We all know plenty of examples of multilinear forms in finitely many variables (e.g. determinants). However, I am missing an interesting example of a form in infinitely many variables, linear in each. The important point here is the word "interesting" ; the example should occur naturally in some nice situation in algebra, or geometry, or analysis... (One can easily find non-interesting examples.) I thought that maybe regularized determinants could provide examples, but I don't know much about these. Thanks for ideas !



Example of a form linear in infinitely many variables ?
Consider the Hilbert space $H=L^2(S^1, \mathbf{C}^n)$ and the subspaces $H\_+, H\_-$ of positive and negative Fourier modes. One can construct the Hilbert space of infinite wedge products $V=\bigwedge(H\_+)\hat{\otimes}\bigwedge(H\_-)^\*$.


Just as $H$ carries an action of $LSU(n)=Maps(S^1, SU(n))$ by pointwise multiplication, $V$ carries an action of the central extension $\widetilde{LSU(n)}$. It turns out that all irreducible positive-energy level 1 representations occur as summands in $V$.


