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


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?

  • 5
    $\begingroup$ 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

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?)

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.