Bipolar Complex Intuitionistic Fuzzy Lie Algebras

We introduce and study bipolar complex intuitionistic fuzzy Lie algebras (BCIFLAs), a new algebraic framework merging bipolar fuzzy theory, complex-valued membership functions in Cartesian form, and intuitionistic fuzzy Lie algebra theory. Adopting the Cartesian coordinate formulation of bipolar complex intuitionistic fuzzy sets (BCIFSs), we define bipolar complex intuitionistic fuzzy Lie subalgebras (BCIFLSAs) and ideals (BCIFLIs) and establish their fundamental properties, including closure under arbitrary meets. We prove that images and preimages of BCIFLSAs and BCIFLIs are preserved under Lie algebra homomorphisms, characterize them via level sets, and show that the sum of two BCIFLIs is, again, a BCIFLI. Non-degenerate three-level examples on sl(2,R) and t(2,R) illustrate the gap between the subalgebra and ideal conditions. This framework strictly generalizes fuzzy, intuitionistic fuzzy, complex intuitionistic fuzzy, and bipolar fuzzy Lie algebras.