# Do all finite $W$-superalgebras have 1-dimensional representations?

Premet proved the famous KW-conjecture in modular Lie algebra. After, Premet introduced the finite $W$-algebra $U(g, e)$. Also, Premet proposed the conjecture every algebra $U(g, e)$ admits a $1$-dimensional representation then, Losev proved this conjecture for g classical.

so, a natural question, for the super version, what about these results when we consider the basis classical Lie superalgebra,i.e, whether every super W-algebra admits a $1$-dim rep??

EDIT: Thanks for Professor José Figueroa-O'Farrill pointing out that in the literature of mathematical physics, the finite W-algebras appeared in the work of de Boer and Tjin from the viewpoint of BRST quantum hamiltonian reduction.

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Not to take any credit away from Premet, but finite W-algebras were introduced by Tjark Tjin in a 1992 paper of that name, published in Phys. Lett. B292, 60 (1992). (ams.org/mathscinet-getitem?mr=1184379) –  José Figueroa-O'Farrill Jun 19 '11 at 11:47
@José Figueroa-O'Farrill , thank you very much. I apologize for inadequate references in this regard. I just want to say the finite $W$-algebras were introduced by Premet into mathematics in a different terminology. –  wison Jun 19 '11 at 12:48
@wison: no problem and thanks for the edit. Let me just emphasise that Tjark's paper, despite being published in a Physics journal, is actually reasonably mathematical. He considers an important special case of what are now known as finite W-algebras. –  José Figueroa-O'Farrill Jun 27 '11 at 18:31