Non-semisimple Hopf algebras over a field of characteristic p ≠ 0 using Personal Computer.

Kenji Yokogawa

§1 Abstract and Introduction

Let K be a field of characteristic p ≠ 0 and H be a Hopf algebra of dimension p over K generated by a p-nilpotent element x, x^{p} = 0 (simply we say a p-nilpotent Hopf algebra. We doesnot assume that p-nilpotent Hopf algebra is cocommutative). A group ring H = K[σ], σ^{p} = 1, is such a Hopf algebra with a p-nilpotent generator σ−1 (we call Hopf algebra of this type as a Hopf algebra of automorphism type). A Hopf algebra generated by a derivation d, d^{p} = 0, is also such a Hopf algebra (we call as a Hopf algebra of derivation type). Another type of p-nilpotent Hopf algebra is not known. So the question whether these two types of Hopf algebra are the all of p-nilpotent Hopf algebras arises. p-nilpotent Hopf algebras are tiny, primitive, fundamental Hopf algebras and it seems easy to classify these algberas. But the recent method using semisimplicity does not work for these algebras and there is no useful methods for these Hopf algebras. Classifying these algberas needs too laborious calculations. Especially to check coassociativity is laborious. And this prevents to get good conjectures to these algebras.

Using a personal computer, the author tried to investigate these algebra in lower characteristic cases. And examining the great amount of calculations in details, the author found some induction rules and succeeded to determine the all p-nilpotent Hopf algebras in an arbitrary non-zero characteristic case. We get for a suitable generator x of a p-nilpotent Hopf algebra, its diagonal map Δ has a following form; Δ(x) = x ⊗ 1 + 1 ⊗ x + cx^{p−1} ⊗ x − 2^{−1}cx^{p−2} ⊗ x^{2} + 3^{−1}cx^{p−3} ⊗ x^{3} + ⋯ − (p − 3)^{−1}cx^{3} ⊗ x^{p−3} + (p − 2)^{−1}cx^{2} ⊗ x^{p−2} − (p − 1)^{−1}cx ⊗ x^{p−1}, c ∈ K. As a Corollary, we get if K is a prime field then there exists exactly p non-isomorphic p-nilpotent Hopf algebras and if K is an algebraically closed field then there exists exactly 2 non-isomorphic p-nilpotent Hopf algebras .

† ここでは, Mathematica の使い方を中心に解説するが, 証明を付けた物はその内発表予定である.

In this paper, we discuss the utilization of a personal computer to mathematics.

Throughout of this paper, K denotes a field of characteristic p ≠ 0, H denotes a p-nilpotent Hopf algebra over K unless otherwise stated. We shall use the terminology of [Sw]. Especially, the diagonal map of H is denoted by Δ, the augmentation map is denoted by ε, and the antipode is denoted by S. ⊗ means ⊗_{K}.

§2 Mathematical Soft Ware

数式処理 Software として、 Mathematica, Maple などがある．これらの Software の主な機能として、数値計算, シンボル計算（文字式の計算） , 組み込み関数を用いた計算, グラフの表示, 行列計算, 連立一次方程式, 微積分, 微分方程式などがある. これらの機能の内, 特にシンボル計算（文字式の計算）に注目する.

Hopf algebra の coassociativity の計算が大変な訳であるが, 具体的にどれくらい大変かを示してみよう. この例では, 後ほど述べる Lemma 1 を用いて, Δ(x) を書き表している. p = 5 とし, Δ : H → H ⊗H, Δ(x) = x⊗1+1⊗x+c_{11}x⊗x+c_{21}x^{2}⊗x+c_{12}x⊗ x^{2}+c_{31}x^{3}⊗x+c_{22}x^{2}⊗x^{2}+c_{13}x⊗x^{3}+c_{41}x^{4}⊗x+c_{32}x^{3}⊗x^{2}+c_{23}x^{2}⊗x^{3}+c_{14}x⊗ x^{4} + c_{42}x^{4} ⊗ x^{2} + c_{33}x^{3} ⊗ x^{3} + c_{24}x^{2} ⊗ x^{4} + c_{43}x^{4} ⊗ x^{3} + c_{34}x^{3} ⊗ x^{4} + c_{44}x^{4} ⊗ x^{4} と書く, 但し x は p-nilpotent generator .

coassociativity を調べるには, (Δ ⊗ 1)Δ(x)， (1 ⊗ Δ)Δ(x) を計算する訳であるが, (Δ ⊗ 1)Δ(x) の部分は (Δ⊗1)Δ(x) = Δ(x)⊗1+1⊗x+c_{11}Δ(x)⊗x+c_{21}Δ(x)^{2} ⊗x+c_{21}Δ(x)⊗x^{2} + c_{31}Δ(x)^{3} ⊗ x + c_{22}Δ(x)^{2} ⊗ x^{2} + c_{13}Δ(x) ⊗ x^{3} + c_{41}Δ(x)^{4} ⊗ x + c_{32}Δ(x)^{3} ⊗ x^{2} + c_{23}Δ(x)^{2} ⊗ x^{3} + c_{14}Δ(x) ⊗ x^{4} + c_{42}Δ(x)^{4} ⊗ x^{2} + c_{33}Δ(x)^{3} ⊗ x^{3} + c_{24}Δ(x)^{2} ⊗ x^{4} + c_{43}Δ(x)^{4} ⊗ x^{3} + c_{34}Δ(x)^{3} ⊗ x^{4} + c_{44}Δ(x)^{4} ⊗ x^{4} となる．最後の項 c_{44}Δ(x)^{4} ⊗x^{4} = c_{44}{x⊗1+1⊗x+c_{11}x⊗x+⋯+c_{44}x^{4} ⊗x^{4}}^{4} ⊗x^{4} である.

膨大な計算量であるのが判るであろう. しかも調べた所では, p = 7 まで計算しないと一般形にたどり着かない. とても人間業では無い．この計算の部分を computer + 数式処理 Software に任せ, その結果から予想を出し、それを証明する訳である.

注 1: 私が使用した数式処理 SoftWare は「Mathematica」であるが, それ以外に「Maple」がある. 色々な点で「Maple」の方が優れているように思われる.

注 2: 最初 p-nilpotent Hopf algebra に cocommutative を仮定し, その後一般の場合を調べた. 数式処理 Software の利用方法を説明するのには, cocommutative の場合の方が適切なので, cocommutative を仮定して説明する.

§3 Preliminary

Lemma 1. Let H be a Hopf algebra generated by a p-nilpotent element x (throughout H denotes such a Hopf algebra) . Write Δ(x) = ∑ c_{ij}x^{i}⊗x^{j} (0 ≤ i, j < p), then c_{00} = 0, c_{10} = 1, c_{i0} = 0(1 ≤ i < p), c_{01} = 1 and c_{0j} = 0 (1 ≤ j < p).

Proof. Use relations (ε ⊗ 1)Δ(x) = (1 ⊗ ε)Δ(x) = x.

Δ(x) = ∑ c_{ij}x^{i} ⊗ x^{j} として, multiplicative, linear に拡張すれば, Lemma 1 から常に Δ(x)^{p} = Δ(x^{p}) = 0 を満たすので, Δ は algebra-homo. になる. 即ち,

Lemma 2. Let c_{ij} be coefficients satisfying the relation of Lemma 1. We write Δ(x) = ∑ c_{ij}x^{i} ⊗ x^{j} and extend to Δ : H → H ⊗ H multiplicatively and linearly, then Δ is an algebra homomorphism.

Lemma 3. y ∈ H is also an p-nilpotent generator if and only if y = a_{1}x + a_{2}x^{2} + ... + a_{p − 1}x^{p − 1} with a_{1} a unit of K.

Lemma 3 から、良い generator を取る事が大切になる訳である. 実際に p = 5 の場合に mathematica を動かした結果を見せ, どのような generator を取れば良いか説明する.

Deltas = Δ(x), H = K[x], H ⊗ H = K[X, Y], X = x ⊗ 1, Y = 1 ⊗ x. degree 4 以下の部分だけを考えるので, c_{41} = c_{32} = c_{23} = c_{14} = c_{43} = c_{34} = 0 とし, Δ(x) = X + Y + c_{11}XY + c_{21}X^{2}Y + c_{12}XY^{2} + c_{31}X^{3}Y + c_{22}X^{2}Y^{2} + c_{13}XY^{3} と置く. 生成元変換 s = a_{1}x + a_{2}x^{2} + a_{3}x^{3} + a_{4}x^{4} を行い, U = s ⊗ 1, V = 1 ⊗ s と置き, Δ(s) を U, V の綺麗な形で表わす訳である. 逆変換を, x = b_{1}s + b_{2}s^{2} + b_{3}s^{3} + b_{4}s^{4}, X = b_{1}U + b_{2}U^{2} + b_{3}U^{3} + b_{4}U^{4}, Y = b_{1}V + b_{2}V^{2} + b_{3}V^{3} + b_{4}V^{4} として置く. （以下 X,Y の代わりに S,T と書く）

以下は, Mathematica の結果であるが, Mathematica では c_{11} を c11, X^{2} を X ＾ 2 等と表わしている.

In[7]:=Deltax = X + Y + c11 X Y + c21 X ＾ 2 Y + c12 X Y ＾ 2 + c31 X ＾ 3 Y + c22 X ＾ 2 Y ＾ 2 + c13 X Y ＾ 3; In[8]:=S = b1 U + b2 U ＾ 2 + b3 U ＾ 3 + b4 U ＾ 4; In[9]:=T = b1 V + b2 V ＾ 2 + b3 V ＾ 3 + b4 V ＾ 4; In[10]:= deltas = a1 Deltax + a2 Deltax ＾ 2 + a3 Deltax ＾ 3 + a4 Deltax ＾ 4 Out[10]= a1 ((X + Y + c11 X Y + c21 X ＾ 2 Y + c31 X ＾ 3 Y + c12 X Y ＾ 2 + c22 X ＾ 2 Y ＾ 2 + c13 X Y ＾ 3)) + a2 ((X + Y + c11 X Y + c21 X ＾ 2 Y + c31 X ＾ 3 Y + c12 X Y ＾ 2 + c22 X ＾ 2 Y ＾ 2 + c13 X Y ＾ 3)) ＾ 2 + a3 ((X + Y + c11 X Y + c21 X ＾ 2 Y + c31 X ＾ 3 Y + c12 X Y ＾ 2 + c22 X ＾ 2 Y ＾ 2 + c13 X Y ＾ 3)) ＾ 3 + a4 ((X + Y + c11 X Y + c21 X ＾ 2 Y + c31 X ＾ 3 Y + c12 X Y ＾ 2 + c22 X ＾ 2 Y ＾ 2 + c13 X Y ＾ 3)) ＾ 4) In[11]:= deltas = deltas /. X −＞ S 注: −＞は代入 Out[11]=a1 ((b1 U + b2 U ＾ 2 + b3 U ＾ 3 + b4 U ＾ 4 + Y + c11 ((b1 U + b2 U ＾ 2 + b3 U ＾ 3 + b4 U ＾ 4)) Y + c21 ((b1 U + b2 U ＾ 2 + b3 U ＾ 3 + b4 U ＾ 4)) ＾ 2 Y + c31 ((b1 U + b2 U ＾ 2 + b3 U ＾ 3 + b4 U ＾ 4)) ＾ 3 Y + c12 ((b1 U + b2 U ＾ 2 + b3 U ＾ 3 + b4 U ＾ 4)) Y ＾ 2 + c22 ((b1 U + b2 U ＾ 2 + b3 U ＾ 3 + b4 U ＾ 4)) ＾ 2 Y ＾ 2 + c13 ((b1 U + b2 U ＾ 2 + b3 U ＾ 3 + b4 U ＾ 4)) Y ＾ 3)) + a2(略) + a3(略) + a4(略) In[12]:= deltas = deltas /. Y −＞ T Out[12]= a1 ((b1 U + b2 U ＾ 2 + b3 U ＾ 3 + b4 U ＾ 4 + b1 V + b2 V ＾ 2 + b3 V ＾ 3 + b4 V ＾ 4 + c11 ((b1 U + b2 U ＾ 2 + b3 U ＾ 3 + b4 U ＾ 4)) ((b1 V + b2 V ＾ 2 + b3 V ＾ 3 + b4 V ＾ 4)) + c21 ((b1 U + b2 U ＾ 2 + b3 U ＾ 3 + b4 U ＾ 4)) ＾ 2 ((b1 V + b2 V ＾ 2 + b3 V ＾ 3 + b4 V ＾ 4)) + c31 ((b1 U + b2 U ＾ 2 + b3 U ＾ 3 + b4 U ＾ 4)) ＾ 3 ((b1 V + b2 V ＾ 2 + b3 V ＾ 3 + b4 V ＾ 4)) + c12 ((b1 U + b2 U ＾ 2 + b3 U ＾ 3 + b4 U ＾ 4)) ((b1 V + b2 V ＾ 2 + b3 V ＾ 3 + b4 V ＾ 4)) ＾ 2 + c22 ((b1 U + b2 U ＾ 2 + b3 U ＾ 3 + b4 U ＾ 4)) ＾ 2 ((b1 V + b2 V ＾ 2 + b3 V ＾ 3 + b4 V ＾ 4)) ＾ 2 + c13 ((b1 U + b2 U ＾ 2 + b3 U ＾ 3 + b4 U ＾ 4)) ((b1 V + b2 V ＾ 2 + b3 V ＾ 3 + b4 V ＾ 4)) ＾ 3)) + + a2(略) + a3(略) + a4(略) + 略注: 以下で 5 乗で cut off In[13]:= deltas = PolynomialRemainder[deltas, U ＾ 5, U]; In[14]:=deltas = PolynomialRemainder[deltas, V ＾ 5, V]; In[15]:=SS = Expand[deltas, U,V] Out[15]= a1 b1 U + a2 b1 ＾ 2 U ＾ 2 + a1 b2 U ＾ 2 + a3 b1 ＾ 3 U ＾ 3 + 2 a2 b1 b2 U ＾ 3 + a1 b3 U ＾ 3 + a4 b1 ＾ 4 U ＾ 4 + 3 a3 b1 ＾ 2 b2 U ＾ 4 + a2 b2 ＾ 2 U ＾ 4 + 2 a2 b1 b3 U ＾ 4 + a1 b4 U ＾ 4 + ((a1 b1 + 2 a2 b1 ＾ 2 U + a1 b1 ＾ 2 c11 U + 3 a3 b1 ＾ 3 U ＾ 2 + 2 a2 b1 b2 U ＾ 2 + 2 a2 b1 ＾ 3 c11 U ＾ 2 +略)V + 長い省略 In[16]:= CoefficientList[SS, {U,V}] Out[16]= {{0, a1 b1 = 1, a2 b1 ＾ 2 + a1 b2, a3 b1 ＾ 3 + 2 a2 b1 b2 + a1 b3, a4 b1 ＾ 4 + 3 a3 b1 ＾ 2 b2 + a2 b2 ＾ 2 + 2 a2 b1 b3 + a1 b4\}, \{a1 b1, \underline{2 a2 b1 ＾ 2 + a1 b1 ＾ 2 c11} , 3 a3 b1 ＾ 3 + 2 a2 b1 b2 + 2 a2 b1 ＾ 3 c11 + a1 b1 b2 c11 + a1 b1 ＾ 3 c12, 4 a4 b1 ＾ 4 + 6 a3 b1 ＾ 2 b2 + 2 a2 b1 b3 + 3 a3 b1 ＾ 4 c11 + 4 a2 b1 ＾ 2 b2 c11 + a1 b1 b3 c11 + 2 a2 b1 ＾ 4 c12 + 2 a1 b1 ＾ 2 b2 c12 + a1 b1 ＾ 4 c13, 長い省略. underline 部分に注目して In[17]:= SSS = SS /. a2 −＞ - (1/2) a1 c11 Out[17]=a1 b1 U + a1 b2 U ＾ 2 - 1/2 a1 b1 ＾ 2 c11 U ＾ 2 + a3 b1 ＾ 3 U ＾ 3 + a1 b3 U ＾ 3 - a1 b1 b2 c11 U ＾ 3 + a4 b1 ＾ 4 U ＾ 4 + 3 a3 b1 ＾ 2 b2 U ＾ 4 + a1 b4 U ＾ 4 - 1/2 a1 b2 ＾ 2 c11 U ＾ 4 - a1 b1 b3 c11 U ＾ 4 + ((\underline{a1 b1 + 3 a3 b1 ＾ 3 U ＾ 2} - a1 b1 ＾ 3 c11 ＾ 2 U ＾ 2 + a1 b1 ＾ 3 c21 U ＾ 2 + 4 a4 b1 ＾ 4 U ＾ 3 + 6 a3 b1 ＾ 2 b2 U ＾ 3 + 3 a3 b1 ＾ 4 c11 U ＾ 3 - 2 a1 b1 ＾ 2 b2 c11 ＾ 2 U ＾ 3 + 2 a1 b1 ＾ 2 b2 c21 U ＾ 3 - a1 b1 ＾ 4 c11 c21 U ＾ 3 + a1 b1 ＾ 4 c31 U ＾ 3 + 12 a4 b1 ＾ 3 b2 U ＾ 4 + 3 a3 b1 b2 ＾ 2 U ＾ 4 + 6 a3 b1 ＾ 2 b3 U ＾ 4 + 4 a4 b1 ＾ 5 c11 U ＾ 4 + 9 a3 b1 ＾ 3 b2 c11 U ＾ 4 - a1 b1 b2 ＾ 2 c11 ＾ 2 U ＾ 4 - 2 a1 b1 ＾ 2 b3 c11 ＾ 2 U ＾ 4 + 3 a3 b1 ＾ 5 c21 U ＾ 4 + a1 b1 b2 ＾ 2 c21 U ＾ 4 + 2 a1 b1 ＾ 2 b3 c21 U ＾ 4 - 3 a1 b1 ＾ 3 b2 c11 c21 U ＾ 4 + 3 a1 b1 ＾ 3 b2 c31 U ＾ 4 - a1 b1 ＾ 5 c11 c31 U ＾ 4)) \underline{V} + 長い省略これで UV の係数は 0 にする事が出来る．次に, UV の係数が 0 と言う性質を変えない変換を考える. その為に最初から XY の係数を 0 とし、 UV の係数が 0 になる変換の条件を調べる. In[18]:=SS = SS /. c11 −＞ 0 Out[18]=a1 b1 U + a2 b1 ＾ 2 U ＾ 2 + a1 b2 U ＾ 2 + a3 b1 ＾ 3 U ＾ 3 + 2 a2 b1 b2 U ＾ 3 + a1 b3 U ＾ 3 + a4 b1 ＾ 4 U ＾ 4 + 3 a3 b1 ＾ 2 b2 U ＾ 4 + a2 b2 ＾ 2 U ＾ 4 + 2 a2 b1 b3 U ＾ 4 + a1 b4 U ＾ 4 + ((a1 b1 + 2 a2 b1 ＾ 2 \underline{U} + 3 a3 b1 ＾ 3 U ＾ 2 + 2 a2 b1 b2 U ＾ 2 + a1 b1 ＾ 3 c21 U ＾ 2 + 4 a4 b1 ＾ 4 U ＾ 3 + 6 a3 b1 ＾ 2 b2 U ＾ 3 + 2 a2 b1 b3 U ＾ 3 + (略))) \underline{V} + 長い省略 UV の係数は 2 a2 b1 ＾ 2 = 0 より, UV の係数が 0 ⇔ a2 = 0 を得る. In[19]:=SS = SS /. a2 −＞ 0; In[20]:=SS = SS /. b2 −＞ 0 Out[20]=a1 b1 U + a3 b1 ＾ 3 U ＾ 3 + a1 b3 U ＾ 3 + a4 b1 ＾ 4 U ＾ 4 + 略 In[21]:= CoefficientList[SS, {U,V}] Out[21]= \{\{0, a1 b1, 0, a3 b1＾3 + a1 b3, a4 b1＾4 + a1 b4\}, \{a1 b1, 0, \underline{3 a3 b1 ＾ 3 + a1 b1 ＾ 3 c12} , 4 a4 b1 ＾ 4 + a1 b1 ＾ 4 c13, 6 a3 b1 ＾ 2 b3 + 3 a3 b1 ＾ 5 c12 + 2 a1 b1 ＾ 2 b3 c12}, {0, 3 a3 b1 ＾ 3 + a1 b1 ＾ 3 c21, \underline{6 a4 b1 ＾ 4 + a1 b1 ＾ 4 c22} , 3 a3 b1 ＾ 2 b3 + 6 a3 b1 ＾ 5 c12 + 3 a3 b1 ＾ 5 c21 + a1 b1 ＾ 2 b3 c21, 12 a4 b1 ＾ 3 b3 + 3 a3 b1 ＾ 2 b4 + 12 a4 b1 ＾ 6 c12 + 6 a3 b1 ＾ 6 c13 + 4 a4 b1 ＾ 6 c21 + a1 b1 ＾ 2 b4 c21 + 3 a3 b1 ＾ 6 c22 + 2 a1 b1 ＾ 3 b3 c22}, 略 U ＾ 2 V の係数 = 3 a3 b1 ＾ 3 + a1 b1 ＾ 3 c21, U ＾ 2 V ＾ 2 の係数 = 6 a4 b1 ＾ 4 + a1 b1 ＾ 4 c22 なので, a3, a4 を適当にとり, U ＾ 2 V, U ＾ 2 V ＾ 2 の係数 = 0 と出来る.

同様に、 U ＾ 2 V, U ＾ 2 V ＾ 2 の係数 = 0 とする変換は, a3 = a4 = b3 = b4 = 0 である事が解る.

この計算結果を参考にして, Theorem 4. There exists a p-nilpotent generator x whose cocoefficients satisfy the relations; c_{11} = c_{21} = c_{22} = c_{32} = c_{33} = ⋯ = c_{q q−1} = c_{qq} = 0 (p = 2q + 1). Further if x and y are p-nilpotent generators which satisfy the above relations then y = ax, a ≠ 0, a ∈ K.

Definition. Theorem 4 の条件を満たす p-nilpotent generator を normal generator と呼ぶ．

§4 Main Theorem

Theorem 5. The cocoefficients of a normal generator x of a p-nilpotent Hopf algebra satisfy the relations; c_{10} = c_{01} = 1, c_{p−1 1} = c, c_{p−k k} = (−1)^{k−1}k^{−1}c, (2 ≤ k ≤ p − 1), c ∈ K, the other terms equal to 0 and S(x) = −x (S is an antipode).

Conversely, an algebra generated by a p-nilpotent element whose cocoefficients and antipode satisfy the above relations becomes a p-nilpotent Hopf algebra.

Corollary 6. If K is a prime field then there exists exactly p nonisomorphic p-nilpotent Hopf algebras, and if K is an algebraically closed field then there exists exactly 2 non-isomorphic p-nilpotent Hopf algebras.

Theorem 5 の証明は Step 1: c_{ij} = 0, (0 ≤ i, j ≤ q) except c_{10} = c_{01} = 0, where p = 2q + 1 Step 2: c_{ij} = 0, (q + 1 ≤ i ≤ 2q − 1, 0 ≤ j ≤ i − 2) Step 3: c_{ij} = 0, (q + 1 ≤ i ≤ 2q, 2q + 2 − i ≤ j ≤ i) Step 4: i + j = p の時, c_{ij} が Theorem 5 の条件を満たす事 Step 5: 逆に c_{ij} が Theorem 5 の条件を満たす時, coassociative になる事の証明に分かれる. p = 7 の場合, Step 1 で下図の S1, Step 2 で下図の S2, Step 3 で下図の S3 が 0 を示し, Step 4 で下図の S4 が条件を満たす事を示すわけです.

[ \begin{matrix}0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & S1 & S2 & S3 & S4 \\ 0 & 0 & 0 & 0 & S2 & S4 & S3 \\ 0 & S1 & 0 & 0 & S4 & S3 & S3 \\ 0 & S2 & S2 & S4 & S3 & S3 & S3 \\ 0 & S2 & S4 & S3 & S3 & S3 & S3 \\ 0 & S4 & S3 & S3 & S3 & S3 & S3 \end{matrix} ]

p = 5 の時は, Step 1 の部分は無いので, p = 7 の時, Mathematica の計算を見せよう． H = K[x], H⊗H⊗H = K[X, Y, Z], X = x⊗1⊗1, Y = 1⊗x⊗1, Z = ⊗1⊗1⊗x と置き, Dif = (Δ ⊗ 1)Δ(x) − (1 ⊗ Δ)Δ(x) を計算する． Dif = 0 ⇔ coassociative に注意する.

In[1]:= De= X + Y + c31 X ＾ 3 Y + c31 X Y ＾ 3 + c41 X ＾ 4 Y + c41 X Y ＾ 4 + c51 X ＾ 5 Y + c42 X ＾ 4 Y ＾ 2 + c42 X ＾ 2 Y ＾ 4 + c51 X Y ＾ 5 + c61 X ＾ 6 Y + c52 X ＾ 5 Y ＾ 2 + c43 X ＾ 4 Y ＾ 3 + c43 X ＾ 3 Y ＾ 4 + c52 X ＾ 2 Y ＾ 5 + c61 X Y ＾ 6 + c62 X ＾ 6 Y ＾ 2 + c53 X ＾ 5 Y ＾ 3 + c44 X ＾ 4 Y ＾ 4 + c53 X ＾ 3 Y ＾ 5 + c62 X ＾ 2 Y ＾ 6 + c63 X ＾ 6 Y ＾ 3 + c54 X ＾ 5 Y ＾ 4 + c54 X ＾ 4 Y ＾ 5 + c63 X ＾ 3 Y ＾ 6 + c64 X ＾ 6 Y ＾ 4 + c55 X ＾ 5 Y ＾ 5 + c64 X ＾ 4 Y ＾ 6 + c65 X ＾ 6 Y ＾ 5 + c65 X ＾ 5 Y ＾ 6 + c66 X ＾ 6 Y ＾ 6; In[2]:= De2 = Expand[De ＾ 2, X]; In[3]:= DX = PolynomialRemainder[De2, X ＾ 7, X]; In[4]:= DY = PolynomialRemainder[DX, Y ＾ 7, Y]; In[5]:= De2 = PolynomialRemainder[DY, Z ＾ 7, Z]; In[6]:= De3 = Expand[De ＾ 3, X]; 以下 De4, De5, De6 を計算しておく．

(Δ ⊗ 1)Δ(x) の計算 In[22]:= DeIdDe =De + Z + c31 De3 * Z + c31 De Z ＾ 3 + c41 De4 * Z + c41 De Z ＾ 4 + c51 De5* Z + c42 De4* Z ＾ 2 + c42 De2* Z ＾ 4 +c51 De Z ＾ 5 + c61 De6 * Z + c52 De5 * Z ＾ 2 + c43 De4* Z ＾ 3 + c43 De3* Z ＾ 4 + c52 De2 * Z ＾ 5 + c61 De Z ＾ 6 + c62 De6 * Z ＾ 2 + c53 De5 * Z ＾ 3 + c44 De4 * Z ＾ 4 + c53 De3 * Z ＾ 5 + c62 De2 * Z ＾ 6 +c63 De6 * Z ＾ 3 + c54 De5 * Z ＾ 4 + c54 De4 * Z ＾ 5 +c63 De3 * Z ＾ 6 + c64 De6 * Z ＾ 4 + c55 De5

* Z ＾ 5 + c64 De4 * Z ＾ 6 + c65 De6 * Z ＾ 5 + c65 De5 * Z ＾ 6 +c66 De6 * Z ＾ 6; In[23]:= ExDeIdDe =Collect[DeIdDe, X]; In[24]:= DX = PolynomialRemainder[ExDeIdDe, X ＾ 7, X]; In[25]:= DY = PolynomialRemainder[DX, Y ＾ 7, Y]; In[26]:= ExDeIdDe = PolynomialRemainder[DY, Z ＾ 7, Z]; In[27]:= ExDeIdDe = Expand[ExDeIdDe, X]; In[28]:= ExDeIdDe = PolynomialMod[ExDeIdDe, 7];

同様に (1 ⊗ Δ)Δ(x) = ExIdDeDe を計算し、 In[56]:= Dif = ExDeIdDe − ExIdDeDe; In[57]:= Dif = PolynomialMod[Dif, 7]; In[58]:= Expand[Dif, X]; In[59]:= CoefficientList[Dif, {X,Y,Z}] Out[59]= {{{0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0},{0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0}}, {{0, 0, 0, 0, 0, 0, 0}, {0, 0, \underline{4c31}, \underline{3c41}, \underline{2 c42+2 c51}, 4c31 ＾ 2+2 c52+c61, 2c62\}, \{0, 0, \underline{c41}, \underline{4 c51}, c31 ＾ 2+3 c43+6 c61, 3 c31 c41+3 c53, 2 c41 ＾ 2 + 4 c31 c42 + 2 c31 c51 + 3 c63}, 長い省略 In[60]:= Dif = Dif /. c31 -＞ 0; In[61]:= Dif = Dif /. c41 -＞ 0; In[62]:= Dif = Dif /. c51 -＞ 0; In[63]:= Dif = Dif /. c42 -＞ 0; p = 7 の場合は、単に c31 = 0 であるが, 一般には, ci1 = 0 から cij = 0 を示す帰納法になる．又、 c41 = 0, c42 = 0, c51 = 0 も一般化され、帰納法で証明できる．但し, 帰納法のルールを見つけるには最低でも, p = 11 の時の計算が必要である. In[64]:= Expand[Dif, X]; In[65]:= CoefficientList[Dif, {X,Y,Z}] Out[65]={{{0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0}}, {{0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 2 c52+c61, 2 c62}, {0, 0, 0, 0, 3 c43 + 6 c61, 3 c53, 3 c63}, {0, 0, 0, 4 c43 + c61, \underline{4 c44, 4 c54, 4 c64}\}, \{0, 0, 5 c52 + 6 c61, \underline{5 c53, 5 c54, 5 c55, 5 c65}\}, \{0, 0, 6 \underline{c62, 6 c63, 6 c64, 6 c65, 6 c66}\}, 長い省略 underline の部分に注目すれば, どこが 0 になるかよく判る. In[66]:= Dif = Dif /. c62 -＞ 0; In[67]:= Dif = Dif /. c53 -＞ 0; In[68]:= Dif = Dif /. c63 -＞ 0; In[69]:= Dif = Dif /. c44 -＞ 0; In[70]:= Dif = Dif /. c54 -＞ 0; In[71]:= Dif = Dif /. c64 -＞ 0;c65, c66 も In[78]:= Expand[Dif, X]; In[79]:= CoefficientList[Dif, {X,Y,Z}] Out[79]= {{{0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0}}, {{0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 2 c52 + c61, 0}, {0, 0, 0, 0, 3 c43 + 6 c61, 0, 0}, {0, 0, 0, 4 c43 + c61, 0, 0, 0}, {0, 0, 5 c52 + 6 c61, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 6 c43 c61 + 3 c52 c61 + 3 c61 ＾ 2}}, {{0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 3 c43 + 2 c52, 0, 0}, {0, 0, 0, 6 c43 + 4 c52, 0, 0, 0}, 長い省略この部分が、副対角線から 1 つずれた所である.

Step 5: は単純計算ではできない. automorphism type の Hopf algebra が Theorem 5 の条件を満たし, coassociative である事を利用して証明する．

References

[Kr] H.F.Kreimer, Quadratic Hopf algebras and Galois extensions, Contemporary Math. 13 (1982), 353–361.

[LR1] R.Larson and D.Radford, Finite dimensional cosemisimple Hopf alge-

bras in characteristic 0 are semisimple, J. Algebra 117 (1988), 267–289.

[LR2] , Semisimple cosemisimple Hopf algebras, Amer. J. Math. 110 (1988), 187–195.

[Sw] M. Sweedler, Hopf algebras, Benjamin, New York, 1969.

Faculty of Science

Okayama University of Science Okayama 700, Japan

Email: yokogawa@mdas.ous.ac.jp