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

0, -128, -5038848, 140737488355328, 5000000000000000000000000, -354400937492545922690672153504784580608, -72317557999158469111384459491956546088110808312359944192, 57896044618658097711785492504343953926634992332820282019728792003956564819968

Offset: 1

Author

Carmen Bruni, Oct 08 2011

Keywords

• easy
• sign

Comments

K(x,y) = 2n*Tr(xy)

References

• J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, 1972, 21-22

Programs

Maple

interface(rtablesize=infinity):
with(LinearAlgebra):
for n from 1 to 12 do
for i from 1 by 1 to n-1 do
   M[i] := Matrix(n);
   M[i](i,i) := 1;
   M[i](i+1,i+1) := -1;
  end do:
  ctr := n:
  for i from 1 by 1 to n do
  for j from 1 by 1 to n do
  if(i <> j) then
    M[ctr] := Matrix(n);
    M[ctr](i,j) := 1;
    ctr := ctr +1;
  end if
  end do:
end do:
A := Matrix(n^2-1):
for i from 1 by 1 to n^2-1 do
  for j from 1 by 1 to n^2-1 do
   A(i,j) := 2*n*Trace(M[i].M[j]):
  end do:
  end do:
  print(Determinant(A));
end do:
# Alternatively, using the second description
  print(0);
  for n from 2 to 20 do
  print((-1)^(binomial(n,2))*2^(n^2-1)*n^(n^2));
  end do:

Formula

a(n) = (-1)^binomial(n,2) *2^(n^2-1)*n^(n^2) for n>= 2