• P.PRIYANKA
• M.Raja Rajeswari
• P.Thiruselvi
• S.priyanka

Algebra is a vector space over a field equipped with an associative, bilinear multiplication. This forces us to examine properties of ring theory under the lens of linear algebra, solvability, as we determine subalgebras, ideals. We completely classify all „Lie theory and special functions‟. Applications to finite-dimensional Lie algebra in problems are indicated. A homogeneous space approach is used to investigate of Lie algebras of vector fields with coefficients in smooth functions. By use of a new bilinear operation, any associative algebra into a non associative abstract Lie algebra, where the new product in terms of the given associative product is the commutator. Finite- dimensional real analytic Lie algebras of vector fields on ℝ2 are completely classified up to changes of local coordinates. Lie algebras of first-order differential operators in two complex variables. We completely classify all „Lie algebras‟ in two complex and real variables.

Lie algebra, Lie groups, Subalgebras, Cohomology, Isomorphism, Two complex and real variables, finite-dimensional, Lie theory, Lie algebras of vector field, first-order differential operators.

"FINITE DIMENSIONAL LIEALGEBRAS OF VECTOR FIELDS", International Journal of Emerging Technologies and Innovative Research (www.jetir.org), ISSN:2349-5162, Vol.12, Issue 11, page no.e58-e59, November-2025, Available :http://www.jetir.org/papers/JETIR2511408.pdf

"FINITE DIMENSIONAL LIEALGEBRAS OF VECTOR FIELDS", International Journal of Emerging Technologies and Innovative Research (www.jetir.org | UGC and issn Approved), ISSN:2349-5162, Vol.12, Issue 11, page no. ppe58-e59, November-2025, Available at : http://www.jetir.org/papers/JETIR2511408.pdf