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Let $L$ be a simple complex Lie algebra. Let $\Phi$ be the root set of $L$ and $\Gamma$ be the set of all simple roots of $L.$ We know that for every root $\alpha \in \Phi$ there is a copy $S_{\alpha}$ of $sl_2(\mathbb C)$ sitting inside $L$ whose generators are given by $h_{\alpha}, x_{\alpha}$ and $y_{\alpha}$ where $h_{\alpha}$ is the generator of the Cartan subalgebra $H_{\alpha}$ of $S_{\alpha}$ and $x_{\alpha}, y_{\alpha}$ are the generators of the root subspaces $L_{\pm \alpha}$ of $S_{\alpha}.$ Now for a subset $E \subseteq \Gamma$ consider the following subspaces of $L.$

$$H_{E} = \bigoplus\limits_{\alpha \in E} H_{\alpha},\ \ L_{E} = \bigoplus\limits_{\alpha \in \widehat {E}} L_{\alpha} \oplus H_{E},\ \ N_{E}^{\pm} = \bigoplus\limits_{\alpha \in \Phi^{+} \setminus \widehat {E}} L_{\pm \alpha}$$

Where $\Phi^{+}$ is the set of all positive roots of $L$ and $\widehat {E} \subseteq \mathbb Z \Gamma$ is the set of all roots of $L$ generated by the roots in $E$ i.e. $\widehat {E} = \mathbb Z E \cap \Phi.$ Let us consider the subspace $P_{E}^{+} : = L_{E} \oplus N_{E}^{+}.$ For a given subspace $C \subseteq L$ let $C^{\perp}$ denote the orthogonal complement of $C$ with respect to the Killing form. Then it is easy to see that $$\color{blue}{\left ( P_{E}^{+} \right )^{\perp} = N_{E}^{+} \oplus H_{E}^{\perp}}.$$ This is the part I didn't follow. It's clear that $H_{E} \subseteq P_{E}^{+}$ and hence we should have $\left (P_E^{+} \right )^{\perp} \subseteq H_{E}^{\perp}.$ Also each root subspace is orthogonal to $H,$ the Cartan subalgebra of $L.$ Hence in particular, $N_{E}^{+}$ is orthogonal to $H_{E}.$ So we must have $N_{E}^{+} \subseteq H_{E}^{\perp}.$ So sum in RHS of the blue equality is clearly not direct. Also the authors mentioned that $H_{E} \oplus H_{E}^{\perp} = H$ which I believe has to be the whole of $L$ because the authors took subspaces of $L$ and hence from the context it's clear that the authors told to take orthogonal complement with respect to the Killing form; not with respect to the restriction of the Killing form to $H.$

These are the few things which confuse me a lot. Could anyone please enlighten me by giving me some insight on these facts?

Thanks a bunch!

Authors $:$ Pavel Etingof and Oliver Schiffmann.

Book $:$ Notes on Quantum Groups.

Pages $:$ $45-46.$

NOTE $:$ I have changed the notations in the book slightly in few places according to Humphreys' book without changing the context.

Anacardium
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  • I have a strong feeling that, in confusing notation, by $H_E^\perp$ they mean only the orthogonal complement of $H_E$ in $H$, whereas for every other subspace, $(\cdot)^\perp$ means the orthogonal complement in the full space $L$. If one interprets it like that, do all your doubts disappear? – Torsten Schoeneberg Mar 23 '23 at 16:31
  • @TorstenSchoeneberg$:$ Then I can see that $N_{E}^{+} \oplus H_{E}^{\perp} \subseteq \left (P_{E}^{+} \right )^{\perp}.$ But I can't see why the reverse inclusion also holds. – Anacardium Mar 23 '23 at 18:24
  • Is not $(N_E^+)^\perp = H \oplus \bigoplus_{-\alpha \notin \Phi^+\setminus \hat{E}} L_{\alpha}$, $(L_E)^\perp = H_E^\perp \oplus \bigoplus_{\alpha \notin \hat{E}} L_{\alpha}$ (with the abused notation for $H_E^\perp$), and $(P_E^+)^\perp$ their intersection? – Torsten Schoeneberg Mar 24 '23 at 03:39
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    @TorstenSchoeneberg$:$ Nicely done. Thanks for your help. Just a small remark. The index of the direct sum in the second summand of $\left (N_{E}^{+} \right )^{\perp}$ should be $$\bigoplus\limits_{\color{blue} {\alpha \notin \Phi^{+} \setminus \widehat {E}}} L_{\alpha}.$$ – Anacardium Mar 24 '23 at 04:52
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    @TorstenSchoeneberg$:$ But it is still unclear why $H_{E} \oplus H_{E}^{\perp} = H.$ Also there's no reason to believe that $H_{E} \cap H_{E}^{\perp} = (0).$ Because the Killing form is not positive definite. – Anacardium Mar 24 '23 at 06:52
  • Right on the first one, thanks for correcting. Also good point on the orthogonal decomposition! But I think: A) There is a general linear algebra fact that we have $V = W \oplus W^\perp$ if the form is non-degenerate both on $V$ and when restricted to $W$. And then I am sure B) the Killing form stays non-degenerate when restricted to $H_E$ (or any subspace generated by standard coroots), although that seems more intricate to prove than it looks at first. It is related to discussions in answers and comments to https://math.stackexchange.com/q/280090/96384. – Torsten Schoeneberg Mar 25 '23 at 15:45

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