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Register a new userBraided Cartan Calculi and Submanifold Algebras
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Authors
Thomas Weber
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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.
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In this thesis we give obstructions for Drinfeld twist deformation quantization on several classes of symplectic manifolds. Motivated from this quantization procedure, we further construct a noncommutative Cartan calculus on any braided commutative algebra, as well as an equivariant Levi-Civita covariant derivative for any non-degenerate equivariant metric. This generalizes and unifies the Cartan calculus on a smooth manifold and the Cartan calculus on twist star product algebras. We prove that the Drinfeld functor leads to equivalence classes in braided commutative geometry and commutes with submanifold algebra projection.
We study covariant derivatives on a class of centered bimodules $mathcal{E}$ over an algebra A. We begin by identifying a $mathbb{Z} ( A ) $-submodule $ mathcal{X} ( A ) $ which can be viewed as the analogue of vector fields in this context; $ mathcal{X} ( A ) $ is proven to be a Lie algebra. Connections on $mathcal{E}$ are in one to one correspondence with covariant derivatives on $ mathcal{X} ( A ). $ We recover the classical formulas of torsion and metric compatibility of a connection in the covariant derivative form. As a result, a Koszul formula for the Levi-Civita connection is also derived.
We establish the relationship among Nichols algebras, Nichols braided Lie algebras and Nichols Lie algebras. We prove two results: (i) Nichols algebra $mathfrak B(V)$ is finite-dimensional if and only if Nichols braided Lie algebra $mathfrak L(V)$ is finite-dimensional if there does not exist any $m$-infinity element in $mathfrak B(V)$; (ii) Nichols Lie algebra $mathfrak L^-(V)$ is infinite dimensional if $ D^-$ is infinite. We give the sufficient conditions for Nichols braided Lie algebra $mathfrak L(V)$ to be a homomorphic image of a braided Lie algebra generated by $V$ with defining relations.
It is shown that if $mathfrak B(V) $ is connected Nichols algebra of diagonal type with $dim V>1$, then $dim (mathfrak L^-(V)) = infty$ $($resp. $ dim (mathfrak L(V)) = infty $$)$ $($ resp. $ dim (mathfrak B(V)) = infty $$)$ if and only if $Delta(mathfrak B(V)) $ is an arithmetic root system and the quantum numbers (i.e. the fixed parameters) of generalized Dynkin diagrams of $V$ are of finite order. Sufficient and necessary conditions for $m$-fold adjoint action in $mathfrak B(V)$ equal to zero, viz. $overline{l}_{x_{i}}^{m}[x_{j}]^ -=0$ for $x_i,~x_jin mathfrak B(V)$, are given.
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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