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Given the standard basis for the Lie algebra $\mathfrak{su}(2)$ of SU(2), $\{i\sigma_1,i\sigma_2,i\sigma_3\}$ where

$\sigma_1=\Biggl(\begin{array}{cc} 0&1\\ 1&0\end{array}\Biggr),\quad\sigma_2=\Biggl(\begin{array}{cc} 0&-i\\ i&0\end{array}\Biggr),\quad\sigma_3=\Biggl(\begin{array}{cc} 1&0\\ 0&-1\end{array}\Biggr),$

I want to find a basis for the universal enveloping algebra, $\mathcal{U}(\mathfrak{su}(2))$. By the Poincare-Birkoff-Witt Theorem I believe we have

$\{i\sigma_1,i\sigma_2,i\sigma_3,-i\sigma_1\sigma_2,-i\sigma_1\sigma_3,-i\sigma_2\sigma_3,-i\sigma_1\sigma_2\sigma_3\}$,

in other words all lexicographically ordered monomials. However, since products of the Pauli matrices are Pauli matrices (ie $\sigma_1\sigma_2=i\sigma_3$) it would seem that the two algebras have the same basis, just with the Lie bracket $[,]$ replaced with matrix multiplication. Can someone tell me if this is correct?

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    $\begingroup$ In a canonical monomial the sequences are allowed to be non-decreasing, not just increasing. There are infinitely many such monomials, so your basis should be infinite. The universal enveloping algebra is not the algebra generated by the matrices $\sigma_i$: although $\sigma_1 \sigma_2 = i \sigma_3$ as matrices, this does not imply that $\rho(\sigma_1) \rho(\sigma_2) = i \rho(\sigma_3)$ in any representation $\rho$ of $\mathfrak{su}(2)$. $\endgroup$ – Qiaochu Yuan Jul 6 '11 at 20:43
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You are confused on a couple of points:

(1) The Poincare-Birkoff-Witt basis is the infinite set $$(i \sigma_1)^a (i \sigma_2)^b (i \sigma_3)^c \ \mbox{for} \ a,\ b,\ c,\ \geq 0.$$ You have only listed the cases where $a$, $b$ and $c$ are $0$ or $1$.

(2) The relation $\sigma_1 \sigma_2 = i \sigma_3$ does not hold in $U(\mathfrak{su}_2)$. That relation holds in the standard two dimensional representation of $\mathfrak{su}_2$, but it doesn't hold in (for example) the $3$ dimensional representation. The relations in $U(\mathfrak{su}_2)$ are those which hold in all representations of $\mathfrak{su}_2$. (Are you clear on what a representation of a Lie algebra means?)

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  • $\begingroup$ This is basically what Qiaochu said, just a little wordier. $\endgroup$ – David E Speyer Jul 6 '11 at 20:50
  • $\begingroup$ Ok yes I see my confusion. But I wasn't saying that $\sigma_1\sigma_2=i\sigma_3$ should hold for all dimensions (or all representations) - and yes I am pretty clear on what a representation of a Lie Algebra is. However, I would still like to have some concrete ways of writing (and using) the representations of $\mathcal{U}(\mathfrak{su}(2))$. Can one say anything further then "here are the basis elements and any relations which hold for all representations of the Lie Algebra also hold for these basis elements"? $\endgroup$ – levitopher Jul 9 '11 at 17:17

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