cp's OEIS Frontend

A196995 Determinant of Killing form K(x,y) of the Lie algebra sl(n,C) for n >=1.

%I A196995 #8 Mar 31 2012 10:31:29
%S A196995 0,-128,-5038848,140737488355328,5000000000000000000000000,
%T A196995 -354400937492545922690672153504784580608,
%U A196995 -72317557999158469111384459491956546088110808312359944192,57896044618658097711785492504343953926634992332820282019728792003956564819968
%N A196995 Determinant of Killing form K(x,y) of the Lie algebra sl(n,C) for n >=1.
%C A196995 K(x,y) = 2n*Tr(xy)
%D A196995 J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, 1972, 21-22
%F A196995 a(n) = (-1)^binomial(n,2) *2^(n^2-1)*n^(n^2) for n>= 2
%p A196995 interface(rtablesize=infinity):
%p A196995 with(LinearAlgebra):
%p A196995 for n from 1 to 12 do
%p A196995 for i from 1 by 1 to n-1 do
%p A196995    M[i] := Matrix(n);
%p A196995    M[i](i,i) := 1;
%p A196995    M[i](i+1,i+1) := -1;
%p A196995   end do:
%p A196995   ctr := n:
%p A196995   for i from 1 by 1 to n do
%p A196995   for j from 1 by 1 to n do
%p A196995   if(i <> j) then
%p A196995     M[ctr] := Matrix(n);
%p A196995     M[ctr](i,j) := 1;
%p A196995     ctr := ctr +1;
%p A196995   end if
%p A196995   end do:
%p A196995 end do:
%p A196995 A := Matrix(n^2-1):
%p A196995 for i from 1 by 1 to n^2-1 do
%p A196995   for j from 1 by 1 to n^2-1 do
%p A196995    A(i,j) := 2*n*Trace(M[i].M[j]):
%p A196995   end do:
%p A196995   end do:
%p A196995   print(Determinant(A));
%p A196995 end do:
%p A196995 # Alternatively, using the second description
%p A196995   print(0);
%p A196995   for n from 2 to 20 do
%p A196995   print((-1)^(binomial(n,2))*2^(n^2-1)*n^(n^2));
%p A196995   end do:
%K A196995 easy,sign
%O A196995 1,2
%A A196995 _Carmen Bruni_, Oct 08 2011