| 1. | This universal property follows from the tensor algebra as a natural transformation.
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| 2. | This distinction is developed in greater detail in the article on tensor algebras.
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| 3. | Additional discussion of this point can be found in the tensor algebra article.
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| 4. | Here juxtaposition is symmetric respectively antisymmetric multiplication in the symmetric and antisymmetric tensor algebra.
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| 5. | The universal property can thus be seen as being inherited from the tensor algebra.
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| 6. | Thus, these two are consistency conditions for the Lie bracket on the tensor algebra.
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| 7. | The construction proceeds by first building the tensor algebra of the underlying vector space of the Lie algebra.
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| 8. | The monomials of these generators then generate the tensor algebra, on which the quotienting may be performed.
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| 9. | The result of the lifting is that the tensor algebra of a Lie algebra is a Poisson algebra.
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| 10. | The construction generalizes in straightforward manner to the tensor algebra of any " commutative " ring.
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