HELP FOR: get_alg/dataformat - the intermediary format of the function get_alg

DESCRIPTION:

• The graphical interface of the function get_alg allows input of an algebra via a quiver Q and a generating set of relations for a suitable ideal of the quiveralgebra kQ over some field k. The corresponding data is presented using a table -let's call it a- having three entries indexed by the keywords 'quiver', 'relations', and 'display'. Consequently, the component a[quiver] describes the quiver Q, the component a[relations] the generating set of relations. The component a[display] contains data enabling the function to redisplay an algebra when it is passed back function for further treatment.

• The format of these components is as follows:
  • a[quiver], describing the quiver of an algebra is either a list
    
    [n]
    
    containing a single entry, the number of vertices of the quiver n in case the quiver contains no arrows, or a pair
    
    [n,l]
    
    where the first entry n again is the number of vertices and the second entry l is a list
    
    l=[l₁, ... , l_v]
    
    containing the arrows of the quiver, each arrow given by a triple
    
    l_j=[s,t,m]
    
    where s is the startvertex, t the terminal vertex and m an integer assigned to the arrow as name. For this purpose, the vertices of the quiver are numbered consecutively from 1 to n. In fact, also the arrows are numbered consecutively starting at 1.

  • a[relations] is a set
    
    {r₁, ..., r_u}
    
    of relations generating an ideal I, as mentioned above. Especially, if I is the zerp ideal, a[relations] will be the empty set. Otherwise, each relation is given by a list
    
    r_i=[r_i,1 , ... , r_i,w(i)]
    
    consisting of pairs
    
    r_i,j=[ k_i,j,p_i,j ]
    
    where k_i,j is an integral coefficient and p_i,j is a path in the quiver Q described by a[quiver].
    The relation r_i is then given by
    
    k{i,j} p{i,j} = 0.
    
    A path p_i,j is given by a list
    
    p_i,j=[p₁, ...,p_y ]
    
    where the p_l are the numbers associated to arrows in the quiver Q. The path p then is the formal composition of these arrows, from left to right.

  • a[display] either consists of an empty list
    
    [ ]
    
    in case the quiver Q described by a[quiver] is the empty quiver, or a list with a single entry
    
    [l₁]
    
    which itself is a list, in case the quiver Q is nonempty but contains no arrows, or a pair of lists
    
    [l₁,l₂]
    
    if there are arrows. Here, l₁ = [ [x₁,y₁], ..., [x_n,y_n] ] is a list of coordinate pairs for the vertices of the quiver Q, and l₂= [ [p₁,q₁], ..., [p_v,q_v] ] is a list of coordinate pairs for displaying the arrows of the quiver Q. Display of arrows is done in such a way, that an arrow p =[s,t,z] from s to t passes through the point in the plane given by the coordinates [p_z,q_z].

EXAMPLES:

> get_alg();

> get_alg(%);

> get_alg(%);

> get_alg(%);