# reference request – Lie powers of a graded vector house and Klyachko’s theorem Answer

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reference request – Lie powers of a graded vector house and Klyachko’s theorem

Let $$V$$ breathe a $$mathbb{Z}_2$$-graded vector house (aka tremendous vector house) and $$L(V)$$ breathe the free $$mathbb{Z}_2$$-graded Lie algebra (aka tremendous Lie algebra). The free tremendous Lie algebra is too graded by the variety of mills (the mills $$V$$ sit in diploma 1, and the Lie bracket itself has diploma zero on this sense). So it has a decomposition by levels, $$L(V) = sum_{n=1}^infty L^n(V)$$. The homogeneous parts $$L^n(V)$$ could breathe known as (tremendous) Lie powers of $$V$$ (by affinity with symmetric or alternating powers) and have an embedding $$L^n(V) subset V^{otimes n}$$, the place the Lie (tremendous)commutators are despatched to tensor (tremendous)commutators, $$[a,b] = aotimes b – (-)^a b otimes a$$.

Q: What is the $$GL(V)$$-representation theoretic description of $$L^n(V)$$?

When $$V$$ is solely plane (has no queer part), the query is answered by Klyachko’s theorem (as defined within the solutions to MO187545). So principally, I’m asking: What is the tremendous Lie algebra analog of Klyachko’s theorem? And what’s a reference that clearly states it?

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