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  • mathematical algorithms
    • FriCAS Algebra
      • LinearOperator
        • ClosedLinearOperator
          • SandBoxRibbonCategory <-- You are here.

Refs:

• http://en.wikipedia.org/wiki/Braided_monoidal_category
• http://en.wikipedia.org/wiki/Strongly_ribbon_category

We need the Axiom LinearOperator library.

(1) -> )library CARTEN ARITY CMONAL CPROP CLOP CALEY

 >> System error:
 The value
 15684
is not of type
 LIST

Use the following macros for convenient notation

-- summation
macro Σ(x,i,n)==reduce(+,[x for i in n])

-- list
macro Ξ(f,i,n)==[f for i in n]

-- subscript and superscripts
macro sb == subscript

macro sp == superscript

L is the domain of 2-dimensional linear operators over the rational functions Q (Expression Integer), i.e. ratio of polynomials with integer coefficients.

dim:=2

macro L == List

macro C == CaleyDickson

macro Q == Expression Integer

L := ClosedLinearOperator(OVAR ['1,'2], Q)

 There are no library operations named ClosedLinearOperator 
 Use HyperDoc Browse or issue
 )what op ClosedLinearOperator
 to learn if there is any operation containing " 
 ClosedLinearOperator " in its name.

 Cannot find a definition or applicable library operation named 
 ClosedLinearOperator with argument type(s) 
 Type
 Type

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.

We want to be able to implement linear operators with two "colors" like the following:

An example starting with Complex Algebra

The basis consists of the real and imaginary units. We use complex multiplication to form the "multiplication table" as a matrix. Then the structure constants can be obtained by dividing each matrix entry by the list of basis vectors.

Split-complex can be specified by Caley-Dickson parameter (q0 = -1)

--q:=1 -- split-complex
q:=sp('i,[2])

QQ := C(Q,'i,q);

 There are no library operations named CaleyDickson 
 Use HyperDoc Browse or issue
 )what op CaleyDickson
 to learn if there is any operation containing " CaleyDickson " in
 its name.

 Cannot find a definition or applicable library operation named 
 CaleyDickson with argument type(s) 
 Type
 Variable(i)
 Symbol

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.

Basis: Each B.i is a complex number

B:L QQ := map(x +-> hyper x,1$SQMATRIX(dim,Q)::L L Q)

 QQ is not a valid type.

Multiplication of arbitrary quaternions a and b

a:=Σ(sb('a,[i])*e.i, i,1..dim)

 There are no library operations named e 
 Use HyperDoc Browse or issue
 )what op e
 to learn if there is any operation containing " e " in its name.
 Cannot find a definition or applicable library operation named e 
 with argument type(s) 
 PositiveInteger

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.
 FriCAS will attempt to step through and interpret the code.
 There are no library operations named e 
 Use HyperDoc Browse or issue
 )what op e
 to learn if there is any operation containing " e " in its name.

 Cannot find a definition or applicable library operation named e 
 with argument type(s) 
 PositiveInteger

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.

Multiplication is Associative

test(
 ( Y! J ) / _
 ( Y ) = _
 ( I Y ) / _
 ( Y! ) )

 There are no library operations named Y! 
 Use HyperDoc Browse or issue
 )what op Y!
 to learn if there is any operation containing " Y! " in its name.

 Cannot find a definition or applicable library operation named Y! 
 with argument type(s) 
 Variable(J)

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.

A scalar product is denoted by the (2,0)-tensor U = \{ u_{ij} \}

U:=Σ(Σ(script('u,[[],[i,j]])*d.i*($/d.j), i,1..dim), j,1..dim)

 There are no library operations named d 
 Use HyperDoc Browse or issue
 )what op d
 to learn if there is any operation containing " d " in its name.
 Cannot find a definition or applicable library operation named d 
 with argument type(s) 
 PositiveInteger

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.
 FriCAS will attempt to step through and interpret the code.
 There are no library operations named d 
 Use HyperDoc Browse or issue
 )what op d
 to learn if there is any operation containing " d " in its name.

 Cannot find a definition or applicable library operation named d 
 with argument type(s) 
 PositiveInteger

 Perhaps you should use "@" to indicate the required return type, 
 or "$" to specify which version of the function you need.

Definition 1

We say that the scalar product is associative if the tensor equation holds:

 Y = Y
 U U

In other words, if the (3,0)-tensor:

(three-point function) is zero.

How should we color this?