Function: Octonion[omul] - octonion product in the octonion non-associative division ring and its infix form '&o'

Calling Sequence:

omul(o1,o2,...on);

o1 &o o2 &o ... &o on;

Parameters:

o1, o2, ..., on - expressions of the type 'octonion'

Description:

Procedure 'omul' and its infix form '&o' give the octonion product in the non-associative division ring of octonions.

Octonions are considered here as para-vectors in the Clifford algebra Cl(0,7), that is, any expression of the form

x0 + x1*e1 + x2*e2 + x3*e3 + x4*e4 + x5*e5 + x6*e6 + x7*e7

where x0, x1, ..., x7 are real numbers, is of type 'octonion'. See `type/octonion` for more information.

The basis elements for the octonion algebra are {1,e1,e2,e3,e4,e5,e6,e7} (sometimes 'Id' is returned instead of '1'). They are collected in a global variable '_octbasis'. The basis elements {e1,e2,e3,e4,e5,e6,e7} give pure octonions and are collected in a global variable _pureoctbasis.

To display environmental variables from CLIFFORD and Octonion, use Cliff5[CLIFFORD_ENV] .

The infix form is given by `&o`, e.g., omul(e1,e2) = e1 &o e2. Remember that 'omul' is non-associative!

Octonionic inverse is computed with Octonion[oinv] .

To speed up computations, set the global variable _prolevel to 'true'. To find out more, see help page on Cliff5[cliparse] .

To see the default multiplication table try Octonion[omultable] and to define your own octonionic mutliplication see Octonion[def_omultable] .

Examples:

> restart:with(Cliff5):with(Octonion);

Warning, new definition for init

The following is the default octonionic mutliplication table:

> omultable();

> o1:=1-2*e1+3*e3+e4-e6+e7;

> o2:=2+e3-4*e6+e7;

> type(o1,octonion),type(o2,octonion);

> omul(o1,o2);

Octonionic multiplication is not commutative:

> o1 &o o2;

> o2 &o o1;

We show now that it is not associative either:

> (e1 &o e2) &o e3;

> e1 &o (e2 &o e3);

> o3:=2-3*e1+e5-e7;

> (o1 &o o2) &o o3;

> o1 &o (o2 &o o3);

The difference between (o1 &o o2) &o o3 and o1 &o (o2 &o o3) is measured by an associator, or see Octonion[associator] :

> associator(o1,o2,o3);

The difference between o1 &o o2 and o2 &o o1 is measured by a commutator, or see Octonion[commutator] :

> commutator(o1,o2);

>

See Also: Cliff5[version] , Octonion[oinv] , Octonion[def_omultable] , Octonion[omultable]