Riemannian manifold plays a vital role as a powerful mathematical tool in computer vision, with important applications in curved shape analysis and classification. Significant progress has recently been made by Riemannian framework based methods that achieved state-of-the-art classification accuracy and robustness. However, these Riemannian manifold and Lie group methods require a very high computational complexity and do not include a description of the shape regions. This paper presents a novel mathematical tool, called Block Diagonal Symmetric Positive Definite Matrix Lie Algebra (BDSPDMLA) to represent curves, which extends the existing Lie group representations to a compact yet informative Lie algebra representation. The proposed Lie algebra based method addresses the computational bottleneck problem of the Riemannian framework based methods. In addition, it allows the natural fusion of various regions information with curved shape features for a more discriminative shape description. Here the region information is represented by values of distance maps, local binary patterns (LBP) and image intensity. Extensive experiments on five publicly available databases demonstrate that the proposed Lie algebra based method can achieve a speed of over ten thousand times faster than the Riemannian manifold and Lie group based baseline methods, while obtaining comparable accuracies for 2D shape classification.