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HDB1

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**Please, I have a question about universal enveloping algebra:**Let ##U=U(\mathfrak{g})## be the quotient of the free associative algebra ##\mathcal{F}## with generators ##\left\{a_i: i \in I\right\}## by the ideal ##\mathcal{I}## generated by all elements of the form ##a_i a_j-a_j a_i-\sum_{k \in I} c_{i, j}^k a_k##. The associative algebra ##U(\mathfrak{g}):=\mathcal{F} / \mathcal{I}## is called the universal enveloping algebra.

**1- Please, could I know what is free algebra here, and why universal enveloping algebra is associative?**

2- and please, if you could explain to me the universal enveloping algebra of lie algebra ##\mathfrak{s l}_2##,Thanks in advance,

2- and please, if you could explain to me the universal enveloping algebra of lie algebra ##\mathfrak{s l}_2##,Thanks in advance,