The word maps \( \tilde{w}:\kern0.5em {\mathrm{GL}}_m{(D)}^{2k}\to {\mathrm{GL}}_n(D) \) and \( \tilde{w}:\kern0.5em {D}^{\ast 2k}\to {D}^{\ast } \) for a word \( w=\prod \limits_{i=1}^k\left[{x}_i,{y}_i\right], \) where D is a division algebra over a field K, are considered. It is proved that if \( \tilde{w}\left({D}^{\ast 2k}\right)=\left[{D}^{\ast },{D}^{\ast}\right], \) then \( \tilde{w}\left({\mathrm{GL}}_n(D)\right)\supset {E}_n(D)\backslash Z\left({E}_n(D)\right), \) where En(D) is the subgroup of GLn(D), generated by transvections, and Z(En(D)) is its center. Furthermore if, in addition, n > 2, then \( \tilde{w}\left({E}_n(D)\right)\supset {E}_n(D)\backslash Z\left({E}_n(D)\right). \) The proof of the result is based on an analog of the “Gauss decomposition with prescribed semisimple part” (introduced and studied in two papers of the second author with collaborators) in the case of the group GLn(D), which is also considered in the present paper.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 470, 2018, pp. 88–104.
Translated by I. Ponomarenko.
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Egorchenkova, E.A., Gordeev, N.L. Products of Commutators on a General Linear Group Over a Division Algebra. J Math Sci 243, 561–572 (2019). https://doi.org/10.1007/s10958-019-04556-8
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DOI: https://doi.org/10.1007/s10958-019-04556-8