Finite dimensional consistently graded Lie superalgebras of depth greater than 2 - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T01:07:03Z http://mathoverflow.net/feeds/question/106526 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106526/finite-dimensional-consistently-graded-lie-superalgebras-of-depth-greater-than-2 Finite dimensional consistently graded Lie superalgebras of depth greater than 2 Andrea Altomani 2012-09-06T16:11:35Z 2012-09-06T21:29:05Z <p>Victor Kac, in the paper </p> <p>"Classification of infinite-dimensional simple linearly compact Lie superalgebras", <a href="http://www.mat.univie.ac.at/~esiprpr/esi605.pdf" rel="nofollow">http://www.mat.univie.ac.at/~esiprpr/esi605.pdf</a> </p> <p>writes at the beginning of section 5 (p.39 in the linked preprint):</p> <blockquote> <p>A consistent $\mathbb Z$-graded Lie superalgebra of depth $\geq 2$ is infinite-dimensional (since otherwise all even elements are exponentiable).</p> </blockquote> <p>here a consistent $\mathbb Z$-graded Lie superalgebra of depth $d$ is a complex, possibly infinite dimensional, Lie superalgebra $$\mathfrak g=\bigoplus_{p\in\mathbb Z,\ p\geq -d} \mathfrak g_p$$ with every $\mathfrak g_p$ finite dimensional, <code>$\mathfrak{g}_{-d}\neq 0$</code>, and $$\mathfrak g_{\bar 0}=\bigoplus_{p\in\mathbb Z} \mathfrak g_{2p},\quad\mathfrak g_{\bar 1}=\bigoplus_{p\in\mathbb Z} \mathfrak g_{2p+1}.$$</p> <p>The superalgebra $\mathfrak g$ is also supposed to be <em>fundamental</em>, i.e. $\mathfrak g_{-1}$ generates $\oplus_{p&lt;0}\mathfrak g_p$, and <em>transitive</em>, i.e. $[X,\mathfrak g_{-1}]=0$ implies $X=0$ for $X\in\mathfrak g_p$, $p\geq 0$.</p> <p>The statement is clearly true if $\mathfrak g$ is also <em>even</em>, i.e. if all even elements that are exponentiable have nonnegative degree. The result indeed is needed only for even superalgebras, however the hypothesis is not explicitly stated. My question is the following:</p> <p><em>Are there finite-dimensional consistently graded fundamental transitive Lie superalgebras of depth $\geq 2$?</em></p>