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Register a new userPre-Lie analogues of Poisson-Nijenhuis structures and Maurer-Cartan equations
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In this paper, we study pre-Lie analogues of Poisson-Nijenhuis structures and introduce ON-structures on bimodules over pre-Lie algebras. We show that an ON-structure gives rise to a hierarchy of pairwise compatible O-operators. We study solutions of the strong Maurer-Cartan equation on the twilled pre-Lie algebra associated to an O-operator, which gives rise to a pair of ON-structures which are naturally in duality. We show that KVN-structures and HN-structures on a pre-Lie algebra g are corresponding to ON-structures on the bimodule $(mathfrak g^*;mathrm{ad}^*,-R^*)$, and $KVOmega$-structures are corresponding to solutions of the strong Maurer-Cartan equation on a twilled pre-Lie algebra associated to an $s$-matrix.
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We study (quasi-)twilled pre-Lie algebras and the associated $L_infty$-algebras and differential graded Lie algebras. Then we show that certain twisting transformations on (quasi-)twilled pre-Lie algbras can be characterized by the solutions of Maurer-Cartan equations of the associated differential graded Lie algebras ($L_infty$-algebras). Furthermore, we show that $mathcal{O}$-operators and twisted $mathcal{O}$-operators are solutions of the Maurer-Cartan equations. As applications, we study (quasi-)pre-Lie bialgebras using the associated differential graded Lie algebras ($L_infty$-algebras) and the twisting theory of (quasi-)twilled pre-Lie algebras. In particular, we give a construction of quasi-pre-Lie bialgebras using symplectic Lie algebras, which is parallel to that a Cartan $3$-form on a semi-simple Lie algebra gives a quasi-Lie bialgebra.
We briefly review our results on the Lie theory underlying vector bundles over Lie groupoids and Lie algebroids, pointing out the role of Poisson geometry in extending these results to double Lie algebroids and LA-groupoids.
In this paper, we give Maurer-Cartan characterizations as well as a cohomology theory for compatible Lie algebras. Explicitly, we first introduce the notion of a bidifferential graded Lie algebra and thus give Maurer-Cartan characterizations of compatible Lie algebras. Then we introduce a cohomology theory of compatible Lie algebras and use it to classify infinitesimal deformations and abelian extensions of compatible Lie algebras. In particular, we introduce the reduced cohomology of a compatible Lie algebra and establish the relation between the reduced cohomology of a compatible Lie algebra and the cohomology of the corresponding compatible linear Poisson structures introduced by Dubrovin and Zhang in their study of bi-Hamiltonian structures. Finally, we use the Maurer-Cartan approach to classify nonabelian extensions of compatible Lie algebras.
We construct a noncommutative Cartan calculus on any braided commutative algebra and study its applications in noncommutative geometry. The braided Lie derivative, insertion and de Rham differential are introduced and related via graded braided commutators, also incorporating the braided Schouten-Nijenhuis bracket. The resulting braided Cartan calculus generalizes the Cartan calculus on smooth manifolds and the twisted Cartan calculus. While it is a necessity of derivation based Cartan calculi on noncommutative algebras to employ central bimodules our approach allows to consider bimodules over the full underlying algebra. Furthermore, equivariant covariant derivatives and metrics on braided commutative algebras are discussed. In particular, we prove the existence and uniqueness of an equivariant Levi-Civita covariant derivative for any fixed non-degenerate equivariant metric. Operating in a symmetric braided monoidal category we argue that Drinfeld twist deformation corresponds to gauge equivalences of braided Cartan calculi. The notions of equivariant covariant derivative and metric are compatible with the Drinfeld functor as well. Moreover, we project braided Cartan calculi to submanifold algebras and prove that this process commutes with twist deformation.
We propose a new method to compute connection matrices of quantum Knizhnik-Zamolodchikov equations associated to integrable vertex models with super algebra and Hecke algebra symmetries. The scheme relies on decomposing the underlying spin representation of the affine Hecke algebra in principal series modules and invoking the known solution of the connection problem for quantum affine Knizhnik-Zamolodchikov equations associated to principal series modules. We apply the method to the spin representation underlying the $mathcal{U}_qbigl(hat{mathfrak{gl}}(2|1)bigr)$ Perk-Schultz model. We show that the corresponding connection matrices are described by an elliptic solution of a supersymmetric version of the dynamical quantum Yang-Baxter equation with spectral parameter.
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