Tensor

Naively, a tensor is a generalized collection of vector and covector using the tensor product. It is an element of a space built from vectors and covectors via tensor products (and linear combinations).

Mathematically, A \((k,\ell)\)-tensor on a vector space $V$ is an element of \(V^{\otimes k}\otimes (V^*)^{\otimes \ell}\) (equivalently a multilinear map \((V^*)^k\times V^\ell\to\mathbb{F}\)) characterized by the universal property of the tensor product.

Physically, a tensor is something obeying physical law under the coordinate transformations. Energy and mass for example are rank-0 tensor, which are called scalars. Force (usually in CM) is rank-1 tensor, which is called vector. To be clear, it is \((1, 0)\)-tensor. Bivector is \((2,0)\)-tensor. Conductivity, elasticity, piezoelastic effect, etc. are rank-2 tensor. Moment of Inertia, metric tensor, Ricci curvature, symplectic form are \((0,2)\)-tensor.

Constructing algebras from a vector space

• free associative algebra $TV=\mathbb{F}\oplus V\oplus(V\otimes V)\oplus\cdots$
• Add constraints such as symmetry/antisymmetry/inner product: write them as equations that become zero, then quotient by the two-sided ideal $I$ generated by those equations. That is, choose a set of relations $R\subset TV$. Create the smallest two-sided ideal $I=\langle R\rangle$ containing $R$. Define the desired algebra as $A=TV/I$.
  • symmetric algebra $SV=TV/\langle v\otimes w-w\otimes v\rangle$
  • exterior algebra $\Lambda V=TV/\langle v\otimes v\rangle$
  • Clifford algebra $\mathrm{Cl}(V,g)=TV/\langle v\otimes v-\langle v,v\rangle1\rangle$
• universal property: Given the data “elements of $V$ satisfy certain relations and map into some algebra $A$”, there exists a unique algebra homomorphism $TV/I\rightarrow A$ (= free algebra with relations)

Tensor algebra

• $T^kV$: $k^{th}$ tensor product of $V$
• $\dim T^kV=n^k$, $\dim TV=\infty$
• $TV=\bigoplus_{k\geq0}T^kV=\mathbb{R}\oplus V\oplus (V\otimes V)\oplus\cdots$
• Inner product extension: if $V$ has a pseudo inner product $\langle,\rangle$, then $\langle v_1\otimes\cdots\otimes v_k,w_1\otimes\cdots\otimes w_k\rangle=\prod_i\langle v_i,w_i\rangle$; extend with orthogonality between different degrees.
• Multi-index objects in QFT (tensors, spinors, etc.) are placed in homogeneous components of $TV$

exterior algebra

• $\Lambda^k V=V\wedge\cdots\wedge V$ ($k$ times), completely anti-symmetric. Since $k>n$ automatically vanishes, $\Lambda V=\bigoplus_{k=0}^n \Lambda^k V$. The product is $A\wedge B\in \Lambda^{j+k} V$ and $A\wedge B=(-1)^{jk}B\wedge A$
• Inner product (on homogeneous $k$-components): for simple $k$-vectors, $\langle v_1\wedge\cdots\wedge v_k,w_1\wedge\cdots\wedge w_k\rangle=\det\langle v_i,w_j\rangle$; different $k$’s are orthogonal.
• $\Lambda^k V=0$ for $k>n$: if $\dim V=n$, then any $k>n$ vectors are linearly dependent (by pigeonhole principle). Since the product is completely anti-symmetric, if there is linear dependence among the factors, the $\wedge$ product becomes 0. Therefore, all $k^{th}$ wedge products $v_1\wedge\cdots\wedge v_k=0$

Combinatorial notations

• Two notations for complete antisymmetry (1) Permutation sign $\mathrm{sign}(\pi)$: $v_1\wedge\cdots\wedge v_k=\mathrm{sign}(\pi)v_{\pi(1)}\wedge\cdots\wedge v_{\pi(k)}$ (2) Levi-Civita symbol: $v_1\wedge\cdots\wedge v_k=\frac{1}{k!}\sum_{i_1,\dots,i_k}\epsilon_{i_1,\dots,i_k} v_{i_1}\wedge\cdots\wedge v_{i_k}$
• Generalized Kronecker delta: $\delta_{\mu_1\cdots\mu_k}^{\nu_1\cdots\nu_k}=\sum_\pi\mathrm{sign}(\pi)\delta_{\mu_1}^{\nu_{\pi(1)}}\cdots\delta_{\mu_k}^{\nu_{\pi(k)}}=\frac{1}{(n-k)!}\epsilon^{\nu_1\cdots\nu_k \lambda}\epsilon_{\mu_1\cdots\mu_k\lambda}$ and other relations
• Basis transformation and determinant: $e_\mu’=M_\mu^\nu e_\nu\Rightarrow e_1’\wedge\cdots\wedge e_n’=\det(M)e_1\wedge\cdots\wedge e_n$

Graded algebras

• $\mathbb{Z}2$ graded algebra $A=A+\oplus A_-$
  • $A_rA_s=A_{r+s}(\text{mod 2}),\quad r,s=0,1,\quad A_0=A_+,\quad A_1=A_-$
  • is graded commutative when : $ab=(-1)^{d(a)d(b)}ba,\quad a,b\in A$ where $d(a)=r$ if $a\in A_r$ is the parity of $a$
• A generalized used in physics: DeWitt algebra
  • algebra $B$ of formal series with a unit $e$ and an infinite number of generators $z^I, I\in \mathbb{N}$ with the property $z^Iz^J=-z^Jz^I$
  • $a=\sum_{p\in \mathbb{N}}a(p),\quad a(p)=\frac{1}{p!}a_{I_1,\dots,I_p}z^{I_1}\cdots z^{I_p}$
    • $a(0)=a_0e$ is the body of $a$.
    • $a_s=\sum_{p\geq 1}a(p)$ is its soul.
    • $a_0,a_{I_0,\dots,I_p}$ are real or complex. $a_{I_0,\dots,I_p}$ is totally antisymmetric in $I_0,\dots,I_p$.
    • $p$ is the degree of $a(p)$.
• Fermionic superalgebra is $\mathbb{Z}_2$-graded Lie superalgebra.

Clifford algebras

Geometric algebra

Tensor algebras on the dual space

the structure of the dual space

Tensors

Tensors as multilinear mappings

Abstract index notation

Tensors as multi-dimensional arrays

Exterior forms

TBA