Abstract
Let k be a field of characteristic 0 and let k<X> be a free associative algebra with finite basis X. Let R=R(k,X) be the universal enveloping algebra of the square of Lie(X), regarded as a subalgebra of k<X> and called the Specht subalgebra of the free algebra. We prove that k<X> is a free (left) R-module, find sufficient conditions for some system of elements in k<X> to be a basis for this module, and obtain an explicit formula that allows us to calculate the R-coefficients of the elements of the free algebra over a special basis of “symmetric monomials.”
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Gavrilov, A.V. A Free Associative Algebra as a Free Module over a Specht Subalgebra. Siberian Mathematical Journal 44, 428–434 (2003). https://doi.org/10.1023/A:1023909829715
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DOI: https://doi.org/10.1023/A:1023909829715