Tensor Categories

Magma allows tensors and tensor spaces to change categories. Unless a user specifies otherwise, all tensors are assigned a category that is natural to the method by which it was created. For example a tensor created from an algebra will be assigned an algebra category, whereas a tensor created by structure constants will be assigned the Albert homotopism category [Alb42]. Tensor categories influence the behavior of commands such as kernels and images as well as the algebraic invariants such as derivation algebras of a tensor.

Our conventions follow [Wil13a]. In particular, given a tensor \(T\) framed by \([U_{\nu},\dots,U_0]\) then a tensor category for \(T\) will specify a function \(A:[\nu]\to \{-1,0,1\}\) along with a partition \({\cal P}\) of \([\nu]\) such that the following rules apply to the tensors and morphisms in the category.

(1)

for each tensor \(T\) framed by \([U_{\nu},\dots,U_0]\), if \(X\in{\cal P}\), then

\[\forall i,j\in X,\quad U_i=U_j.\]

(2)

Given a second tensor \(S\) framed by \([V_v,\dots,V_0]\), a morphism \(f:T\to S\) (Magma type Hmtp) will be a list \([f_{\nu},\dots,f_0]\) of homomorphisms as follows:

•

(Covariant) if \(A(i)=1\) then \(f_i:U_i\to V_i\);

•

(Constant) if \(A(i)=0\) then \(U_i=V_i\) and \(f_i=1_{U_i}\); or else

•

(Contravariant) \(A(i)=-1\) and \(f_i:U_i\leftarrow V_i\). So if \(A(0)=1\) then

\[\left\langle \sum_{i\in A^{-1}(1)} u_i f_i +\sum_{j\not\in A^{-1}(-1)} v_j\right\rangle_S = \left\langle \sum_{i\in A^{-1}(1)} u_i +\sum_{j\not\in A^{-1}(-1)} v_j f_j\right\rangle_T f_0;\]

if \(A(0)=0\) then

\[\left\langle \sum_{i\in A^{-1}(1)} u_i f_i +\sum_{j\not\in A^{-1}(-1)} v_j\right\rangle_S = \left\langle \sum_{i\in A^{-1}(1)} u_i +\sum_{j\not\in A^{-1}(-1)} v_j f_j\right\rangle_T;\]

else \(A(0)=-1\) and

\[\left\langle \sum_{i\in A^{-1}(1)} u_i f_i +\sum_{j\not\in A^{-1}(-1)} v_j\right\rangle_S f_0 = \left\langle \sum_{i\in A^{-1}(1)} u_i +\sum_{j\not\in A^{-1}(-1)} v_j f_j\right\rangle_T.\]

Magma manages internally the differences between vectors and covectors and more generally tensors and cotensors. Both types are issued the Magma type TenSpcElt. For operations sensitive to the difference, Magma stores a value of co/contra-variance of the tensor as a property of the tensor category. This is the third general property stored in Magma’s tensor category type TenCat.

We use the phrase tensor category exclusively for categories that describe tensors and tensor spaces. In other words, the data structure of a tensor category is a function \(A:[\nu]\rightarrow \{-1,0,1\}\) and a partition \({\cal P}\) of \([\nu]\). It is useful to distinguish from tensors and cotensors at the categorical level, so a tensor category is either covariant or contravariant as well (in the latter case, referred to as a cotensor category).

Constructing Tensor Categories

TensorCategory(A, P): [RngIntElt], {SetEnum} -> TenCat
TensorCategory(A, P): Map, {SetEnum} -> TenCat: Sets up a covariant tensor space category with specified direction of arrows \(A\), and a partition \({\cal P}\) indicating variables to be treated as equivalent. The fiber \(A^{-1}(1)\) denotes the covariant variables, \(A^{-1}(0)\) identifies the constant variables, and \(A^{-1}(-1)\) marks the contra-variant variables.

CotensorCategory(A, P): [RngIntElt], {SetEnum} -> TenCat
CotensorCategory(A, P): Map, {SetEnum} -> TenCat: Sets up a contra-variant tensor space category with specified direction of arrows \(A\), and a partition \({\cal P}\) indicating variables to be treated as equivalent. The fiber \(A^{-1}(1)\) denotes the covariant variables, \(A^{-1}(0)\) identifies the constant variables, and \(A^{-1}(-1)\) marks the contra-variant variables.

Example: Basic Tensor Category Construction (ex-2873fe)

We demonstrate the basic tensor category constructor. The difference between TensorCategory and CotensorCategory is only that the former is covariant and the latter is contravariant.

> C := TensorCategory([1,0,-1], {{0},{1},{2}});
> C;
Tensor category of valence 3 (->,==,<-) ({ 1 },{ 2 },{ 0 })
> IsCovariant(C);
true
>
> arrows := map< {1..5} -> {1} | x :-> 1 >;
> C := CotensorCategory(arrows, {{1..5}});
> C;
Cotensor category of valence 6 (->,->,->,->,->,==) ({ 0 },{ 1 .. 5 })
> IsContravariant(C);
true

Tensor Categories#

Constructing Tensor Categories#

Operations on Tensor Categories#

Categorical Operations#

Categorical Operations on Tensors#

Categorical Operations on Tensor Spaces#

Homotopisms#

Constructions of Homotopisms#

Basic Operations with Homotopisms#

Basic Properties of Homotopisms#

Tensor Categories

Constructing Tensor Categories

Operations on Tensor Categories

Categorical Operations

Categorical Operations on Tensors

Categorical Operations on Tensor Spaces

Homotopisms

Constructions of Homotopisms

Basic Operations with Homotopisms

Basic Properties of Homotopisms