I am a physics student, and I am trying to understand the Casimir operator from a formal perspective; therefore, I come to learn what's Universal enveloping algebra.
Two definitions of Universal enveloping algebra $U(\mathfrak{g})$ of a lie algebra $\mathfrak{g}$ are shown in Wikipedia page of Universal enveloping algebra. The Universal enveloping algebra is defined on the tensor algebra $$T(\mathfrak{g})=\bigoplus_{k=0}^{\infty} T^k V=K \oplus \mathfrak{g} \oplus(\mathfrak{g} \otimes \mathfrak{g}) \oplus(\mathfrak{g} \otimes \mathfrak{g} \otimes \mathfrak{g}) \oplus \cdots$$ where $K$ is the field over which the Lie algebra is defined. Then, one definition is we can "lift" the lie bracket of the original by lie algebra to the tensor algebra by defining $$a \otimes b-b \otimes a=[a, b],$$ $$[a \otimes b, c]=a \otimes[b, c]+[a, c] \otimes b,$$ $$ [a, b \otimes c]=[a, b] \otimes c+b \otimes[a, c].$$ Then the universal enveloping algebra $U({\mathfrak {g}})$ of ${\mathfrak {g}}$ is defined as the quotient space $$U(\mathfrak{g})=T(\mathfrak{g}) / \sim$$ where $T(\mathfrak{g})$ is the tensor algebra of the lie algebra, and the equivalence relation $\sim$ is given by $$ a \otimes b-b \otimes a=[a, b]. $$ My first question is, Can you explicitly show me some examples of how two elements of the tensor algebra are equivalent under the given equivalent relation? Does this mean the elements like (a,b,...) and (0, a,...,b) are equivalent as they give the same element under the lie bracket? Or something else?
In the Wikipedia page, there is one sentence explain the effect of the quotient:
The result of this lifting is explicitly a Poisson algebra. It is a unital associative algebra with a Lie bracket that is compatible with the Lie algebra bracket; it is compatible by construction. It is not the smallest such algebra, however; it contains far more elements than needed. One can get something smaller by projecting back down.
My second question is, why quotient this equivalent relation can give the smallest such algebra?