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1 Correction to: Geom Dedicata (2016) 111:193–212 https://doi.org/10.1007/s10711-015-0119-z
The paper [2] is devoted to the classification of generalized Wallach spaces. A generalized Wallach space is a homogeneous spaces G/H of a connected compact semisimple Lie group G (H is a compact subgroup of G), such that there is a \(\langle \cdot , \cdot \rangle \)-orthogonal and \({\text {Ad}}(H)\)-invariant decomposition \({\mathfrak {g}}={\mathfrak {h}}\oplus {\mathfrak {p}}_1\oplus {\mathfrak {p}}_2\oplus {\mathfrak {p}}_3\), where \({\mathfrak {g}}\) and \({\mathfrak {h}}\) are Lie algebras of G and H respectively, \(\langle \cdot , \cdot \rangle \) is the minus Killing form of \({\mathfrak {g}}\), the modules \({\mathfrak {p}}_i\) are \({\text {Ad}}(H)\)-irreducible, \([{\mathfrak {p}}_i,{\mathfrak {p}}_i]\subset {\mathfrak {h}}\) for \(i=1,2,3\). Here we will make a correction to the obtained classification. In what follows, we use the notation from [2].
The main result of the paper [2] (Theorem 1) should be stated as follows (in fact, we just add the item (4)):
Theorem 1
Let G/H be a connected and simply connected compact homogeneous space. Then G/H is a generalized Wallach space if and only if it is of one of the following types:
-
(1)
G/H is a direct product of three irreducible symmetric spaces of compact type \((A=a_1=a_2=a_3=0\) in this case);
-
(2)
The group G is simple and the pair \(({\mathfrak {g}}, {\mathfrak {h}})\) is one of the pairs in Table 1 of [2] (the embedding of \({\mathfrak {h}}\) to \({\mathfrak {g}}\) is determined by the following requirement: the corresponding pairs \(({\mathfrak {g}}, {\mathfrak {k}}_i)\) and \(({\mathfrak {k}}_i,{\mathfrak {h}})\), \(i=1,2,3\), in Table 2 of [2] are symmetric);
-
(3)
\(G=F\times F\times F \times F\) and \(H={\text {diag}}(F)\subset G\) for some connected simply connected compact simple Lie group F, with the following description on the Lie algebra level:
$$\begin{aligned} ({\mathfrak {g}}, {\mathfrak {h}})= \bigl ({\mathfrak {f}}\oplus {\mathfrak {f}}\oplus {\mathfrak {f}}\oplus {\mathfrak {f}},\,{\text {diag}}({\mathfrak {f}})=\{(X,X,X,X)\,|\, X\in f\}\bigr ), \end{aligned}$$where \({\mathfrak {f}}\) is the Lie algebra of F, and (up to permutation) \({\mathfrak {p}}_1=\{(X,X,-X,-X)\,|\, X\in f\}\), \({\mathfrak {p}}_2\!=\!\{(X,-X,X,-X)\,|\, X\in f\}\), \({\mathfrak {p}}_3\!=\!\{(X,-X,-X,X)\,|\, X\in f\}\) (\(a_1\!=\!a_2\!=\!a_3\!=\!1/4\) in this case).
-
(4)
\(H={{\,\mathrm{diag}\,}}(K)\subset K\times K\subset F\times F=G\), where \(({\mathfrak {f}},{\mathfrak {k}})\) is a compact irreducible symmetric pair with simple \({\mathfrak {f}}\) and with simple or one-dimensional \({\mathfrak {k}}\), (up to permutation) \({\mathfrak {p}}_1=\{(X,X)\,|\, X\in {\mathfrak {q}}\}\), \({\mathfrak {p}}_2=\{(X,-X)\,|\, X\in {\mathfrak {q}}\}\), \({\mathfrak {p}}_3=\{(Y,-Y)\,|\, Y\in {\mathfrak {k}}\}\), and \({\mathfrak {q}}\) is the orthogonal complement to \({\mathfrak {k}}\) in \({\mathfrak {f}}\) with respect to the Killing form of the Lie algebra \({\mathfrak {f}}\).
The same item (4) should be added in the statement of Theorem 3 in [2]. The reason for the above correction is the fact that Corollary 2 in [2] is not correct in general, but it is true under some additional assumptions. The correct version of this corollary is as follows.
Corollary 1
If \(p \ge 2\) and at least one of the modules \({\mathfrak {p}}_i\), \(i=1,2,3\), is situated in some simple ideal \({\mathfrak {g}}_j\) of the Lie algebra \({\mathfrak {g}}\), then \(A=0\), consequently, G/H locally is a direct product of three irreducible symmetric spaces of compact type.
Proof
Without loss of generality we may suppose that \({\mathfrak {p}}_1\subset {\mathfrak {g}}_1\), then \([{\mathfrak {p}}_1,{\mathfrak {p}}_2]\subset {\mathfrak {p}}_3 \cap {\mathfrak {g}}_1\) and \([{\mathfrak {p}}_1,{\mathfrak {p}}_3]\subset {\mathfrak {p}}_2 \cap {\mathfrak {g}}_1\). If \([{\mathfrak {p}}_1,{\mathfrak {p}}_2]=0\) or \([{\mathfrak {p}}_1,{\mathfrak {p}}_3]=0\), we get \(A=0\). Otherwise, \({\mathfrak {p}}_2,{\mathfrak {p}}_3 \subset {\mathfrak {g}}_1\) (note that all the modules \({\mathfrak {p}}_2\), \({\mathfrak {p}}_3\), \({\mathfrak {p}}_2 \cap {\mathfrak {g}}_1\), and \({\mathfrak {p}}_3 \cap {\mathfrak {g}}_1\) are \({{\,\mathrm{Ad}\,}}(H)\)-irreducible), which implies \(p=1\). \(\square \)
This result should be completed with the following proposition (that provides the case (4) for Theorems 1 and 3 in [2]).
Proposition 1
If \(p \ge 2\) and no one module \({\mathfrak {p}}_i\), \(i=1,2,3\), is in some simple ideal \({\mathfrak {g}}_j\) of \({\mathfrak {g}}\), then \(p=2\) and \(({\mathfrak {g}}, {\mathfrak {h}})=({\mathfrak {f}}\oplus {\mathfrak {f}}, {{\,\mathrm{diag}\,}}({\mathfrak {k}}))\), where \(({\mathfrak {f}},{\mathfrak {k}})\) is a compact irreducible symmetric pair with simple \({\mathfrak {f}}\) and with simple or one-dimensional \({\mathfrak {k}}\). Moreover, up to permutation of indices, we have \({\mathfrak {p}}_1=\{(X,X)\,|\, X\in {\mathfrak {q}}\}\), \({\mathfrak {p}}_2=\{(X,-X)\,|\, X\in {\mathfrak {q}}\}\), \({\mathfrak {p}}_3=\{(Y,-Y)\,|\, Y\in {\mathfrak {k}}\}\), where \({\mathfrak {q}}\) is the orthogonal complement to \({\mathfrak {k}}\) in \({\mathfrak {f}}\) with respect to the Killing form of the Lie algebra \({\mathfrak {f}}\).
Proof
Recall that \(\varphi _i({\mathfrak {h}})\) is the \(\langle \cdot , \cdot \rangle \)-orthogonal projection of \({\mathfrak {h}}\) to \({\mathfrak {g}}_i\). Let \({\mathfrak {q}}_i\) be the \(\langle \cdot , \cdot \rangle \)-orthogonal complement to \(\varphi _i({\mathfrak {h}})\) in \({\mathfrak {g}}_i\), \(1\le i \le p\). It is clear that \({\mathfrak {q}}_1 \oplus {\mathfrak {q}}_2 \oplus \cdots \oplus {\mathfrak {q}}_p\subset {\mathfrak {p}}\). Obviously, we have \(p\le 3\). If \(p=3\), then all \({\mathfrak {q}}_i\), \(i=1,2,3\), are \({{\,\mathrm{Ad}\,}}(H)\)-irreducible and \(\varphi _1({\mathfrak {h}})\oplus \varphi _2({\mathfrak {h}})\oplus \varphi _2({\mathfrak {h}})\subset {\mathfrak {h}}\). Since \([\varphi _i({\mathfrak {h}}),{\mathfrak {q}}_i]\ne 0\) and \([\varphi _i({\mathfrak {h}}),{\mathfrak {q}}_j]=0\) for \(i\ne j\), the \({{\,\mathrm{Ad}\,}}(H)\)-modules \({\mathfrak {q}}_i\), \(i=1,2,3\), are pairwise non-isomorphic, hence, they coincides with the corresponding modules \({\mathfrak {p}}_i\), \(i=1,2,3\). By the above corollary we have \(A=0\) in this case.
If \(p=2\), then there are some isomorphic \({{\,\mathrm{Ad}\,}}(H)\)-irreducible submodules \({\mathfrak {q}}_1^\prime \subset {\mathfrak {q}}_1\) and \({\mathfrak {q}}_2^\prime \subset {\mathfrak {q}}_2\). Therefore, by the above arguments, \(\varphi _1({\mathfrak {h}})\not \subset {\mathfrak {h}}\) and \(\varphi _2({\mathfrak {h}})\not \subset {\mathfrak {h}}\) (otherwise, \({\mathfrak {q}}_1^\prime \) is not isomorphic to \({\mathfrak {q}}_2^\prime \)). Hence, \({\mathfrak {h}} \subsetneq \varphi _1({\mathfrak {h}})\oplus \varphi _2({\mathfrak {h}})\), \({\mathfrak {q}}_1^\prime ={\mathfrak {q}}_1\), and \({\mathfrak {q}}_2^\prime = {\mathfrak {q}}_2\). Without loss of generality, we may assume that \({\mathfrak {p}}_1 \oplus {\mathfrak {p}}_2={\mathfrak {q}}_1 \oplus {\mathfrak {q}}_2\), and \({\mathfrak {h}}\oplus {\mathfrak {p}}_3=\varphi _1({\mathfrak {h}})\oplus \varphi _2({\mathfrak {h}})\). Therefore, \(\bigl (\varphi _1({\mathfrak {h}})\oplus \varphi _2({\mathfrak {h}}),{\mathfrak {h}}\bigr )\) is a compact irreducible symmetric pair, which has the form \(\bigl ({\mathfrak {k}} \oplus {\mathfrak {k}}, {{\,\mathrm{diag}\,}}({\mathfrak {k}})\bigr )\), where \({\mathfrak {k}}\) is a compact simple Lie algebra or \({\mathbb {R}}\) [1, Theorem 7.81]. Hence, \(\varphi _1\) and \(\varphi _2\) determine Lie algebra isomorphisms between \({\mathfrak {h}}\) and \(\varphi _i({\mathfrak {h}})\), \(i=1,2\). Let us consider \(\theta : \varphi _1({\mathfrak {h}}) \mapsto \varphi _2({\mathfrak {h}})\), such that \(\theta =\varphi _2\circ \varphi _1^{-1}\). It is clear that \({\mathfrak {h}}=\{(Y,\theta (Y))\,|\, Y\in \varphi _1({\mathfrak {h}}) \}\).
Now, let us consider the \(\langle \cdot , \cdot \rangle \)-orthogonal projections \(\pi _i:{\mathfrak {p}}_1 \rightarrow {\mathfrak {q}}_i\), \(i=1,2\). We may assume that \(\pi _1\) is a bijection (otherwise, we can take \({\mathfrak {p}}_2\) instead of \({\mathfrak {p}}_1\)). Now, let us consider the \({{\,\mathrm{Ad}\,}}(H)\)-equivariant linear map \(\psi :=\pi _2\circ \pi _1^{-1}:{\mathfrak {q}}_1 \mapsto {\mathfrak {q}}_2\). We have \({\mathfrak {p}}_1=\{(X,\psi (X))\,|\, X\in {\mathfrak {q}}_1\}\). Since \([{\mathfrak {h}},{\mathfrak {p}}_1]\subset {\mathfrak {p}}_1\) and \([{\mathfrak {p}}_1,{\mathfrak {p}}_1]\subset {\mathfrak {h}}\), we get \([{\mathfrak {q}}_1, {\mathfrak {q}}_1]\subset \varphi _1({\mathfrak {h}})\), \(\psi ([Y,X])=[\theta (Y),\psi (X)]\) and \(\theta ([X,Z])=[\psi (X),\psi (Z)]\) for every \(Y\in \varphi _1({\mathfrak {h}})\) and for every \(X,Z \in {\mathfrak {q}}_1\). In particular, \(({\mathfrak {q}}_1, \varphi _1({\mathfrak {h}}))\) is a compact irreducible symmetric pair with simple \({\mathfrak {q}}_1\) and and with simple or one-dimensional \(\varphi _1({\mathfrak {h}})\).
If we extend the linear map \(\psi \) from \({\mathfrak {q}}_1\) to \({\mathfrak {g}}_1\) setting \(\psi (X):=\theta (X)\) for any \(X\in \varphi _1({\mathfrak {h}})\), we obtain the isomorphism \(\psi \) between \({\mathfrak {g}}_1\) and \({\mathfrak {g}}_2\). Indeed, \(\psi ([X,Y])=[\psi (X),\psi (Y)]\) for every \(X,Y \in {\mathfrak {g}}_1\), \(\varphi _2({\mathfrak {h}})\subset \psi ({\mathfrak {g}}_1)\), and \({\mathfrak {g}}_1\) is simple. Therefore, \(\psi ({\mathfrak {g}}_1)\) is a simple Lie subalgebra in \({\mathfrak {g}}_2\), and, moreover, \(\psi ({\mathfrak {g}}_1)={\mathfrak {g}}_2\), since \(\varphi _2({\mathfrak {h}})\subset \psi ({\mathfrak {g}}_1)\) and \({\mathfrak {p}}_2\) is \({{\,\mathrm{ad}\,}}({\mathfrak {h}})\)-irreducible. Note that \({\mathfrak {p}}_2=\{(X,-\psi (X))\,|\, X\in {\mathfrak {q}}_1\}\) and \({\mathfrak {p}}_3=\{(Y,-\psi (Y))\,|\, Y\in \varphi _1({\mathfrak {h}})\}\). Therefore, we may consider \({\mathfrak {g}}_2\) as the copy \({\mathfrak {g}}_1\) under the isomorphism \(\psi \). The proposition is proved. \(\square \)
The list of all generalized Wallach spaces of the type as in the Proposition 1 follows directly from the list of compact irreducible symmetric spaces, see e. g. [1, 7.102]. Using structure of symmetric spaces and the Casimir operators for the isotropy representations (see e. g. [1, Chapter 7]), one can easily compute the values A, \(a_1\), \(a_2\), and \(a_3\) (see the formulas (5) and (6) in [2]) for the spaces in Proposition 1: \(A=\frac{1}{4} \bigl (\dim ({\mathfrak {f}})- \dim ({\mathfrak {k}})\bigr )=\frac{1}{4} \dim ({\mathfrak {p}}_1)=\frac{1}{4} \dim ({\mathfrak {p}}_2)\), \(a_1=\frac{A}{\dim ({\mathfrak {p}}_1)}=a_2=\frac{A}{\dim ({\mathfrak {p}}_2)}=1/4\), and \(a_3=\frac{A}{\dim ({\mathfrak {p}}_3)}=\frac{\dim ({\mathfrak {f}})- \dim ({\mathfrak {k}})}{4\dim ({\mathfrak {k}})}\le 1/2\).
References
Besse, A.L.: Einstein Manifolds. Springer, Berlin, p. XII+510 (1987)
Nikonorov, Yu.G.: Classification of generalized Wallach spaces. Geometriae Dedicata 111(1), 193–212 (2016). https://doi.org/10.1007/s10711-015-0119-z
Acknowledgements
The author would sincerely thank Huibin Chen and Zhiqi Chen for pointing out an omission in the statement of the classification theorem for generalized Wallach spaces in [2].
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Nikonorov, Y.G. Correction to: Classification of generalized Wallach spaces. Geom Dedicata 214, 849–851 (2021). https://doi.org/10.1007/s10711-021-00604-3
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DOI: https://doi.org/10.1007/s10711-021-00604-3