On [A, A] / [A, [A, A]] and on a Wn-action on the consecutive commutators of free associative algebras

Boris Feigin; Boris Shoikhet

doi:10.4310/MRL.2007.v14.n5.a7

On [A, A] / [A, [A, A]] and on a W_n-action on the consecutive commutators of free associative algebras

Boris Feigin^*
, Boris Shoikhet

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

21 Scopus citations

Abstract

We consider the lower central series of the free associative algebra A _n with n generators as a Lie algebra. We consider the associated graded Lie algebra. It is shown that this Lie algebra has a huge center which belongs to the cyclic words, and on the quotient Lie algebra by the center there acts the Lie algebra W_n of polynomial vector fields on ℂⁿ. We compute the space [A_n, A_n)/[A _n, [A_n, A_n)) and show that it is isomorphic to the space Ω_closed²(ℂⁿ) ⊕ Ω_closed⁴(ℂⁿ) ⊕Ω _closed⁶ (ℂⁿ) ⊕ . . . .

Original language	English
Pages (from-to)	781-795
Number of pages	15
Journal	Mathematical Research Letters
Volume	14
Issue number	5-6
DOIs	https://doi.org/10.4310/MRL.2007.v14.n5.a7
State	Published - 2007
Externally published	Yes

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10.4310/MRL.2007.v14.n5.a7

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title = "On [A, A] / [A, [A, A]] and on a Wn-action on the consecutive commutators of free associative algebras",

abstract = "We consider the lower central series of the free associative algebra A n with n generators as a Lie algebra. We consider the associated graded Lie algebra. It is shown that this Lie algebra has a huge center which belongs to the cyclic words, and on the quotient Lie algebra by the center there acts the Lie algebra Wn of polynomial vector fields on ℂn. We compute the space [An, An)/[A n, [An, An)) and show that it is isomorphic to the space Ωclosed2(ℂn) ⊕ Ωclosed4(ℂn) ⊕Ω closed6 (ℂn) ⊕ . . . .",

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AU - Feigin, Boris

AU - Shoikhet, Boris

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N2 - We consider the lower central series of the free associative algebra A n with n generators as a Lie algebra. We consider the associated graded Lie algebra. It is shown that this Lie algebra has a huge center which belongs to the cyclic words, and on the quotient Lie algebra by the center there acts the Lie algebra Wn of polynomial vector fields on ℂn. We compute the space [An, An)/[A n, [An, An)) and show that it is isomorphic to the space Ωclosed2(ℂn) ⊕ Ωclosed4(ℂn) ⊕Ω closed6 (ℂn) ⊕ . . . .

AB - We consider the lower central series of the free associative algebra A n with n generators as a Lie algebra. We consider the associated graded Lie algebra. It is shown that this Lie algebra has a huge center which belongs to the cyclic words, and on the quotient Lie algebra by the center there acts the Lie algebra Wn of polynomial vector fields on ℂn. We compute the space [An, An)/[A n, [An, An)) and show that it is isomorphic to the space Ωclosed2(ℂn) ⊕ Ωclosed4(ℂn) ⊕Ω closed6 (ℂn) ⊕ . . . .

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On [A, A] / [A, [A, A]] and on a W_n-action on the consecutive commutators of free associative algebras

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