2

Given $(e, h, f)$ the standard basis for $sl(2)$, I have to calculate the dual basis $(x, y, z)$ with respect to the Killing form. Now, following the definition, I am determining $x$ by imposing these conditions: $k(e,x) = 1$ $k(h, x)= 0$ $k(f,x) = 0$

By developing these equations, I have obtained some conditions on some coefficients of $ad(x)$, but not on all of them. What I am missing? Is this procedure correct? And if so, how do I obtain $x$ after computing all the entries of $ad(x)$?

cip
  • 1,127
  • 1
    Writing $x=ae+bh+cf$, where $a,b,c\in\mathbb{R}$, your three conditions should give you three linear equations in the three unknowns $a,b,c$, which you can then solve (since the Killing form is non-degenerate). – user17945 Jan 23 '20 at 01:39
  • Thank you. Another related question: suppose you have $V$ and $W$ vectorial spaces over the field $K$, $(v_i)$ basis of $V$ and you want to compute the dual basis $(w_i)$ of $W$ with respect to a generic bilinear form $k : V \times W \to K$. In this case I can't write the elements of the dual basis as linear combinations of the first basis. So what is the trick in this more general case? – cip Jan 23 '20 at 13:03
  • In that case, there may be no dual bases (for example, suppose $k$ is identically zero), or there may be more than one set of dual bases. You would need a condition saying that the matrix of entries $[k(v_i,w_j)]$ with respect to one (and hence every) bases $\lbrace v_i\rbrace$ of $V$ and $\lbrace w_j\rbrace$ of $W$ is invertible (which in particular requires $V$ and $W$ to have the same dimension). – user17945 Jan 24 '20 at 00:40
  • 1
    Compare https://math.stackexchange.com/q/1286100/96384. – Torsten Schoeneberg Jan 24 '20 at 04:34
  • Thank you very much! – cip Jan 24 '20 at 08:53
  • @TorstenSchoeneberg Generalizing this, is there an easy way to compute the dual basis of $sl(n,F)$ relative to Killing form? – user5826 May 01 '20 at 00:39
  • 1
    @Aljebr: The dual basis to which basis? – Torsten Schoeneberg May 01 '20 at 05:28
  • @TorstenSchoeneberg standard basis for $sl(n,F)$: ${e_{ij}}$ where $1 \le i \ne j \le n$ together with ${e_{ii}-e_{i+1, i+1}}$ for $1 \le i \le n-1$. – user5826 May 02 '20 at 01:39
  • @AlJebr: See https://math.stackexchange.com/q/3877102/96384. – Torsten Schoeneberg Oct 23 '20 at 05:47

0 Answers0