Let $L_n$ ($n\in\mathbb{Z}$) and $c$ be the standard generators of the Virasoro algebra ${\rm Vit}$. In the literature one usually considers the involutive authomorphism given by $\tau(L_n)=-L_{-n}$, $\tau(c)=c$. Let $\mathfrak{g}\subset {\rm Vit}$ be the subalgebra consisting of the fixed points of $\tau$. Has the Lie algebra $\mathfrak{g}$ some interest? E.g. does it appear as the symmetry algebra of some PDE or is it interesting from the point of view of Physics? The same question is about the similar subalgebra of the Witt algebra.
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In the boundary state approach to D-brane states in closed string theory one has two copies of Virasoro , $L_n$ and $\tilde L_n$ and boundary states $|B \rangle$ are annihilated by $L_n+\tau(\tilde L_n)$. This is not quite your set-up but rather involves a sub algebra of two copies of the Virasoro algebra with the same central charge.
answered Apr 10, 2015 at 22:51
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$\begingroup$ Thank you, but I would like to understand the usage of the Lie algebra smaller then the Virasoro or the Witt algebra. The involution is considered quite often, and the should be some meaning of the Lie algebra of its fixed points. $\endgroup$ Commented Apr 11, 2015 at 15:57