In this paper, we offer a detailed classification of Lie bialgebra structures on nondegenerate flat Lie algebras. A flat Lie algebra is defined as the Lie algebra of a Lie group equipped with a left-invariant Riemannian metric, whose Levi-Civita connection is flat. These algebras are distinguished by an orthogonal decomposition into an abelian subalgebra and an abelian ideal, naturally occurring in the analysis of homogeneous spaces with flat metrics. By breaking down cocycles on these algebras into coboundary and invariant components, we systematically classify the potential Lie bialgebra structures. Additionally, we provide explicit examples from various types of flat Lie algebras, demonstrating the range of structures that can arise. Our findings enhance the understanding of Poisson-Lie group quantization and have significant implications for the field of noncommutative geometry.

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Here are five related questions you might find interesting:

1. What are the key properties of nondegenerate flat Lie algebras?
2. How does the orthogonal decomposition of flat Lie algebras impact the classification of Lie bialgebra structures?
3. Can you explain the significance of decomposing cocycles into coboundary and invariant components in this context?
4. What are some specific examples of flat Lie algebras, and how do they differ in terms of Lie bialgebra structures?
5. How do the results of this classification contribute to the study of noncommutative geometry?

Tip: When studying Lie algebras, it's helpful to understand their connection to differential geometry, particularly how Riemannian metrics and connections influence their structure.