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# Weak Mirror Symmetry of Complex Symplectic Algebras

**Article**

*in*Journal of Geometry and Physics 61(8) · April 2010

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DOI: 10.1016/j.geomphys.2011.03.018 · Source: arXiv

Cite this publicationAbstract

A complex symplectic structure on a Lie algebra $\lie h$ is an integrable
complex structure $J$ with a closed non-degenerate $(2,0)$-form. It is
determined by $J$ and the real part $\Omega$ of the $(2,0)$-form. Suppose that
$\lie h$ is a semi-direct product $\lie g\ltimes V$, and both $\lie g$ and $V$
are Lagrangian with respect to $\Omega$ and totally real with respect to $J$.
This note shows that $\lie g\ltimes V$ is its own weak mirror image in the
sense that the associated differential Gerstenhaber algebras controlling the
extended deformations of $\Omega$ and $J$ are isomorphic. The geometry of
$(\Omega, J)$ on the semi-direct product $\lie g\ltimes V$ is also shown to be
equivalent to that of a torsion-free flat symplectic connection on the Lie
algebra $\lie g$. By further exploring a relation between $(J, \Omega)$ with
hypersymplectic algebras, we find an inductive process to build families of
complex symplectic algebras of dimension $8n$ from the data of the
$4n$-dimensional ones.

- ... The pair (ω, J) is called • a pseudo-Kähler structure on g if ω(Jx, Jy) = ω(x, y) for all x, y ∈ g. • a complex-symplectic structure on g if ω(Jx, Jy) = −ω(x, y) for all x, y ∈ g. See [19] for this definition. ...... The pair (ω, J) is called @BULLET a pseudo-Kähler structure on g if ω(Jx, Jy) = ω(x, y) for all x, y ∈ g. @BULLET a complex-symplectic structure on g if ω(Jx, Jy) = −ω(x, y) for all x, y ∈ g. See [19] for this definition. ...... Lie algebras g endowed with a symplectic structure and a torsion-free flat symplectic connection give rise to hypersymplectic structures on g ⋉ V where V is the underlying vector space to g, as proved in [19]. (See proofs and definitions there). ...We study the problem of extending a complex structure to a given Lie algebra g, which is firstly defined on an ideal h of g. We consider the next situations: h is either complex or it is totally real. The next question is to equip g with an additional structure, such as a (non)-definite metric or a symplectic structure and to ask either h is non-degenerate, isotropic, etc. with respect to this structure, by imposing a compatibility assumption. We show that this implies certain constraints on the algebraic structure of g. Constructive examples illustrating this situation are shown, in particular computations in dimension six are given.
- ... This topic does not seem to have got much attention in the literature so far and only a very restrictive class of complex symplectic structures, called special Lagrangian, on not necessarily nilpotent or solvable Lie algebras was studied in [14] in detail. ...... Compared to this, the existence of complex symplectic structures on nilpotent Lie algebras seems to be less restrictive. On the other hand, the 8-dimensional Lie alge- bra (0,0,12,13,14,15,16,17) has nilpotency step 7 and carries the symplectic structure ω = e 18 + e 27 − e 36 + e 45 . Moreover, the 8-dimensional Lie algebra obtained by setting A = E = F = H = K = M = P = s = 0, B = −1 and C = D = G = L = N = 1 in Theorem 5.9 (i) has ascending type (1,3,5,6,8) and nilpotency step 5; its center is 1-dimensional, hence the complex structure is strongly non-nilpotent and the Lie algebra admits no complex symplectic struc- ture. ...Preprint
- Nov 2018

We investigate Lie algebras endowed with a complex symplectic structure and develop a method, called \emph{complex symplectic oxidation}, to construct certain complex symplectic Lie algebras of dimension $4n+4$ from those of dimension $4n$. We specialize this construction to the nilpotent case and apply complex symplectic oxidation to classify eight-dimensional nilpotent complex symplectic Lie algebras. - ... @BULLET They generalize previous results concerning complex product structures [2, 7], complex and symplectic structures related to tangent algebras [1, 6, 17], complex and paracomplex structures on homogeneous manifolds [11]. @BULLET The existence of LSA structures imposes a clear obstruction. ...... We observe that for such J, the subspaces h and k are totally real, that is, they satisfy Jh ∩ h = {0} and the same holds for k. Such an almost complex structure was called totally real with respect to the decomposition g = h ⊕ k in [17]. Conversely, if an almost complex structure on g = h ⊕ k satisfies Jh = k, then the map j := J| h : h → k is a linear isomorphism. ...Article
- Apr 2016
- Manuscripta Math

Let $G=H\ltimes K$ denote a semidirect product Lie group with Lie algebra $\mathfrak g=\mathfrak h \oplus \mathfrak k$, where $\mathfrak k$ is an ideal and $\mathfrak h$ is a subalgebra of the same dimension as $\mathfrak k$. There exist some natural split isomorphisms $S$ with $S^2=\pm \,Id$ on $\mathfrak g$: given any linear isomorphism $j:\mathfrak h \to \mathfrak k$, we have the almost complex structure $J(x,v)=(-j^{-1}v, jx)$ and the almost paracomplex structure $E(x,v)=(j^{-1}v, jx)$. In this work we show that the integrability of the structures $J$ and $E$ above is equivalent to the existence of a left-invariant torsion-free connection $\nabla$ on $G$ such that $\nabla J=0=\nabla E$ and also to the existence of an affine structure on $H$. Applications include complex, paracomplex and symplectic geometries. - ... These structures play important roles in algebra, geometry and mathematical physics, and are widely studied. See [1][2][3][4][5]8,9,12,[14][15][16]18,19,29,35] for more details. ...In this paper, first we introduce the notion of a phase space of a 3-Lie algebra and show that a 3-Lie algebra has a phase space if and only if it is sub-adjacent to a 3-pre-Lie algebra. Then we introduce the notion of a product structure on a 3-Lie algebra using the Nijenhuis condition as the integrability condition. A 3-Lie algebra enjoys a product structure if and only if it is the direct sum (as vector spaces) of two subalgebras. We find that there are four types special integrability conditions, and each of them gives rise to a special decomposition of the original 3-Lie algebra. They are also related to $\huaO$-operators, Rota-Baxter operators and matched pairs of 3-Lie algebras. Parallelly, we introduce the notion of a complex structure on a 3-Lie algebra and there are also four types special integrability conditions. Finally, we add compatibility conditions between a complex structure and a product structure, between a symplectic structure and a paracomplex structure, between a symplectic structure and a complex structure, to introduce the notions of a complex product structure, a para-K\"{a}hler structure and a pseudo-K\"{a}hler structure on a 3-Lie algebra. We use 3-pre-Lie algebras to construct these structures. Furthermore, a Levi-Civita product is introduced associated to a pseudo-Riemannian 3-Lie algebra and deeply studied.
- ... REMARK. The results above and those in [8] suggest that totally real complex structures are interesting objects to be consider in presence of compatible structures. ...This paper deals with left invariant complex structures on simply connected Lie groups, the Lie algebra of which is of the type $\mathrm{T}_{\pi} \mathfrak{h}=\mathfrak{h} \ltimes_{\pi} V$, where $\pi$ is either the adjoint or the coadjoint representation. The main topic is the existence question of complex structures on $\mathrm{T}_{\pi} \mathfrak{h}$ for $\mathfrak{h}$ a three dimensional real Lie algebra. First it was proposed the study of complex structures $J$ satisfying the constraint $J\mathfrak{h} = V$. Whenever $\pi$ is the adjoint representation this kind of complex structures are associated to non-singular derivations of $\mathfrak{h}$. This fact allows different kinds of applications.
- ... If the homomorphism induces an isomorphism at cohomology level, these two DGAs are said to be quasi-isomorphic. In such case, the generalized complex structure J and the deformed one are also said to form a weak mirror pair [23], [6], [7]. Let Φ : L → L be a vector bundle homomorphism depending on Γ. ...Associated to every generalized complex structure is a differential Gerstenhaber algebra (DGA). When the generalized complex structure deforms, so does the associated DGA. In this paper, we identify the infinitesimal conditions when the DGA is invariant as the generalized complex structure deforms. We prove that the infinitesimal condition is always integrable. When the underlying manifold is a holomorphic Poisson nilmanifolds, or simply a group in the general, and the geometry is invariant, we find a general construction to solve the infinitesimal conditions under some geometric conditions. Examples and counterexamples of existence of solutions to the infinitesimal conditions are given.
- It is well known that on any Lie group, a left-invariant Riemannian structure can be defined. For other left-invariant geometric structures, for example, complex, symplectic, or contact structures, there are difficult obstructions for their existence, which have still not been overcome, although a lot of works were devoted to them. In recent years, substantial progress in this direction has been made; in particular, classification theorems for low-dimensional groups have been obtained. This paper is a brief review of left-invariant complex, symplectic, pseudo-Kählerian, and contact structures on low-dimensional Lie groups and classification results for Lie groups of dimension 4, 5, and 6.

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- COMMUN MATH PHYS

We construct neutral Calabi-Yau metrics and hypersymplectic structures on some Kodaira manifolds. Our structures are symmetric with respect to the central tori. - Hypersymplectic quotients, ACTA Academiae Scientiarum Taurinensis
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N. J. Hitchin, Hypersymplectic quotients, ACTA Academiae Scientiarum Taurinensis, Supplemento al numero 124 (1990), 169–180. - Article
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We present the extended Kuranishi space for a Kodaira surface as a nontrivial example to Kontsevich and Barannikov's extended deformation theory. We provide a non-trivial example of Hertling-Manin's weak Frobenius manifold. In addition, we find that Kodaira surface is its own mirror image in the sense of Merkulov. The calculations of extended deformation and the weak Frobenius structure are based on Merkulov's perturbation method. Our computation of cohomology is done in the context of compact nilmanifolds. - In this paper we begin the development of a relative version of T-duality in generalized complex geometry which we propose as a manifestation of mirror symmetry. Let M be an n-dimensional smooth real manifold, V a rank n real vector bundle on M, and ∇ a flat connection on V. We define the notion of a ∇-semi-flat generalized almost complex structure on the total space of V. We show that there is an explicit bijective correspondence between ∇-semi-flat generalized almost complex structures on the total space of V and ∇∨-semi-flat generalized almost complex structures on the total space of V∨. We show that semi-flat generalized complex structures give rise to a pair of transverse Dirac structures on the base manifold. We also study the ways in which our results generalize some aspects of T-duality such as the Buscher rules. We show explicitly how spinors are transformed and discuss the induces correspondence on branes under certain conditions.
- In this paper we continue the development of a relative version of T-duality in generalized complex geometry which we propose as a manifestation of mirror symmetry. We discuss the integrability of the transform from Part I in terms of data on the base manifold. We work with semi-flat generalized complex structures on real n-torus bundles with section over an n-dimensional base and use the transform on vector bundles developed in Part I of this paper to discuss the bijective correspondence between semi-flat generalized complex structures on pairs of dual torus bundles. We give interpretations of these results in terms of relationships between the cohomology of torus bundles and their duals. We comment on the ways in which our results generalize some well established aspects of mirror symmetry. Along the way, we give methods of constructing generalized complex structures on the total spaces of the bundles we consider.
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- J GEOM PHYS

We characterize real Lie algebras carrying a hypersymplectic structure as bicrossproducts of two symplectic Lie algebras endowed with a compatible flat torsion-free connection. In particular, we obtain the classification of all hypersymplectic structures on 4-dimensional Lie algebras, and we describe the associated metrics on the corresponding Lie groups. - Article
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This article provides a complete description of the differential Gerstenhaber algebras of all nilpotent complex structures on any real six-dimensional nilpotent algebra. As an applica-tion, we classify all pseudo-Kählerian complex structures on six-dimensional nilpotent algebras whose differential Gerstenhaber algebra is quasi-isomorphic to that of the symplectic structure. In a weak sense of mirror symmetry, this gives a classification of pseudo-Kähler structures on six-dimensional nilpotent algebras whose mirror images are themselves. - In this paper we give a procedure to construct hypersymplectic structures on ℝ4n beginning with affine-symplectic data on ℝ2n. These structures are shown to be invariant by a 3-step nilpotent double Lie group and the resulting metrics are complete and not necessarily flat. Explicit examples of this construction are exhibited.
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S. A. Merkulov, A note on extended complex manifolds. Symplectic and contact topology: interactions and perspectives (Toronto, ON/Montreal, QC, 2001), 145-155, Fields Inst. Commun., 35, Amer. Math. Soc., Providence, RI, 2003. - Article
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It is argued that every Calabi-Yau manifold X with a mirror Y admits a family of supersymmetric toroidal 3-cycles. Moreover the moduli space of such cycles together with their flat connections is precisely the space Y. The mirror transformation is equivalent to T-duality on the 3-cycles. The geometry of moduli space is addressed in a general framework. Several examples are discussed. - Article
- Oct 1999

http://www.tulips.tsukuba.ac.jp/mylimedio/dl/page.do?issueid=397883&tocid=100080715&page=321-332 - We construct a generalization of the variations of Hodge structures on Calabi-Yau manifolds. It gives a Mirror partner for the theory of genus=0 Gromov-Witten invariants
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- Jul 1999
- ACTA MATH SIN

We explain how deformation theories of geometric objects such as complex structures, Poisson structures and holomorphic bundle structures lead to differential Gerstenhaber or Poisson algebras. We use homological perturbation theory to construct A∞ algebra structures on the cohomology, and their canonically defined deformations. Such constructions are used to formulate a version of A∞ algebraic mirror symmetry. - Article
- Mar 2004
- J REINE ANGEW MATH

We present the extended Kuranishi space for Kodaira surface as a non-trivial example to Kontsevich and Barannikov's extended deformation theory. We provide a non-trivial example of Hertling-Manin's weak Frobenius manifold. In addition, we find that Kodaira surface is its own mirror image. Our computation is done in the context of compact nilmanifolds. The calculation of extended deformation and the weak Frobenius structure is based on Merkulov's perturbation method. The approach to relating symplectic structure and complex structure is inspired by Gualtieri's work on generalized complex structures. - A study is made of real Lie algebras admitting compatible complex and product structures, including numerous 4-dimensional examples. Any Lie algebra g with such a structure is even-dimensional and its complexification has a hypercomplex structure. In addition, g splits into the direct sum of two Lie subalgebras of the same dimension, and each of these is shown to have a left-symmetric algebra (LSA) structure. Interpretations of these results are obtained that are relevant to the theory of both hypercomplex and hypersymplectic manifolds and their associated connections.
- ArticleFull-text available
- Jun 2000

We introduce a category of extended complex manifolds, and prove that the functor describing deformations of a classical compact complex manifold $M$ within this category is versally representable by (an analytic subspace in) $H^*(M,T_M)$. By restricting the associated versal family of extended complex manifolds over $H^*(M,T_M)$ to the subspace $H^1(M,T_M)$ one gets a correct limit to the classical picture. - The existence of a flat torsion-free connection, or left symmetric algebra structure on a Lie algebra g gives rise to a canonically defined complex structure on g+g and a symplectic structure on g+g^*. We verify that the associated differential Gerstenhaber algebras controlling the deformation theories of the complex and symplectic form are isomorphic. This provides a class of examples of "weak mirror symmetry" as suggested by Merkulov. For nilpotent algebras in dimension 4 and 6 the isomorphism classes of the semi-direct products g+g and g+g^* are listed. A one-parameter family of inequivalent pseudo-K\"ahler structures is given.