α8ai1⋯i8(y1ps)i1(y2ps)i2y2pr+s+1(y3ps)i3-1⋯(y8ps)i8
+∑i1=0α1⋯∑i8=0α8ai1⋯i8(y1ps)i1(y2ps)i2(y3ps)y2pr+s+1(y3ps)i3-2⋯(y8ps)i8
(B.8)+⋯+∑i1=0α1⋯∑i8=0α8ai1⋯i8(y1ps)i1(y2ps)i2(y3ps)i3-1y2pr+s+1⋯(y8ps)i8.
It is not difficult to see that there will be many newly produced higher-degree terms in (B.8).
Note that (B.8) heavily depends on the situation [⋅,⋅]∘. These new higher-degree terms
will disappear in (B.8) automatically. Thus we conclude that
y2pr+s+1y3ps=y3psy2pr+s+1,
y2pr+s+1y4ps=y4psy2pr+s+1 modulo new higher-degree terms in (B.8),
y2pr+s+1y5ps=y5psy2pr+s+1 modulo new higher-degree terms in (B.8),
y2pr+s+1y6ps=y6psy2pr+s+1 modulo new higher-degree terms in (B.8),
y2pr+s+1y7ps=y7psy2pr+s+1 modulo new higher-degree terms in (B.8),
y2pr+s+1y8ps=y8psy2pr+s+1 modulo new higher-degree terms in (B.8).
So we have
∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2y2pr+s+1(y3ps)i3-1⋯(y8ps)i8
+∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2(y3ps)y2pr+s+1(y3ps)i3-2⋯(y8ps)i8
+…+∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2(y3ps)i3-1y2pr+s+1⋯(y8ps)i8
=∂wm∂y3psy2pr+s+1.
Once again, here “=” does make sense against the backdrop of [⋅,⋅]∘. Thus (B.1) can be rewritten as
[y1pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2(y3ps)i3y1pr(y4ps)i4⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y1pr}∘+∂wm∂y3psy2pr+s+1.
We treat (y4ps)i4 as we treated (y3ps)i3, and continue the process. After i4 steps, we get
[y1pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1…(y4ps)i4y1pr(y5ps)i5⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y1pr}∘
+∂wm∂y3psy2pr+s+1-2∂wm∂y4psy1pr+s+1.
Let us do the same thing with respect to (y5ps)i5, (y6ps)i6, (y7ps)i7 and (y8ps)i8. We eventually obtain
[y1pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1⋯(y5ps)i5y1pr⋯(y8ps)i8-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y1pr}∘
+∂wm∂y3psy2pr+s+1-2∂wm∂y4psy1pr+s+1+∂wm∂y5psy1pr+s+1
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1⋯(y6ps)i6y1pr(y7ps)i7(y8ps)i8-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y1pr}∘
+∂wm∂y3psy2pr+s+1-2∂wm∂y4psy1pr+s+1+∂wm∂y5psy1pr+s+1+∂wm∂y6psy4pr+s+1
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1⋯(y7ps)i7y1pr(y8ps)i8-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y1pr}∘
+∂wm∂y3psy2pr+s+1-2∂wm∂y4psy1pr+s+1+∂wm∂y5psy1pr+s+1+∂wm∂y6psy4pr+s+1-∂wm∂y7psy8pr+s+1.
This shows that
[y1pr,wm]∘=∂wm∂y3psy2pr+s+1-2∂wm∂y4psy1pr+s+1+∂wm∂y5psy1pr+s+1+∂wm∂y6psy4pr+s+1-∂wm∂y7psy8pr+s+1,
and the result follows.
The computational techniques and methods of other situations are very similar to the one above, and these calculations are briefly listed below.
The calculation of [y2pr,wm]∘:
By invoking the relation [y1pr,y2ps]∘=[y2pr,y3ps]∘=0 (cf. (3.2))
we have
[y2pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8y2pr(y1ps)i1(y2ps)i2⋯(y8ps)i8-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y2pr}∘
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2(y3ps)i3y2pr(y4ps)i4⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y2pr}∘.
Putting the relation [y2pr,y4ps]∘=-y2pr+s+1 (cf. (3.2)) into the above
equation gives
[y2pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2(y3ps)i3(y4ps)y2pr(y4ps)i4-1⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y2pr}∘
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2(y3ps)i3y2pr+s+1(y4ps)i4-1⋯(y8ps)i8.
Repeating the above process until y2pr commutes with all y4ps in the term (y4ps)i4, we arrive at
[y2pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1⋯(y4ps)i4y2pr⋯(y8ps)i8-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y2pr}∘
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2(y3ps)i3y2pr+s+1(y4ps)i4-1⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1⋯(y4ps)y2pr+s+1(y4ps)i4-2⋯(y8ps)i8+⋯.
It should be remarked that “=” makes sense against the backdrop of [⋅,⋅]∘.
Due to those relations in (3.2), y2pr+s+1 commutes with
y4ps, y5ps, y6ps, y7ps and y8ps in the sense of modulo newly produced higher-degree terms.
So the above equation can be rewritten as
[y2pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1⋯(y4ps)i4y2pr⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y2pr}∘-∂wm∂y4psy2pr+s+1.
Let us do the same thing with respect to (y5ps)i5, (y6ps)i6, (y7ps)i7 and (y8ps)i8. We eventually obtain
[y2pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1⋯(y5ps)i5y2pr(y6ps)i6(y7ps)i7(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y2pr}∘-∂wm∂y4psy2pr+s+1-∂wm∂y5psy2pr+s+1
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1⋯(y6ps)i6y2pr(y7ps)i7(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y2pr}∘-∂wm∂y4psy2pr+s+1-∂wm∂y5psy2pr+s+1-∂wm∂y6psy3pr+s+1
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1⋯(y7ps)i7y2pr(y8ps)i8-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y2pr}∘
-∂wm∂y4psy2pr+s+1-∂wm∂y5psy2pr+s+1-∂wm∂y6psy3pr+s+1+∂wm∂y7ps(y4pr+s+1+y5pr+s+1).
This illustrates that
[y2pr,wm]∘=-∂wm∂y4psy2pr+s+1-∂wm∂y5psy2pr+s+1-∂wm∂y6psy3pr+s+1+∂wm∂y7ps(y4pr+s+1+y5pr+s+1)+∂wm∂y8psy1pr+s+1.
The calculation of [y3pr,wm]∘:
It follows from the relation [y1pr,y3ps]∘=y2pr+s+1 (cf. (3.2)) that
[y3pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8y3pr(y1ps)i1(y2ps)i2⋯(y8ps)i8-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y3pr}∘
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)y3pr(y1ps)i1-1(y2ps)i2⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y3pr}∘-∑i1=0α1…∑i8=0α8ai1…i8y2pr+s+1(y1ps)i1-1⋯(y8ps)i8.
Repeating the above process until y3pr commutes with all y1ps in the term (y1ps)i1, we obtain
[y3pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1y3pr(y2ps)i2⋯(y8ps)i8-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y3pr}∘
-∑i1=0α1…∑i8=0α8ai1…i8y2pr+s+1(y1ps)i1-1⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)y2pr+s+1(y1ps)i1-2(y2ps)i2⋯(y8ps)i8+⋯.
We must be careful of the fact that “=” makes sense against the backdrop of [⋅,⋅]∘.
In light of those relations in (3.2), y2pr+s+1 commutes with
yjps (j=1,…,8) in the sense of modulo newly produced higher-degree terms.
Then the above equation can be rewritten as
[y3pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1y3pr(y2ps)i2⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y3pr}∘-∂wm∂y1psy2pr+s+1.
Let us do the same thing with respect to (y2ps)i2, (y3ps)i3,…,(y8ps)i8. We eventually obtain
[y3pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2(y3ps)i3y3pr(y4ps)i4⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y3pr}∘-∂wm∂y1psy2pr+s+1
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1⋯(y4ps)i4y3pr(y5ps)i5⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y3pr}∘-∂wm∂y1psy2pr+s+1+∂wm∂y4psy3pr+s+1
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1⋯(y5ps)i5y3pr(y6ps)i6(y7ps)i7(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y3pr}∘-∂wm∂y1psy2pr+s+1+∂wm∂y4psy3pr+s+1-2∂wm∂y5psy3pr+s+1
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1⋯(y7ps)i7y3pr(y8ps)i8-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y3pr}∘
-∂wm∂y1psy2pr+s+1+∂wm∂y4psy3pr+s+1-2∂wm∂y5psy3pr+s+1+∂wm∂y7psy6pr+s+1.
This implies that
[y3pr,wm]∘=-∂wm∂y1psy2pr+s+1+∂wm∂y4psy3pr+s+1-2∂wm∂y5psy3pr+s+1+∂wm∂y7psy6pr+s+1+∂wm∂y8psy5pr+s+1.
The calculation of [y4pr,wm]∘:
In view of the relation [y1pr,y4ps]∘=-2y1pr+s+1 (cf. (3.2)), we assert that
[y4pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8y4pr(y1ps)i1(y2ps)i2⋯(y8ps)i8-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y4pr}∘
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)y4pr(y1ps)i1-1(y2ps)i2⋯(y8ps)i8-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y4pr}∘
+2∑i1=0α1…∑i8=0α8ai1…i8y1pr+s+1(y1ps)i1-1⋯(y8ps)i8.
Repeating the above process until y4pr commutes with all y1ps in the term
(y1ps)i1, we conclude that
[y4pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1y4pr(y2ps)i2⋯(y8ps)i8-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y4pr}∘
+2∑i1=0α1…∑i8=0α8ai1…i8y1pr+s+1(y1ps)i1-1⋯(y8ps)i8
+2∑i1=0α1…∑i8=0α8ai1…i8(y1ps)y1pr+s+1(y1ps)i1-2(y2ps)i2⋯(y8ps)i8+⋯.
It should be pointed out that “=” makes sense against the backdrop of [⋅,⋅]∘.
Taking into account those relations in (3.2), y1pr+s+1 commutes with
yjps (j=1,…,8) in the sense of modulo newly produced higher-degree terms.
Thus the above equation becomes
[y4pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1y4pr(y2ps)i2⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y4pr}∘+2∂wm∂y1psy1pr+s+1.
Let us do the same thing with respect to (y2ps)i2, (y3ps)i3,…,(y8ps)i8. We eventually get
[y4pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2y4pr(y3ps)i3⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y4pr}∘+2∂wm∂y1psy1pr+s+1+∂wm∂y2psy2pr+s+1
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2(y3ps)i3y4pr(y4ps)i4⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y4pr}∘+2∂wm∂y1psy1pr+s+1+∂wm∂y2psy2pr+s+1-∂wm∂y3psy3pr+s+1
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1⋯(y5ps)i5y4pr(y6ps)i6(y7ps)i7(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y4pr}∘+2∂wm∂y1psy1pr+s+1+∂wm∂y2psy2pr+s+1-∂wm∂y3psy3pr+s+1
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1⋯(y6ps)i6y4pr(y7ps)i7(y8ps)i8-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y4pr}∘
+2∂wm∂y1psy1pr+s+1+∂wm∂y2psy2pr+s+1-∂wm∂y3psy3pr+s+1-2∂wm∂y6psy6pr+s+1
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1⋯(y7ps)i7y4pr(y8ps)i8-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y4pr}∘
+2∂wm∂y1psy1pr+s+1+∂wm∂y2psy2pr+s+1-∂wm∂y3psy3pr+s+1-2∂wm∂y6psy6pr+s+1-∂wm∂y7psy7pr+s+1.
This infers
[y4pr,wm]∘=2∂wm∂y1psy1pr+s+1+∂wm∂y2psy2pr+s+1-∂wm∂y3psy3pr+s+1-2∂wm∂y6psy6pr+s+1-∂wm∂y7psy7pr+s+1+∂wm∂y8psy8pr+s+1.
The calculation of [y5pr,wm]∘:
Considering the relation [y1pr,y5ps]∘=y1pr+s+1 (cf. (3.2)), we have
[y5pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8y5pr(y1ps)i1(y2ps)i2⋯(y8ps)i8-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y5pr}∘
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)y5pr(y1ps)i1-1(y2ps)i2⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y5pr}∘
-∑i1=0α1…∑i8=0α8ai1…i8y1pr+s+1(y1ps)i1-1⋯(y8ps)i8.
Repeating the above process until y5pr commutes with all y1ps in the term (y1ps)i1, we obtain
[y5pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1y5pr(y2ps)i2⋯(y8ps)i8-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y5pr}∘
-∑i1=0α1…∑i8=0α8ai1…i8y1pr+s+1(y1ps)i1-1⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)y1pr+s+1(y1ps)i1-2(y2ps)i2⋯(y8ps)i8+⋯.
Note that “=” does make sense against the backdrop of [⋅,⋅]∘.
For the sake of those relations in (3.2), y1pr+s+1 commutes with
yjps (j=1,…,8) in the sense of modulo newly produced higher-degree terms.
Thus the above equation can be rewritten as
[y5pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1y5pr(y2ps)i2⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y5pr}∘-∂wm∂y1psy1pr+s+1.
Let us do the same thing with respect to (y2ps)i2, (y3ps)i3,…,(y8ps)i8. We eventually get
[y5pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2y5pr(y3ps)i3⋯(y8ps)i8-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y5pr}∘
-∂wm∂y1psy1pr+s+1+∂wm∂y2psy2pr+s+1
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2(y3ps)i3y5pr⋯(y8ps)i8-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y5pr}∘
-∂wm∂y1psy1pr+s+1+∂wm∂y2psy2pr+s+1+2∂wm∂y3psy3pr+s+1
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1⋯(y5ps)i5y5pr(y6ps)i6(y7ps)i7(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y5pr}∘
-∂wm∂y1psy1pr+s+1+∂wm∂y2psy2pr+s+1+2∂wm∂y3psy3pr+s+1
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1⋯(y6ps)i6y5pr(y7ps)i7(y8ps)i8-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y5pr}∘
-∂wm∂y1psy1pr+s+1+∂wm∂y2psy2pr+s+1+2∂wm∂y3psy3pr+s+1+∂wm∂y6psy6pr+s+1
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1⋯(y7ps)i7y5pr(y8ps)i8-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y5pr}∘
-∂wm∂y1psy1pr+s+1+∂wm∂y2psy2pr+s+1+2∂wm∂y3psy3pr+s+1+∂wm∂y6psy6pr+s+1-∂wm∂y7psy7pr+s+1.
This gives that
[y5pr,wm]∘=-∂wm∂y1psy1pr+s+1+∂wm∂y2psy2pr+s+1+2∂wm∂y3psy3pr+s+1+∂wm∂y6psy6pr+s+1-∂wm∂y7psy7pr+s+1-2∂wm∂y8psy8pr+s+1.
The calculation of [y6pr,wm]∘:
It follows from the relation [y1pr,y6ps]∘=y4pr+s+1 (cf. (3.2)) that
[y6pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8y6pr(y1ps)i1(y2ps)i2⋯(y8ps)i8-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y6pr}∘
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)y6pr(y1ps)i1-1(y2ps)i2⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y6pr}∘
-∑i1=0α1…∑i8=0α8ai1…i8y4pr+s+1(y1ps)i1-1⋯(y8ps)i8.
Repeating the above process until y6pr commutes with all y1ps in the term (y1ps)i1, we assert that
[y6pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1y6pr(y2ps)i2⋯(y8ps)i8-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y6pr}∘
-∑i1=0α1…∑i8=0α8ai1…i8y4pr+s+1(y1ps)i1-1⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)y4pr+s+1(y1ps)i1-2(y2ps)i2⋯(y8ps)i8+⋯.
One should remark that “=” makes sense against the backdrop of [⋅,⋅]∘.
In light of those relations in (3.2), y4pr+s+1 commutes with
yjps (j=1,…,8) in the sense of modulo newly produced higher-degree terms.
And hence the above equation becomes
[y6pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1y6pr(y2ps)i2⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y6pr}∘-∂wm∂y1psy4pr+s+1.
Let us do the same thing with respect to (y2ps)i2, (y3ps)i3,…,(y8ps)i8. We conclude that
[y6pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2y6pr(y3ps)i3⋯(y8ps)i8-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y6pr}∘
-∂wm∂y1psy4pr+s+1+∂wm∂y2psy3pr+s+1
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1…(y4ps)i4y6pr⋯(y8ps)i8-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y6pr}∘
-∂wm∂y1psy4pr+s+1+∂wm∂y2psy3pr+s+1+2∂wm∂y4psy6pr+s+1
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1⋯(y5ps)i5y6pr(y6ps)i6(y7ps)i7(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y6pr}∘
-∂wm∂y1psy4pr+s+1+∂wm∂y2psy3pr+s+1+2∂wm∂y4psy6pr+s+1-∂wm∂y5psy6pr+s+1.
This demonstrates that
[y6pr,wm]∘=-∂wm∂y1psy4pr+s+1+∂wm∂y2psy3pr+s+1+2∂wm∂y4psy6pr+s+1-∂wm∂y5psy6pr+s+1-∂wm∂y8psy7pr+s+1.
The calculation of [y7pr,wm]∘:
By invoking the relation [y1pr,y7ps]∘=-y8pr+s+1 (cf. (3.2)), we get
[y7pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8y7pr(y1ps)i1(y2ps)i2⋯(y8ps)i8-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y7pr}∘
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)y7pr(y1ps)i1-1(y2ps)i2⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y7pr}∘+∑i1=0α1…∑i8=0α8ai1…i8y8pr+s+1(y1ps)i1-1⋯(y8ps)i8.
Repeating the above process until y7pr commutes with all y1ps in the term (y1ps)i1, we obtain
[y7pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1y7pr(y2ps)i2⋯(y8ps)i8-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y7pr}∘
+∑i1=0α1…∑i8=0α8ai1…i8y8pr+s+1(y1ps)i1-1⋯(y8ps)i8
+∑i1=0α1…∑i8=0α8ai1…i8(y1ps)y8pr+s+1(y1ps)i1-2(y2ps)i2⋯(y8ps)i8+⋯.
One should remark that “=” makes sense against the backdrop [⋅,⋅]∘.
According to those relations in (3.2), y8pr+s+1 commutes with
yjps (j=1,…,8) in the sense of modulo newly produced higher-degree terms.
So the above equation can be rewritten as
[y7pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1y7pr(y2ps)i2⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y7pr}∘+∂wm∂y1psy8pr+s+1.
Let us do the same thing with respect to (y2ps)i2, (y3ps)i3,…,(y8ps)i8.
We eventually arrive at
[y7pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2y7pr(y3ps)i3⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y7pr}∘
+∂wm∂y1psy8pr+s+1-∂wm∂y2ps(y4pr+s+1+y5pr+s+1)
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2(y3ps)i3y7pr⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y7pr}∘
+∂wm∂y1psy8pr+s+1-∂wm∂y2ps(y4pr+s+1+y5pr+s+1)-∂wm∂y3psy6pr+s+1
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1⋯(y4ps)i4y7pr(y5ps)i5⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y7pr}∘
+∂wm∂y1psy8pr+s+1-∂wm∂y2ps(y4pr+s+1+y5pr+s+1)-∂wm∂y3psy6pr+s+1+∂wm∂y4psy7pr+s+1
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1⋯(y5ps)i5y7pr(y6ps)i6(y7ps)i7(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y7pr}∘
+∂wm∂y1psy8pr+s+1-∂wm∂y2ps(y4pr+s+1+y5pr+s+1)-∂wm∂y3psy6pr+s+1+∂wm∂y4psy7pr+s+1+∂wm∂y5psy7pr+s+1
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1⋯(y7ps)i7(y8ps)i8y7pr-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y7pr}∘
+∂wm∂y1psy8pr+s+1-∂wm∂y2ps(y4pr+s+1+y5pr+s+1)-∂wm∂y3psy6pr+s+1+∂wm∂y4psy7pr+s+1+∂wm∂y5psy7pr+s+1.
This illustrates that
[y7pr,wm]∘=∂wm∂y1psy8pr+s+1-∂wm∂y2ps(y4pr+s+1+y5pr+s+1)-∂wm∂y3psy6pr+s+1+∂wm∂y4psy7pr+s+1+∂wm∂y5psy7pr+s+1.
The calculation of [y8pr,wm]∘:
By the relation [y1pr,y8ps]∘=0 (cf. (3.2)) we know that
[y8pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8y8pr(y1ps)i1(y2ps)i2⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y8pr}∘.
It follows from the relations [y1pr,y8ps]=0 and [y2pr,y8ps]∘=y1pr+s+1 (cf. (3.2)) that
[y8pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)y8pr(y2ps)i2-1(y3ps)i3⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y8pr}∘
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1y1pr+s+1(y2ps)i2-1⋯(y8ps)i8
Repeating the above process until y8pr commutes with all y2ps in the term (y2ps)i2,
we see that
[y8pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2y8pr⋯(y8ps)i8-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y8pr}∘
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1y1pr+s+1(y2ps)i2-1⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)y1pr+s+1(y2ps)i2-2⋯(y8ps)i8+⋯.
Note that “=” does make sense against the backdrop of [⋅,⋅]∘.
Considering those relations in (3.2), y1pr+s+1 commutes with
yjps (j=1,…,8) in the sense of modulo newly produced higher-degree terms.
Thus we can rewrite the above equation as
[y8pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2y8pr⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y8pr}∘-∂wm∂y2psy1pr+s+1.
Let us do the same thing with respect to (y2ps)i2, (y3ps)i3,…,(y8ps)i8.
We eventually obtain
[y8pr,wm]∘={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2(y3ps)i3y8pr⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y8pr}∘
-∂wm∂y2psy1pr+s+1-∂wm∂y3psy5pr+s+1
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1⋯(y4ps)i4y8pr⋯(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y8pr}∘
-∂wm∂y2psy1pr+s+1-∂wm∂y3psy5pr+s+1-∂wm∂y4psy8pr+s+1
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1⋯(y5ps)i5y8pr(y6ps)i6(y7ps)i7(y8ps)i8
-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y8pr}∘
-∂wm∂y2psy1pr+s+1-∂wm∂y3psy5pr+s+1-∂wm∂y4psy8pr+s+1+2∂wm∂y5psy8pr+s+1
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1⋯(y6ps)i6y8pr(y7ps)i7(y8ps)i8-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y8pr}∘
-∂wm∂y2psy1pr+s+1-∂wm∂y3psy5pr+s+1-∂wm∂y4psy8pr+s+1+2∂wm∂y5psy8pr+s+1+∂wm∂y6psy7pr+s+1
={∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1⋯(y6ps)i6(y7ps)i7(y8ps)i8y8pr-∑i1=0α1…∑i8=0α8ai1…i8(y1ps)i1(y2ps)i2⋯(y8ps)i8y8pr}∘
-∂wm∂y2psy1pr+s+1-∂wm∂y3psy5pr+s+1-∂wm∂y4psy8pr+s+1+2∂wm∂y5psy8pr+s+1+∂wm∂y6psy7pr+s+1.
This implies that
[y8pr,wm]∘=-∂wm∂y2psy1pr+s+1-∂wm∂y3psy5pr+s+1-∂wm∂y4psy8pr+s+1+2∂wm∂y5psy8pr+s+1+∂wm∂y6psy7pr+s+1.
