Let J be an almost complex structure on a 4-dimensional and unimodular Lie algebra g. We show that there exists a symplectic form taming J if and only if there is a symplectic form compatible with J. We also introduce groups HJ+(g) and HJ-(g) as the subgroups of the Chevalley-Eilenberg cohomology classes which can be represented by J-invariant, respectively J-anti-invariant, 2-forms on g. and we prove a cohomological J-decomposition theorem following Drǎghici etal. (2010) : H2(g)=HJ+(g)⊕HJ-(g). We discover that tameness of J can be characterized in terms of the dimension of HJ±(g), just as in the complex surface case. We also describe the tamed and compatible symplectic cones. Finally, two applications to homogeneous J on 4-manifolds are obtained.
Bibliographical noteFunding Information:
The first author would like to thank the Department of Mathematics at University of Parma and the Mathematical Science Center at Tsinghua University, and the second author would like to thank the School of Mathematics at University of Minnesota, for their warm hospitality. The authors are also grateful to Daniele Angella, Tedi Draˇghici and Weiyi Zhang for useful discussions. The second author was supported by GNSAGA of INdAM .
- Almost kähler structure
- Symplectic lie algebra
- Tamed almost complex structure
- Unimodular lie algebra