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  浙江大学学报(理学版)  2018, Vol. 45 Issue (6): 651-655, 672  DOI:10.3785/j.issn.1008-9497.2018.06.001
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引用本文 [复制中英文]

胡献国, 郭梦甜, 吕家凤. 微分分次Poisson Hopf代数的张量积[J]. 浙江大学学报(理学版), 2018, 45(6): 651-655, 672. DOI: 10.3785/j.issn.1008-9497.2018.06.001.
[复制中文]
HU Xianguo, GUO Mengtian, LYU Jiafeng. The tensor product of differential graded Poisson Hopf algebras[J]. Journal of Zhejiang University(Science Edition), 2018, 45(6): 651-655, 672. DOI: 10.3785/j.issn.1008-9497.2018.06.001.
[复制英文]

基金项目

国家自然科学基金资助项目(11571316);浙江省自然科学基金资助项目(LY16A010003)

作者简介

胡献国(1992-), ORCID:http://orcid.org/0000-0002-8084-9199, 男, 硕士研究生, 主要从事代数学研究

通信作者

吕家凤, ORCID:http://orcid.org/0000-0002-2637-142X, E-mail:jiafenglv@zjnu.edu.cn

文章历史

收稿日期:2017-11-09
微分分次Poisson Hopf代数的张量积
胡献国 , 郭梦甜 , 吕家凤     
浙江师范大学 数学系, 浙江 金华 321004
摘要: 证明了任意2个p次微分分次Poisson Hopf代数的张量积仍为p次微分分次Poisson Hopf代数.作为应用,证明了p次微分分次Poisson Hopf代数构成的范畴dg-PHA是对称monoidal范畴.
关键词: p次微分分次Poisson Hopf代数    张量积    对称monoidal范畴    
The tensor product of differential graded Poisson Hopf algebras
HU Xianguo, GUO Mengtian, LYU Jiafeng     
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang Province, China
Abstract: This paper studies p-differential graded Poisson Hopf algebras and proves that the tensor product of p-differential graded Poisson Hopf algebras is also a p-differential graded Poisson Hopf algebra. As an application, we show that the category of p-differential graded Poisson Hopf algebras, denoted by dg-PHA, belongs to symmetric monoidal category.
Key Words: p-differential graded Poisson Hopf algebras    tensor product    symmetric monoidal category    
0 引言

Poisson代数的概念起源于Poisson几何, 可简单看作交换代数与Lie代数的结合.近年来,随着Poisson代数的广泛应用, 得到了多种Poisson代数的推广形式[1-4].特别地, DRINFEL’D[5]定义了Poisson Hopf代数,详细研究了这类代数在Poisson-Lie群上的应用.此外, 吕家凤等[6]给出了Poisson Hopf代数及其泛包络代数的基本性质.微分分次代数起源于代数拓扑与表示理论, 在交换代数与非交换代数领域有重要作用[7-8].作为其推广, LYU等[9]引入了微分分次Poisson代数, 研究了这类代数的张量积及相关性质和应用.受此启发, 本文尝试将Poisson Hopf代数的概念推广到微分分次的情形, 定义了p次微分分次Poisson Hopf代数, 并推广了文献[9]的相关结果:证明了任意2个p次微分分次Poisson Hopf代数的张量积仍为p次微分分次Poisson Hopf代数; 证明了p次微分分次Poisson Hopf代数构成的范畴dg-PHA是对称monoidal范畴.

本文的主要结果如下:

定理1  (1)设(A, uA, ηA, ΔA, εA, SA, {·, ·}A, dA)和(B, uB, ηB, ΔB, εB, SB, {·, ·}B, dB)是任意2个p次微分分次Poisson Hopf代数,则

$ \left( {A \otimes B,u,\eta ,\Delta ,\varepsilon ,S,\left\{ { \cdot , \cdot } \right\},d} \right) $

也是p次微分分次Poisson Hopf代数.相关运算定义为:

$ S\left( {a \otimes b} \right): = {S_A}\left( a \right) \otimes {S_B}\left( b \right), $
$ \eta \left( {{1_k}} \right): = {1_A} \otimes {1_B},\varepsilon \left( {a \otimes b} \right): = {\varepsilon _A}\left( a \right){\varepsilon _B}\left( b \right), $
$ u\left( {\left( {a \otimes b} \right) \otimes \left( {a' \otimes b'} \right)} \right): = {\left( { - 1} \right)^{\left| {a'} \right|\left| b \right|}}aa' \otimes bb', $
$ \Delta \left( {a \otimes b} \right): = {\left( { - 1} \right)^{\left| {{a_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right|}}{a_{\left( 1 \right)}} \otimes {b_{\left( 1 \right)}} \otimes {a_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}}, $
$ d\left( {a \otimes b} \right): = {d_A}\left( a \right) \otimes b + {\left( { - 1} \right)^{\left| a \right|}}a \otimes {d_B}\left( b \right), $
$ \begin{array}{*{20}{c}} {\left\{ {a \otimes b,a' \otimes b'} \right\}: = {{\left( { - 1} \right)}^{\left( {\left| {a'} \right| + p} \right)\left| b \right|}}\left( {{{\left\{ {a,a'} \right\}}_A} \otimes bb'} \right) + }\\ {{{\left( { - 1} \right)}^{\left( {\left| b \right| + p} \right)\left| {a'} \right|}}\left( {aa' \otimes {{\left\{ {b,b'} \right\}}_B}} \right),} \end{array} $

其中, uA(aa′):=aa′, uB(bb′):=bb′, ΔA(a):=a(1)a(2), ΔB(b):=b(1)b(2), ||表示齐次元的次数, a, a′∈A, b, b′∈B为齐次元.

(2) 记dg-PHAp次微分分次Poisson Hopf代数构成的范畴, 则dg-PHA是对称monoidal范畴, 其左单位元与右单位元均为基础域k.

(3) 设AopBop分别为ABp次微分分次Poisson Hopf反代数.则

$ {\left( {A \otimes B} \right)^{{\rm{op}}}} = {A^{{\rm{op}}}} \otimes {B^{{\rm{op}}}}. $
1 p次微分分次Poisson Hopf代数

首先,回顾一些后面要用到的概念.

如无特别说明, 文中所有代数均含有单位元1, k表示特征为0的基域, 所涉及的对象都是域k上的向量空间, 所涉及的分次均为Z-分次.对任给的分次向量空间VW, 扭转映射是指

$ T:V \otimes W \to W \otimes V:T\left( {v \otimes w} \right) = {\left( { - 1} \right)^{\left| v \right|\left| w \right|}}w \otimes v, $

其中,vV, wW.

文中的分次代数AZ-非负分次代数(A, u, η), 其中A=⊕n≥0An满足A0=k, uAAAηkA分别被称为A的乘法与单位.方便起见, 记u(ab)为ab, ∀a, bA.而微分分次代数是指具有次数是1的微分dAA分次代数, 并且其是一个分次导子.

定义1  设(A, ·)是分次k-代数.若存在k-齐次线性映射:

$ \left\{ { \cdot , \cdot } \right\}:A \otimes A \to A,\;\;\;\;\left| {\left\{ { \cdot , \cdot } \right\}} \right| = p, $

对于任意的齐次元a, b, cA, 满足

(ⅰ) (A, {·, ·})是p次分次Lie代数, 即

$ \left( {{\rm{ia}}} \right)\left\{ {a,b} \right\} = - {\left( { - 1} \right)^{\left( {\left| a \right| + p} \right)\left( {\left| b \right| + p} \right)}}\left\{ {b,a} \right\}, $
$ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( {{\rm{ib}}} \right)\left\{ {a,\left\{ {b,c} \right\}} \right\} = \left\{ {\left\{ {a,b} \right\},c} \right\} + \\ {\left( { - 1} \right)^{\left( {\left| a \right| + p} \right)\left( {\left| b \right| + p} \right)}}\left\{ {b,\left\{ {a,c} \right\}} \right\}, $

(ⅱ)分次交换性:a·b=(-1)|a||b|b·a,

(ⅲ)双导子性质:{a, b·c}={a, bc+(-1)(|a|+p)|b|b·{a, c},

则称(A, ·, {·, ·})为p次分次Poisson代数[10].若在此基础上, 存在1次k-线性映射dAA, 满足d2=0与

$ \left( {{\rm{iv}}} \right)d\left( {\left\{ {a,b} \right\}} \right) = \left\{ {d\left( a \right),b} \right\} + {\left( { - 1} \right)^{\left( {\left| a \right| + p} \right)}}\left\{ {a,d\left( b \right)} \right\}, $
$ \left( {\rm{v}} \right)d\left( {a \cdot b} \right) = d\left( a \right) \cdot b + {\left( { - 1} \right)^{\left| a \right|}}a \cdot d\left( b \right), $

其中a, bA为齐次元, 则称Ap次微分分次Poisson代数, 可表示为(A, ·, {·, ·}, d).在不引起混淆的情况下, 可表示为(A, {·, ·}, d)或A.

文中的分次余代数(C, Δ, ε)为非负分次向量空间C, 具有次数为0的k-齐次线性映射

$ \Delta :C \to C \otimes C\;\;\;与\;\;\;\;\varepsilon :C \to k, $

使得通常的图表可交换[11-12].注意到

$ k \otimes C \cong C \cong C \otimes k $

是明显同构的, 因此,非负分次向量空间C是分次余代数当且仅当(C, Δ, ε)满足:

$ \left( {\Delta \otimes I} \right)\Delta = \left( {I \otimes \Delta } \right)\Delta , $
$ \left( {\varepsilon \otimes I} \right)\Delta = I = \left( {I \otimes \varepsilon } \right)\Delta , $

其中,ICC恒等同态, Δ与ε分别称为C的余乘法与余单位.在不引起混淆的情况下, 可将(C, Δ, ε)记为C.

对于任给的cC, 参照文献[12]中的记号, 记Δ(c):=∑(c)c(1)c(2).在使用过程中, 求和符号经常省略,因此, 对任给的齐次元cC, 余乘法与余单位可分别表示为

$ \begin{array}{l} \left( {\Delta \otimes I} \right)\Delta \left( c \right) = \left( {I \otimes \Delta } \right)\Delta \left( c \right) = {c_{\left( 1 \right)}} \otimes {c_{\left( 2 \right)}} \otimes {c_{\left( 3 \right)}}\\ 与\;\;\;\;\;\;\;\;\;c = \varepsilon \left( {{c_{\left( 1 \right)}}} \right){c_{\left( 2 \right)}} = {c_{\left( 1 \right)}}\varepsilon \left( {{c_{\left( 2 \right)}}} \right). \end{array} $

设(C, ΔC, εC)与(D, ΔD, εD)是2个分次余代数, 如果f满足条件:

$ \left( {f \otimes f} \right) \circ {\Delta _C} = {\Delta _D} \circ f,\;\;\;\;{\varepsilon _D} \circ f = {\varepsilon _C}, $

则称次数为0的分次线性映射fCD为分次余代数同态.

定义2  设H是分次k-向量空间.若存在1次k-线性映射dHH, 满足d2=0与

(ⅰ) (H, Δ, ε, d)是微分分次余代数, 即

(ⅰa) (H, Δ, ε)是分次余代数,

(ⅰb) d是次数为1的分次余导子,即有εd=0与Δd=(dI+T(dI)T)Δ,

(ⅱ) (H, u, η, d)是微分分次代数,

(ⅲ) Δ与ε是分次代数同态,

则称H为微分分次双代数.若在此基础上, 存在次数为0的齐次线性映射SHH, 满足

$ u\left( {I \otimes S} \right)\Delta = u\left( {S \otimes I} \right)\Delta = \eta \varepsilon , $

则称(H, u, η, Δ, ε, S, d)为微分分次Hopf代数[13], 并称SH的对极.

注记1  在定义2的(ⅰb)中, 若使用文献[12]中的记号, 对任给的齐次元xH,

$ \Delta \left( {d\left( x \right)} \right) = d\left( {{x_{\left( 1 \right)}}} \right) \otimes {x_{\left( 2 \right)}} + {\left( { - 1} \right)^{\left| {{x_{\left( 1 \right)}}} \right|}}{x_{\left( 1 \right)}} \otimes d\left( {{x_{\left( 2 \right)}}} \right). $

设(C, ΔC, εC)与(D, ΔD, εD)是2个分次双代数, 若f既是分次代数同态, 又是分次余代数同态,则称次数为0的分次线性映射fCD为分次双代数同态.更进一步, 若(C, SC)与(D, SD)都是分次Hopf代数,则易证f为分次Hopf代数同态, 即SD°f=f°SC(参见文献[12]引理4.0.4).

定义3  设A是分次k-向量空间.若存在k-齐次线性映射:

$ \begin{array}{*{20}{c}} {\left\{ { \cdot , \cdot } \right\}:A \otimes A \to A,\;\;\;\;\left| {\left\{ { \cdot , \cdot } \right\}} \right| = p;}\\ {d:A \to A,\;\;\;\;\left| d \right| = 1,} \end{array} $

满足d2=0与

(ⅰ) (A, u, η, {·, ·}, d)是p次微分分次Poisson代数,

(ⅱ) (A, u, η, Δ, ε, d)是微分分次双代数,

(ⅲ) Δ({a, b}A)={Δ(a), Δ(b)}AA, ∀a, bA

其中{·, ·}AA定义为

$ \begin{array}{l} {\left\{ {a \otimes a',b \otimes b'} \right\}_{A \otimes A}}: = {\left( { - 1} \right)^{\left( {\left| b \right| + p} \right)\left| {a'} \right|}}\left( {\left\{ {a,b} \right\} \otimes a'b'} \right) + \\ \;\;\;\;\;\;\;{\left( { - 1} \right)^{\left( {\left| {a'} \right| + p} \right)\left| b \right|}}\left( {ab \otimes \left\{ {a',b'} \right\}} \right), \end{array} $

a, b, a′, b′∈A为齐次元,

则称(A, u, η, Δ, ε, {·, ·}, d)为p次微分分次Poisson双代数.若在此基础上, 存在次数为0的k-齐次线性映射SAA, 满足

$ u\left( {I \otimes S} \right)\Delta = u\left( {S \otimes I} \right)\Delta = \eta \varepsilon , $

则称Ap次微分分次Poisson Hopf代数, 表示为(A, u, η, Δ, ε, S, {·, ·}, d).

定义4  设AB是任意2个p次微分分次Poisson Hopf代数, 若对任给的齐次元a, bA, 有fdA=dBf, f({a, b}A)={f(a), f(b)}B,则称次数为0的分次双代数同态fAB为微分分次Poisson Hopf代数同态.进一步, 若微分分次Poisson Hopf代数同态fAB是双射的, 则称作为微分分次Poisson Hopf代数的AB是同构的, 记为A$ \cong $B.

dg-PHAp次微分分次Poisson Hopf代数构成的范畴, 其中的态射为微分分次Poisson Hopf代数同态.

下面给出具体的例子.

例1  设(A, u, η, Δ, ε, S, {·, ·}, d)为任给的p次微分分次Poisson Hopf代数.那么

$ \left( {{A^{{\rm{op}}}},{u^{{\rm{op}}}},\eta ,{\Delta ^{{\rm{op}}}},\varepsilon ,S,{{\left\{ { \cdot , \cdot } \right\}}^{{\rm{op}}}},d} \right) $

也为p次微分分次Poisson Hopf代数, 其中,

$ {u^{{\rm{op}}}}\left( {a \otimes b} \right) = {\left( { - 1} \right)^{\left| a \right|\left| b \right|}}b \cdot a = a \cdot b = u\left( {a \otimes b} \right), $
$ \begin{array}{*{20}{c}} {{{\left\{ {a,b} \right\}}^{{\rm{op}}}} = {{\left( { - 1} \right)}^{\left( {\left| a \right| + p} \right)\left( {\left| b \right| + p} \right)}}\left\{ {b,a} \right\} = - \left\{ {a,b} \right\},}\\ {{\Delta ^{{\rm{op}}}} = T\Delta ,} \end{array} $

a, bA为齐次元, TAAAA为扭转映射, 即T(ab)=(-1)|a||b|ba.

例2  设A:=$ \frac{{k\left\langle {x, y} \right\rangle }}{{{{\left( {xy - yx, y^2} \right)}}}}$, |x|=2, |y|=3.定义1次k-线性映射d

$ d\left( x \right) = y,d\left( y \right) = 0, $

且{x, y}=-{y, x}=y2, {x, x}={y, y}=0.定义分次Hopf代数的结构如下:

$ \Delta \left( x \right) = x \otimes 1 + 1 \otimes x,\Delta \left( y \right) = y \otimes 1 + 1 \otimes y, $
$ \varepsilon \left( x \right) = 0,\varepsilon \left( y \right) = 0,S\left( x \right) = - x,S\left( y \right) = - y. $

注意到在A中有y2=0.易证A是1次微分分次Poisson Hopf代数.

2 定理1的证明

简单起见, 在不引起混淆的情况下, 下文中常省去下标.所取的元素都是对应代数中的齐次元.

根据定义3, 定理1的证明可以分解成以下几个引理.

引理1  由定理1中的定义, 有(AB, Δ, ε)为分次余代数.

证明  因为(A, Δ, ε)是分次余代数, 由分次余代数的定义, 有

$ {a_{\left( {11} \right)}} \otimes {a_{\left( {12} \right)}} \otimes {a_{\left( 2 \right)}} = {a_{\left( 1 \right)}} \otimes {a_{\left( {21} \right)}} \otimes {a_{\left( {22} \right)}}, $
$ \varepsilon \left( {{a_{\left( 1 \right)}}} \right){a_{\left( 2 \right)}} = a = {a_{\left( 1 \right)}}\varepsilon \left( {{a_{\left( 2 \right)}}} \right). $

对分次余代数(B, Δ, ε), 有类似的等式成立.从而

$ \begin{array}{l} \left( {\Delta \otimes I} \right)\Delta \left( {a \otimes b} \right) = \\ \left( {\Delta \otimes I} \right)\left( {{{\left( { - 1} \right)}^{\left| {{a_{\left( 2 \right)}}} \right|\left| {{a_{\left( 1 \right)}}} \right|}}{a_{\left( 1 \right)}} \otimes {b_{\left( 1 \right)}} \otimes {a_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}}} \right) = \\ {\left( { - 1} \right)^{\left| {{a_{\left( 2 \right)}}} \right|\left( {\left| {{b_{\left( {11} \right)}}} \right| + \left| {{b_{\left( {12} \right)}}} \right|} \right) + \left| {{a_{\left( {12} \right)}}} \right|\left| {{b_{\left( {11} \right)}}} \right|}}{a_{\left( {11} \right)}} \otimes {b_{\left( {11} \right)}} \otimes \\ {a_{\left( {12} \right)}} \otimes {b_{\left( {12} \right)}} \otimes {a_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}} = \\ {\left( { - 1} \right)^{\left| {{a_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right| + \left| {{a_{\left( 3 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right| + \left| {{a_{\left( 3 \right)}}} \right|\left| {{b_{\left( 2 \right)}}} \right|}}{a_{\left( 1 \right)}} \otimes {b_{\left( 1 \right)}} \otimes \\ {a_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}} \otimes {a_{\left( 3 \right)}} \otimes {b_{\left( 3 \right)}}, \end{array} $
$ \begin{array}{l} \left( {I \otimes \Delta } \right)\Delta \left( {a \otimes b} \right) = \\ \left( {I \otimes \Delta } \right)\left( {{{\left( { - 1} \right)}^{\left| {{a_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right|}}{a_{\left( 1 \right)}} \otimes {b_{\left( 1 \right)}} \otimes {a_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}}} \right) = \\ {\left( { - 1} \right)^{\left( {\left| {{a_{\left( {21} \right)}}} \right| + \left| {{a_{\left( {22} \right)}}} \right|} \right)\left| {{b_{\left( 1 \right)}}} \right| + \left| {{a_{\left( {22} \right)}}} \right|\left| {{b_{\left( {21} \right)}}} \right|}}{a_{\left( 1 \right)}} \otimes {b_{\left( 1 \right)}} \otimes \\ {a_{\left( {21} \right)}} \otimes {b_{\left( {21} \right)}} \otimes {a_{\left( {22} \right)}} \otimes {b_{\left( {22} \right)}} = \\ {\left( { - 1} \right)^{\left| {{a_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right| + \left| {{a_{\left( 3 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right| + \left| {{a_{\left( 3 \right)}}} \right|\left| {{b_{\left( 2 \right)}}} \right|}}{a_{\left( 1 \right)}} \otimes {b_{\left( 1 \right)}} \otimes \\ {a_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}} \otimes {a_{\left( 3 \right)}} \otimes {b_{\left( 3 \right)}}, \end{array} $
$ \begin{array}{l} \left( {I \otimes \varepsilon } \right)\Delta \left( {a \otimes b} \right) = \\ \left( {I \otimes \varepsilon } \right)\left( {{{\left( { - 1} \right)}^{\left| {{a_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right|}}{a_{\left( 1 \right)}} \otimes {b_{\left( 1 \right)}} \otimes {a_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}}} \right) = \\ {\left( { - 1} \right)^{\left| {{a_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right|}}\left( {{a_{\left( 1 \right)}} \otimes {b_{\left( 1 \right)}}} \right) \cdot \varepsilon \left( {{a_{\left( 2 \right)}}} \right)\varepsilon \left( {{b_{\left( 2 \right)}}} \right) = \\ {\left( { - 1} \right)^{\left| {{a_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right|}}{a_{\left( 1 \right)}}\varepsilon \left( {{a_{\left( 2 \right)}}} \right) \otimes {b_{\left( 1 \right)}}\varepsilon \left( {{b_{\left( 2 \right)}}} \right) \end{array} $

$ \begin{array}{l} \left( {\varepsilon \otimes I} \right)\Delta \left( {a \otimes b} \right) = \\ \left( {\varepsilon \otimes I} \right)\left( {{{\left( { - 1} \right)}^{\left| {{a_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right|}}{a_{\left( 1 \right)}} \otimes {b_{\left( 1 \right)}} \otimes {a_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}}} \right) = \\ {\left( { - 1} \right)^{\left| {{a_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right|}}\varepsilon \left( {{a_{\left( 1 \right)}}} \right)\varepsilon \left( {{b_{\left( 1 \right)}}} \right) \cdot \left( {{a_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}}} \right) = \\ {\left( { - 1} \right)^{\left| {{a_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right|}}\varepsilon \left( {{a_{\left( 1 \right)}}} \right){a_{\left( 2 \right)}} \otimes \varepsilon \left( {{b_{\left( 1 \right)}}} \right){b_{\left( 2 \right)}}. \end{array} $

进而有

$ \left( {\Delta \otimes {I_{A \otimes B}}} \right)\Delta = \left( {{I_{A \otimes B}} \otimes \Delta } \right)\Delta . $

注意到εAεB均为次数为0的齐次线性映射, 且k为次数聚集在0处的分次向量空间, 所以对任意i>0, 有εA(Ai)=0与εB(Bi)=0, 其中A=⊕i≥0AiB=⊕i≥0Bi.同理可得, Δ(An)⊆⊕i+j=nAiAj, 其中n≥0.因此, 只有在a(2)A0时, 才有εA(a(2))≠0, 此时(-1)|a(2)||b(1)|=1.故

$ \begin{array}{l} \left( {I \otimes \varepsilon } \right)\Delta \left( {a \otimes b} \right) = {\left( { - 1} \right)^{\left| {{a_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right|}}{a_{\left( 1 \right)}}\varepsilon \left( {{a_{\left( 2 \right)}}} \right) \otimes \\ \;\;\;\;\;\;\;\;{b_{\left( 1 \right)}}\varepsilon \left( {{b_{\left( 2 \right)}}} \right) = {a_{\left( 1 \right)}}\varepsilon \left( {{a_{\left( 2 \right)}}} \right) \otimes {b_{\left( 1 \right)}}\varepsilon \left( {{b_{\left( 2 \right)}}} \right) = a \otimes b. \end{array} $

类似地, 有  (εI)Δ(ab)=ab.

由此可得, (AB, Δ, ε)为分次余代数.

引理2  由定理1中的定义, 有εd=0, 且

$ \Delta d = \left( {d \otimes {I_{A \otimes B}} + T\left( {d \otimes {I_{A \otimes B}}} \right)T} \right)\Delta . $

证明  注意到εAdA=0与εBdB=0.因此

$ \begin{array}{l} \varepsilon d\left( {a \otimes b} \right) = \varepsilon \left( {d\left( a \right) \otimes b + {{\left( { - 1} \right)}^{\left| a \right|}}a \otimes d\left( b \right)} \right) = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\varepsilon d\left( a \right)\varepsilon \left( b \right) + {\left( { - 1} \right)^{\left| a \right|}}\varepsilon \left( a \right)\varepsilon d\left( b \right) = 0. \end{array} $

下面证明Δ与微分的交换性.根据Δ与d的结构, 可得

$ \begin{array}{l} \Delta d\left( {a \otimes b} \right) = \Delta \left( {d\left( a \right) \otimes b + {{\left( { - 1} \right)}^{\left| a \right|}}a \otimes d\left( b \right)} \right) = \\ {\left( { - 1} \right)^{\left| {d{{\left( a \right)}_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right|}}d{\left( a \right)_{\left( 1 \right)}} \otimes {b_{\left( 1 \right)}} \otimes d{\left( a \right)_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}} + \\ {\left( { - 1} \right)^{\left| {{a_{\left( 1 \right)}}} \right| + \left| {{a_{\left( 2 \right)}}} \right| + \left| {{a_{\left( 2 \right)}}} \right|\left| {d{{\left( b \right)}_{\left( 1 \right)}}} \right|}}{a_{\left( 1 \right)}} \otimes d{\left( b \right)_{\left( 1 \right)}} \otimes {a_{\left( 2 \right)}} \otimes \\ d{\left( b \right)_{\left( 2 \right)}} = {\left( { - 1} \right)^{\left| {{a_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right| + \left| {{a_{\left( 1 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right|}}{a_{\left( 1 \right)}} \otimes {b_{\left( 1 \right)}} \otimes \\ d\left( {{a_{\left( 2 \right)}}} \right) \otimes {b_{\left( 2 \right)}} + {\left( { - 1} \right)^{\left| {{a_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right|}}d\left( {{a_{\left( 1 \right)}}} \right) \otimes {b_{\left( 1 \right)}} \otimes \\ {a_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}} + {\left( { - 1} \right)^{\left| {{a_{\left( 1 \right)}}} \right| + \left| {{a_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right|}}{a_{\left( 1 \right)}} \otimes d\left( {{b_{\left( 1 \right)}}} \right) \otimes \\ {a_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}} + {\left( { - 1} \right)^{\left| {{a_{\left( 1 \right)}}} \right| + \left| {{a_{\left( 2 \right)}}} \right| + \left| {{b_{\left( 1 \right)}}} \right| + \left| {{a_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right|}}{a_{\left( 1 \right)}} \otimes {b_{\left( 1 \right)}} \otimes \\ {a_{\left( 2 \right)}} \otimes d\left( {{b_{\left( 2 \right)}}} \right), \end{array} $
$ \begin{array}{l} \left( {d \otimes I + T\left( {d \otimes I} \right)T} \right)\Delta \left( {a \otimes b} \right) = \left( {d \otimes I + T\left( {d \otimes } \right.} \right.\\ \left. {\left. I \right)T} \right)\left( {{{\left( { - 1} \right)}^{\left| {{a_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right|}}{a_{\left( 1 \right)}} \otimes {b_{\left( 1 \right)}} \otimes {a_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}}} \right) = \\ {\left( { - 1} \right)^{\left| {{a_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right|}}d\left( {{a_{\left( 1 \right)}} \otimes {b_{\left( 1 \right)}}} \right) \otimes {a_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}}+\\ {\left( { - 1} \right)^{\left| {{a_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right| + \left| {{a_{\left( 1 \right)}} \otimes {b_{\left( 1 \right)}}} \right|}}{a_{\left( 1 \right)}} \otimes {b_{\left( 1 \right)}} \otimes d\left( {{a_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}}} \right) = \\ {\left( { - 1} \right)^{\left| {{a_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right|}}\left[ {\left( {d\left( {{a_{\left( 1 \right)}}} \right) \otimes {b_{\left( 1 \right)}} + {{\left( { - 1} \right)}^{\left| {{a_{\left( 1 \right)}}} \right|}}{a_{\left( 1 \right)}} \otimes } \right.} \right.\\ \left. {d\left( {{b_{\left( 1 \right)}}} \right)} \right) \otimes {a_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}} + {\left( { - 1} \right)^{\left| {{a_{\left( 1 \right)}}} \right| + \left| {{b_{\left( 1 \right)}}} \right|}}{a_{\left( 1 \right)}} \otimes \\ \left. {{b_{\left( 1 \right)}} \otimes \left( {d\left( {{a_{\left( 2 \right)}}} \right) \otimes {b_{\left( 2 \right)}} + {{\left( { - 1} \right)}^{\left| {{a_{\left( 2 \right)}}} \right|}}{a_{\left( 2 \right)}} \otimes d\left( {{b_{\left( 2 \right)}}} \right)} \right)} \right]. \end{array} $

$ \Delta d = \left( {d \otimes {I_{A \otimes B}} + T\left( {d \otimes {I_{A \otimes B}}} \right)T} \right)\Delta . $

引理3  由定理1中的定义, 有Δ与ε为分次代数同态.

证明  先证Δ为分次代数同态.注意到对任给的齐次元a, a′∈A, 有

$ \begin{array}{l} \Delta \left( {aa'} \right) = \Delta \left( a \right)\Delta \left( {a'} \right) = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;{\left( { - 1} \right)^{\left| {{{a'}_{\left( 1 \right)}}} \right|\left| {{a_{\left( 2 \right)}}} \right|}}{a_{\left( 1 \right)}}{{a'}_{\left( 1 \right)}} \otimes {a_{\left( 2 \right)}}{{a'}_{\left( 2 \right)}}. \end{array} $

同理, 由ΔB为分次代数同态, 可推出

$ \Delta \left( {bb'} \right) = {\left( { - 1} \right)^{\left| {{{b'}_{\left( 1 \right)}}} \right|\left| {{b_{\left( 2 \right)}}} \right|}}{b_{\left( 1 \right)}}{{b'}_{\left( 1 \right)}} \otimes {b_{\left( 2 \right)}}{{b'}_{\left( 2 \right)}}, $

其中b, b′∈B为齐次元.

从而有

$ \begin{array}{l} \Delta \left( {\left( {a \otimes b} \right)\left( {a' \otimes b'} \right)} \right) = \Delta \left( {{{\left( { - 1} \right)}^{\left| {a'} \right|\left| b \right|}}aa' \otimes bb'} \right) = \\ {\left( { - 1} \right)^{\left| {a'} \right|\left| b \right| + \left| {{{\left( {aa'} \right)}_{\left( 2 \right)}}} \right|\left| {{{\left( {bb'} \right)}_{\left( 1 \right)}}} \right|}}{\left( {aa'} \right)_{\left( 1 \right)}} \otimes {\left( {bb'} \right)_{\left( 1 \right)}} \otimes \\ {\left( {aa'} \right)_{\left( 2 \right)}} \otimes {\left( {bb'} \right)_{\left( 2 \right)}} = \\ {\left( { - 1} \right)^{\left| {{a_{\left( 2 \right)}}} \right|\left| {{{a'}_{\left( 1 \right)}}{b_{\left( 1 \right)}}{{b'}_{\left( 1 \right)}}} \right| + \left| {{{b'}_{\left( 1 \right)}}} \right|\left| {{{a'}_{\left( 2 \right)}}{b_{\left( 2 \right)}}} \right| + \left| {{{a'}_{\left( 2 \right)}}} \right|\left| {{b_{\left( 2 \right)}}} \right|}} \times \\ {\left( { - 1} \right)^{\left| {{{a'}_{\left( 1 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right| + \left| {{{a'}_{\left( 1 \right)}}} \right|\left| {{b_{\left( 2 \right)}}} \right|}}{a_{\left( 1 \right)}}{{a'}_{\left( 1 \right)}} \otimes {b_{\left( 1 \right)}}{{b'}_{\left( 1 \right)}} \otimes \\ {a_{\left( 2 \right)}}{{a'}_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}}{{b'}_{\left( 2 \right)}}, \end{array} $
$ \begin{array}{l} \Delta \left( {a \otimes b} \right)\Delta \left( {a' \otimes b'} \right) = {\left( { - 1} \right)^{\left| {{a_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right|}}{a_{\left( 1 \right)}} \otimes {b_{\left( 1 \right)}} \otimes \\ {a_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}} \times {\left( { - 1} \right)^{\left| {{{a'}_{\left( 2 \right)}}} \right|\left| {{{b'}_{\left( 1 \right)}}} \right|}}{{a'}_{\left( 1 \right)}} \otimes {{b'}_{\left( 1 \right)}} \otimes {{a'}_{\left( 2 \right)}} \otimes {{b'}_{\left( 2 \right)}} = \\ {\left( { - 1} \right)^{\left| {{a_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right| + \left| {{{a'}_{\left( 2 \right)}}} \right|\left| {{{b'}_{\left( 1 \right)}}} \right| + \left| {{a_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}}} \right| + \left| {{{a'}_{\left( 1 \right)}} \otimes {{b'}_{\left( 1 \right)}}} \right|}}\left( {{a_{\left( 1 \right)}} \otimes } \right.\\ \left. {{b_{\left( 1 \right)}}} \right)\left( {{{a'}_{\left( 1 \right)}} \otimes {{b'}_{\left( 1 \right)}}} \right) \otimes \left( {{a_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}}} \right)\left( {{{a'}_{\left( 2 \right)}} \otimes {{b'}_{\left( 2 \right)}}} \right) = \\ {\left( { - 1} \right)^{\left| {{a_{\left( 2 \right)}}} \right|\left| {{{a'}_{\left( 1 \right)}}{b_{\left( 1 \right)}}{{b'}_{\left( 1 \right)}}} \right| + \left| {{{b'}_{\left( 1 \right)}}} \right|\left| {{{a'}_{\left( 2 \right)}}{b_{\left( 2 \right)}}} \right| + \left| {{{a'}_{\left( 2 \right)}}} \right|\left| {{b_{\left( 2 \right)}}} \right|}} \times \\ {\left( { - 1} \right)^{\left| {{{a'}_{\left( 1 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right| + \left| {{{a'}_{\left( 1 \right)}}} \right|\left| {{b_{\left( 2 \right)}}} \right|}}{a_{\left( 1 \right)}}{{a'}_{\left( 1 \right)}} \otimes {b_{\left( 1 \right)}}{{b'}_{\left( 1 \right)}} \otimes \\ {a_{\left( 2 \right)}}{{a'}_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}}{{b'}_{\left( 2 \right)}}. \end{array} $

因此

$ \Delta \left( {\left( {a \otimes b} \right)\left( {a' \otimes b'} \right)} \right) = \Delta \left( {a \otimes b} \right)\Delta \left( {a' \otimes b'} \right). $

接下来证ε为分次代数同态.

$ \begin{array}{l} \varepsilon \left( {\left( {a \otimes b} \right)\left( {a' \otimes b'} \right)} \right) = \varepsilon \left( {{{\left( { - 1} \right)}^{\left| {a'} \right|\left| b \right|}}aa' \otimes bb'} \right) = \\ \;\;\;\;\;\;{\left( { - 1} \right)^{^{\left| {a'} \right|\left| b \right|}}}\varepsilon \left( {aa'} \right)\varepsilon \left( {bb'} \right) = \\ \;\;\;\;\;\;{\left( { - 1} \right)^{^{\left| {a'} \right|\left| b \right|}}}\varepsilon \left( a \right)\varepsilon \left( {a'} \right)\varepsilon \left( b \right)\varepsilon \left( {b'} \right) = \\ \;\;\;\;\;\;\varepsilon \left( a \right)\varepsilon \left( b \right)\varepsilon \left( {a'} \right)\varepsilon \left( {b'} \right) = \varepsilon \left( {a \otimes b} \right)\varepsilon \left( {a' \otimes b'} \right). \end{array} $

引理4  由定理1中的定义, 有

$ \Delta \left( {\left\{ {a \otimes b,a' \otimes b'} \right\}} \right) = \left\{ {\Delta \left( {a \otimes b} \right),\Delta \left( {a' \otimes b'} \right)} \right\}. $

证明  注意到对任给的齐次元a, a′∈A, 有

$ \begin{array}{l} \Delta \left( {\left\{ {a,a'} \right\}} \right) = \left\{ {\Delta \left( a \right),\Delta \left( {a'} \right)} \right\} = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;{\left( { - 1} \right)^{\left( {\left| {{{a'}_{\left( 1 \right)}}} \right| + p} \right)\left| {{a_{\left( 2 \right)}}} \right|}}\left\{ {{a_{\left( 1 \right)}},{{a'}_{\left( 1 \right)}}} \right\} \otimes {a_{\left( 2 \right)}}{{a'}_{\left( 2 \right)}} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;{\left( { - 1} \right)^{\left( {\left| {{a_{\left( 2 \right)}}} \right| + p} \right)\left| {{{a'}_{\left( 1 \right)}}} \right|}}{a_{\left( 1 \right)}}{{a'}_{\left( 1 \right)}} \otimes \left\{ {{a_{\left( 2 \right)}},{{a'}_{\left( 2 \right)}}} \right\}. \end{array} $

同理, 有

$ \begin{array}{l} \Delta \left( {\left\{ {b,b'} \right\}} \right) = {\left( { - 1} \right)^{\left( {\left| {{{b'}_{\left( 1 \right)}}} \right| + p} \right)\left| {{b_{\left( 2 \right)}}} \right|}}\left\{ {{b_{\left( 1 \right)}},{{b'}_{\left( 1 \right)}}} \right\} \otimes \\ {b_{\left( 2 \right)}}{{b'}_{\left( 2 \right)}} + {\left( { - 1} \right)^{\left( {\left| {{b_{\left( 2 \right)}}} \right| + p} \right)\left| {{{b'}_{\left( 1 \right)}}} \right|}}{b_{\left( 1 \right)}}{{b'}_{\left( 1 \right)}} \otimes \left\{ {{b_{\left( 2 \right)}},{{b'}_{\left( 2 \right)}}} \right\}, \end{array} $

其中b, b′∈B为齐次元.

从而有

$ \begin{array}{l} \left\{ {\Delta \left( {a \otimes b} \right),\Delta \left( {a' \otimes b'} \right)} \right\} = \left\{ {{{\left( { - 1} \right)}^{\left| {{a_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right|}}{a_{\left( 1 \right)}} \otimes } \right.\\ {b_{\left( 1 \right)}} \otimes {a_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}},{\left( { - 1} \right)^{\left| {{{a'}_{\left( 2 \right)}}} \right|\left| {{{b'}_{\left( 1 \right)}}} \right|}}{{a'}_{\left( 1 \right)}} \otimes {{b'}_{\left( 1 \right)}} \otimes {{a'}_{\left( 2 \right)}} \otimes \\ \left. {{{b'}_{\left( 2 \right)}}} \right\} = {\left( { - 1} \right)^{\left| {{a_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right| + \left| {{{a'}_{\left( 2 \right)}}} \right|\left| {{{b'}_{\left( 1 \right)}}} \right|}} \times \\ \left[ {{{\left( { - 1} \right)}^{\left| {{a_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}}} \right|\left( {\left| {{{a'}_{\left( 1 \right)}} \otimes {{b'}_{\left( 1 \right)}}} \right| + p} \right)}}\left\{ {{a_{\left( 1 \right)}} \otimes {b_{\left( 1 \right)}},{{a'}_{\left( 1 \right)}} \otimes } \right.} \right.\\ \left. {{{b'}_{\left( 1 \right)}}} \right\} \otimes \left( {{a_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}}} \right) \cdot \left( {{{a'}_{\left( 2 \right)}} \otimes {{b'}_{\left( 2 \right)}}} \right) + \\ {\left( { - 1} \right)^{\left( {\left| {{a_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}}} \right| + p} \right)\left| {{{a'}_{\left( 1 \right)}} \otimes {{b'}_{\left( 1 \right)}}} \right|}}\left( {{a_{\left( 1 \right)}} \otimes {b_{\left( 1 \right)}}} \right) \cdot \left( {{{a'}_{\left( 1 \right)}} \otimes } \right.\\ \left. {\left. {{{b'}_{\left( 1 \right)}}} \right) \otimes \left\{ {{a_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}},{{a'}_{\left( 2 \right)}} \otimes {{b'}_{\left( 2 \right)}}} \right\}} \right] = \\ {\left( { - 1} \right)^{\left| {{a_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right| + \left| {{{a'}_{\left( 2 \right)}}} \right|\left| {{{b'}_{\left( 1 \right)}}} \right| + \left| {{a_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}}} \right|\left| {{{a'}_{\left( 1 \right)}} \otimes {{b'}_{\left( 1 \right)}}} \right|}} \times \\ \left[ {{{\left( { - 1} \right)}^{p\left| {{a_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}}} \right| + \left| {{{a'}_{\left( 2 \right)}}} \right|\left| {{b_{\left( 2 \right)}}} \right| + \left| {{{a'}_{\left( 1 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right|}}\left( {{{\left( { - 1} \right)}^{p\left| {{b_{\left( 1 \right)}}} \right|}} \times } \right.} \right.\\ \left\{ {{a_{\left( 1 \right)}},{{a'}_{\left( 1 \right)}}} \right\} \otimes {b_{\left( 1 \right)}}{{b'}_{\left( 1 \right)}} + {\left( { - 1} \right)^{p\left| {{{a'}_{\left( 1 \right)}}} \right|}}{a_{\left( 1 \right)}}{{a'}_{\left( 1 \right)}} \otimes \\ \left. {\left\{ {{b_{\left( 1 \right)}},{{b'}_{\left( 1 \right)}}} \right\}} \right) \otimes {a_{\left( 2 \right)}}{{a'}_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}}{{b'}_{\left( 2 \right)}} + \\ {\left( { - 1} \right)^{p\left| {{{a'}_{\left( 1 \right)}} \otimes {{b'}_{\left( 1 \right)}}} \right| + \left| {{{a'}_{\left( 1 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right| + \left| {{{a'}_{\left( 2 \right)}}} \right|\left| {{b_{\left( 2 \right)}}} \right|}}{a_{\left( 1 \right)}}{{a'}_{\left( 1 \right)}} \otimes \\ {b_{\left( 1 \right)}}{{b'}_{\left( 1 \right)}} \otimes \left( {{{\left( { - 1} \right)}^{p\left| {{b_{\left( 2 \right)}}} \right|}}\left\{ {{a_{\left( 2 \right)}},{{a'}_{\left( 2 \right)}}} \right\} \otimes {b_{\left( 2 \right)}}{{b'}_{\left( 2 \right)}} + } \right.\\ \left. {\left. {{{\left( { - 1} \right)}^{p\left| {{{a'}_{\left( 2 \right)}}} \right|}}{a_{\left( 2 \right)}}{{a'}_{\left( 2 \right)}} \otimes \left\{ {{b_{\left( 2 \right)}},{{b'}_{\left( 2 \right)}}} \right\}} \right)} \right] \end{array} $

$ \begin{array}{l} \Delta \left( {\left\{ {a \otimes b,a' \otimes b'} \right\}} \right) = \Delta \left( {{{\left( { - 1} \right)}^{\left( {\left| {a'} \right| + p} \right)\left| b \right|}}\left\{ {a,a'} \right\} \otimes } \right.\\ \left. {bb' + {{\left( { - 1} \right)}^{\left( {\left| b \right| + p} \right)\left| {a'} \right|}}aa' \otimes \left\{ {b,b'} \right\}} \right) = \\ {\left( { - 1} \right)^{\left( {\left| {a'} \right| + p} \right)\left| b \right|}}\Delta \left( {\left\{ {a,a'} \right\} \otimes bb'} \right) + \\ {\left( { - 1} \right)^{\left( {\left| b \right| + p} \right)\left| {a'} \right|}}\Delta \left( {aa' \otimes \left\{ {b,b'} \right\}} \right) = \\ {\left( { - 1} \right)^{\left( {\left| {a'} \right| + p} \right)\left| b \right| + \left| {{{\left\{ {a,a'} \right\}}_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}{{b'}_{\left( 1 \right)}}} \right| + \left| {{b_{\left( 2 \right)}}} \right|\left| {{{b'}_{\left( 1 \right)}}} \right|}} \times \\ {\left\{ {a,a'} \right\}_{\left( 1 \right)}} \otimes {b_{\left( 1 \right)}}{{b'}_{\left( 1 \right)}} \otimes {\left\{ {a,a'} \right\}_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}}{{b'}_{\left( 2 \right)}} + \\ {\left( { - 1} \right)^{\left( {\left| b \right| + p} \right)\left| {a'} \right| + \left| {{a_{\left( 2 \right)}}{{a'}_{\left( 2 \right)}}} \right|\left| {{{\left\{ {b,b'} \right\}}_{\left( 1 \right)}}} \right| + \left| {{a_{\left( 2 \right)}}} \right|\left| {{{a'}_{\left( 1 \right)}}} \right|}} \times \\ {a_{\left( 1 \right)}}{{a'}_{\left( 1 \right)}} \otimes {\left\{ {b,b'} \right\}_{\left( 1 \right)}} \otimes {a_{\left( 2 \right)}}{{a'}_{\left( 2 \right)}} \otimes {\left\{ {b,b'} \right\}_{\left( 2 \right)}} = \\ {\left( { - 1} \right)^{\left( {\left| {a'} \right| + p} \right)\left| b \right| + \left| {{a_{\left( 2 \right)}}{{a'}_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}{{b'}_{\left( 1 \right)}}} \right| + \left| {{b_{\left( 2 \right)}}} \right|\left| {{{b'}_{\left( 1 \right)}}} \right|}} \times \\ \left[ {{{\left( { - 1} \right)}^{\left( {\left| {{{a'}_{\left( 1 \right)}}} \right| + p} \right)\left| {{a_{\left( 2 \right)}}} \right|}}\left\{ {{a_{\left( 1 \right)}},{{a'}_{\left( 1 \right)}}} \right\} \otimes {b_{\left( 1 \right)}}{{b'}_{\left( 1 \right)}} \otimes } \right.\\ {a_{\left( 2 \right)}}{{a'}_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}}{{b'}_{\left( 2 \right)}} + {\left( { - 1} \right)^{p\left| {{b_{\left( 1 \right)}}{{b'}_{\left( 1 \right)}}} \right| + \left( {\left| {{a_{\left( 2 \right)}}} \right| + p} \right)\left| {{{a'}_{\left( 1 \right)}}} \right|}} \times \\ \left. {{a_{\left( 1 \right)}}{{a'}_{\left( 1 \right)}} \otimes {b_{\left( 1 \right)}}{{b'}_{\left( 1 \right)}} \otimes \left\{ {{a_{\left( 2 \right)}},{{a'}_{\left( 2 \right)}}} \right\} \otimes {b_{\left( 2 \right)}}{{b'}_{\left( 2 \right)}}} \right] + \\ {\left( { - 1} \right)^{\left( {\left| b \right| + p} \right)\left| {a'} \right| + \left| {{a_{\left( 2 \right)}}{{a'}_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}{{b'}_{\left( 1 \right)}}} \right| + \left| {{a_{\left( 2 \right)}}} \right|\left| {{{a'}_{\left( 1 \right)}}} \right|}} \times \\ \left[ {{{\left( { - 1} \right)}^{p\left| {{a_{\left( 2 \right)}}{{a'}_{\left( 2 \right)}}} \right| + \left( {\left| {{{b'}_{\left( 1 \right)}}} \right| + p} \right)\left| {{b_{\left( 2 \right)}}} \right|}}{a_{\left( 1 \right)}}{{a'}_{\left( 1 \right)}} \otimes } \right.\\ \left\{ {{b_{\left( 1 \right)}},{{b'}_{\left( 1 \right)}}} \right\} \otimes {a_{\left( 2 \right)}}{{a'}_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}}{{b'}_{\left( 2 \right)}} + {\left( { - 1} \right)^{\left( {\left| {{b_{\left( 2 \right)}}} \right| + p} \right)\left| {{{b'}_{\left( 1 \right)}}} \right|}} \times \\ \left. {{a_{\left( 1 \right)}}{{a'}_{\left( 1 \right)}} \otimes {b_{\left( 1 \right)}}{{b'}_{\left( 1 \right)}} \otimes {a_{\left( 2 \right)}}{{a'}_{\left( 2 \right)}} \otimes \left\{ {{b_{\left( 2 \right)}},{{b'}_{\left( 2 \right)}}} \right\}} \right]. \end{array} $

分别比较各项系数, 可得

$ \Delta \left( {\left\{ {a \otimes b,a' \otimes b'} \right\}} \right) = \left\{ {\Delta \left( {a \otimes b} \right),\Delta \left( {a' \otimes b'} \right)} \right\}. $

引理5  由定理1中的定义, 有SAB的对极, 即

$ u\left( {I \otimes S} \right)\Delta = u\left( {S \otimes I} \right)\Delta = \eta \varepsilon . $

证明  注意到

$ \begin{array}{l} u\left( {I \otimes S} \right)\Delta \left( {a \otimes b} \right) = \\ u\left( {I \otimes S} \right)\left( {{{\left( { - 1} \right)}^{\left| {{a_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right|}}{a_{\left( 1 \right)}} \otimes {b_{\left( 1 \right)}} \otimes {a_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}}} \right) = \\ {\left( { - 1} \right)^{\left| {{a_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right|}}\left( {{a_{\left( 1 \right)}} \otimes {b_{\left( 1 \right)}}} \right)S\left( {{a_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}}} \right) = \\ {\left( { - 1} \right)^{\left| {{a_{\left( 2 \right)}}} \right|\left| {{b_{\left( 1 \right)}}} \right|}}\left( {{a_{\left( 1 \right)}} \otimes {b_{\left( 1 \right)}}} \right)\left( {S\left( {{a_{\left( 2 \right)}}} \right) \otimes S\left( {{b_{\left( 2 \right)}}} \right)} \right) = \\ {a_{\left( 1 \right)}}S\left( {{a_{\left( 2 \right)}}} \right) \otimes {b_{\left( 1 \right)}}S\left( {{b_{\left( 2 \right)}}} \right) = \varepsilon \left( a \right)\varepsilon \left( b \right){1_{A \otimes B}}, \end{array} $

从而

$ \begin{array}{*{20}{c}} {\eta \varepsilon \left( {a \otimes b} \right) = \eta \left( {\varepsilon \left( a \right)\varepsilon \left( b \right)} \right) = \varepsilon \left( a \right)\varepsilon \left( b \right){1_{A \otimes B}} = }\\ {u\left( {I \otimes S} \right)\left( {a \otimes b} \right).} \end{array} $

类似地, 有

$ u\left( {S \otimes I} \right)\Delta = \eta \varepsilon . $

定理1的证明  (1)由文献[9]可知, (AB, u, η, d, {·, ·})是p次微分分次Poisson代数.再由引理1~引理5可得, (AB, u, η, Δ, ε, S, {·, ·}, d)是p次微分分次Poisson Hopf代数.故(1)成立.

(2) 定义映射

$ \begin{array}{*{20}{c}} {\varphi :A \otimes B \to B \otimes A,}\\ {\varphi \left( {a \otimes b} \right) = {{\left( { - 1} \right)}^{\left| a \right|\left| b \right|}}b \otimes a,} \end{array} $

其中,aA, bB为齐次元.要证dg-PHA为对称monoidal范畴, 只须证φ是同构映射.由于φ°φ=1, 故只需证φ是微分分次Poisson Hopf代数同态.注意到εAεB都是次数为0的齐次线性映射, 所以有

$ \begin{array}{l} {\Delta _{B \otimes A}}\varphi \left( {a \otimes b} \right) = {\Delta _{B \otimes A}}\left( {{{\left( { - 1} \right)}^{\left| a \right|\left| b \right|}}b \otimes a} \right) = \\ {\left( { - 1} \right)^{\left| a \right|\left| b \right| + \left| {{a_{\left( 1 \right)}}} \right|\left| {{b_{\left( 2 \right)}}} \right|}}{b_{\left( 1 \right)}} \otimes {a_{\left( 1 \right)}} \otimes {b_{\left( 2 \right)}} \otimes {a_{\left( 2 \right)}}, \end{array} $
$ \begin{array}{l} \left( {\varphi \otimes \varphi } \right){\Delta _{A \otimes B}}\left( {a \otimes b} \right)\\ \left( {\varphi \otimes \varphi } \right)\left( {{{\left( { - 1} \right)}^{\left| {{a_{\left( 2 \right)}}} \right|\left| {{a_{\left( 1 \right)}}} \right|}}{a_{\left( 1 \right)}} \otimes {b_{\left( 1 \right)}} \otimes {a_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}}} \right) = \\ {\left( { - 1} \right)^{\left| a \right|\left| {{b_{\left( 1 \right)}}} \right| + \left| {{a_{\left( 2 \right)}}} \right|\left| {{b_{\left( 2 \right)}}} \right|}}{b_{\left( 1 \right)}} \otimes {a_{\left( 1 \right)}} \otimes {b_{\left( 2 \right)}} \otimes {a_{\left( 2 \right)}} \end{array} $

$ \begin{array}{l} {\varepsilon _{B \otimes A}}\varphi \left( {a \otimes b} \right) = {\varepsilon _{B \otimes A}}\left( {{{\left( { - 1} \right)}^{\left| a \right|\left| b \right|}}b \otimes a} \right) = \\ {\left( { - 1} \right)^{\left| a \right|\left| b \right|}}\varepsilon \left( b \right)\varepsilon \left( a \right) = \varepsilon \left( a \right)\varepsilon \left( b \right) = {\varepsilon _{A \otimes B}}\left( {a \otimes b} \right). \end{array} $

因此, φ是分次余代数同态.注意到φ是微分分次Poisson代数同态, kp次微分分次Poisson Hopf代数, 故结论成立.

(3) 由例1可知, Aop, Bop与(AB)op都是p次微分分次Poisson Hopf代数.注意到(AB)opAopBop的代数结构均由(uop, η, Δop, ε, S, {·, ·}op, d)所决定, 其中,

$ \eta \left( {{1_k}} \right): = {1_A} \otimes {1_B}, $
$ \varepsilon \left( {a \otimes b} \right): = {\varepsilon _A}\left( a \right){\varepsilon _B}\left( b \right), $
$ S\left( {a \otimes b} \right): = {S_A}\left( a \right) \otimes {S_B}\left( b \right), $
$ d\left( {a \otimes b} \right): = {d_A}\left( a \right) \otimes b + {\left( { - 1} \right)^{\left| a \right|}}a \otimes {d_B}\left( b \right), $
$ {u^{{\rm{op}}}}\left( {\left( {a \otimes b} \right) \otimes \left( {a' \otimes b'} \right)} \right): = {\left( { - 1} \right)^{\left| {a'} \right|\left| b \right|}}aa' \otimes bb', $
$ \begin{array}{l} {\left\{ {a \otimes b,a' \otimes b'} \right\}^{{\rm{op}}}}: = - {\left( { - 1} \right)^{\left( {\left| {a'} \right| + p} \right)\left| b \right|}}\left( {{{\left\{ {a,a'} \right\}}_A} \otimes } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {bb'} \right) - {\left( { - 1} \right)^{\left( {\left| b \right| + p} \right)\left| {a'} \right|}}\left( {aa' \otimes {{\left\{ {b,b'} \right\}}_B}} \right), \end{array} $
$ \begin{array}{l} {\Delta ^{{\rm{op}}}}\left( {a \otimes b} \right): = {\left( { - 1} \right)^{\left| {{a_{\left( 1 \right)}}} \right|\left| {{a_{\left( 2 \right)}}} \right| + \left| {{a_{\left( 1 \right)}}} \right|\left| {{b_{\left( 2 \right)}}} \right| + \left| {{b_{\left( 1 \right)}}} \right|\left| {{b_{\left( 2 \right)}}} \right|}} \times \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{a_{\left( 2 \right)}} \otimes {b_{\left( 2 \right)}} \otimes {a_{\left( 1 \right)}} \otimes {b_{\left( 1 \right)}}, \end{array} $

a, a′∈A, b, b′∈B为齐次元.故

$ {\left( {A \otimes B} \right)^{{\rm{op}}}} = {A^{{\rm{op}}}} \otimes {B^{{\rm{op}}}}. $

证毕!

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