Rethinking RoPE: A Mathematical Blueprint for N-dimensional Positional Encoding

Rotary Position Embedding (RoPE) is widely adopted in Transformers due to its ability to encode relative positions with high efficiency and extrapolation capability. However, existing RoPE variants lack a unified theoretical foundation, especially in higher dimensions. In this paper, we propose a systematic mathematical framework for RoPE grounded in Lie group and Lie algebra theory. We identify two core properties of RoPE, named relativity and reversibility, and derive general constraints and constructions for valid RoPE in 1D, 2D, and N-dimensional (ND). We prove that RoPE must lie in the basis of a maximal abelian subalgebra (MASA) of the special orthogonal Lie algebra, and show that standard RoPE corresponds to the maximal toral subalgebra. Furthermore, we propose to model inter-dimensional interactions by learning an orthogonal basis transformation. Our framework unifies and explains existing RoPE designs, while enabling principled extensions to new modalities and tasks.