/*  Heizenberg Group                                                     */
/*  simpliest non-abelian group :  < a b | [[a,b],a] = [[a,b],b] = e >   */

n : ident (3) $
z : zeromatrix(3,3) $
     
a : ematrix (3, 3,  1,  1, 2) + n ;
b : ematrix (3, 3,  1,  2, 3) + n ;
c : ematrix (3, 3,  1,  1, 3) + n ;

/*      x ∙ y - y ∙ x        */
lie_braket (x , y) := x . y - y . x $

/*      x ∙ y ∙ x⁻ ∙ y⁻      */
commutat (x , y) := x . y . invert (x) . invert (y) $
     
     /*          test suitee           */
     
"Jacoby equity :" $ 
j1 : lie_braket (lie_braket(a , b) , c) $ 
j2 : lie_braket (lie_braket(b , c) , a) $ 
j3 : lie_braket (lie_braket(c , a) , b) $
print (is (equal ((j1 + j2 + j3) , z))) $

" [a , b] = c where [] - commutator" $ 
print (is (equal (commutat (a, b) , c))) $
     
" [a , c] = n where [] - commutator" $ 
print (is (equal (commutat (a, c) , n))) $

" [b , c] = n where [] - commutator" $     
print (is (equal (commutat (b, c) , n))) $