# 张量代数

：本文中所有代数都假设是有单位的结合

## 构造

${\displaystyle V}$ ${\displaystyle K}$ 上一个向量空间。对任何非负整数${\displaystyle k}$ ，我们定以${\displaystyle V}$ ${\displaystyle k}$ 次张量积${\displaystyle V}$ 与自己的${\displaystyle k}$ 张量积

${\displaystyle T^{k}V=V^{\otimes k}={\underset {k}{\underbrace {V\otimes V\otimes \cdots \otimes V} }}}$

${\displaystyle T(V)}$ 为所有${\displaystyle T^{k}V}$ ${\displaystyle k=0,1,2,\ldots }$ ）的直和

${\displaystyle T(V)=\bigoplus _{k=0}^{\infty }T^{k}V=K\oplus V\oplus (V\otimes V)\oplus (V\otimes V\otimes V)\oplus \cdots }$

${\displaystyle T(V)}$ 中的乘法由典范同构确定：

${\displaystyle T^{k}V\otimes T^{\ell }V\to T^{k+\ell }V}$

## 余代数结构

${\displaystyle \Delta (v_{1}\otimes \dots \otimes v_{m}):=\sum _{i=0}^{m}(v_{1}\otimes \dots \otimes v_{i})\otimes (v_{i+1}\otimes \dots \otimes v_{m})}$

${\displaystyle T^{m}V\to \bigoplus _{i+j=m}T^{i}V\otimes T^{j}V}$

${\displaystyle \varepsilon }$ 也与分次相容。

${\displaystyle \Delta (x_{1}\otimes \dots \otimes x_{m})=\sum _{p=0}^{m}\sum _{\sigma \in \mathrm {Sh} _{p,m-p}}\left(v_{\sigma (1)}\otimes \dots \otimes v_{\sigma (p)}\right)\otimes \left(v_{\sigma (p+1)}\otimes \dots \otimes v_{\sigma (m)}\right)}$

${\displaystyle S(x_{1}\otimes \dots \otimes x_{m})=(-1)^{m}x_{m}\otimes \dots \otimes x_{1}}$

## 参考文献

• 陈维桓. 微分流形初步 第二版. 北京: 高等教育出版社. 2001年8月.
• Mac Lane, Saunders. Categories for the Working Mathematician(2nd ed.). GTM5. Spinger, 1998