Liealgebraan

Liealgebraan are a class of algebraic structures used to study continuous symmetries in mathematics and physics. Each Liealgebraan is a vector space g over a field F (commonly F = R or C) equipped with a bilinear operation [ , ]: g × g → g called the Lie bracket. The bracket is antisymmetric [x,y] = −[y,x] and satisfies the Jacobi identity [x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0 for all x, y, z in g. This structure encodes a notion of infinitesimal symmetry, and the bracket acts as a derivation in each argument.

Examples: The space of n×n matrices with the commutator [A,B] = AB − BA forms a Liealgebra called gl(n,F). The subspace of traceless matrices gives sl(n,F). Classical Liealgebraan such as so(n,F) and sp(2n,F) arise as brackets on skew-symmetric or symplectic matrix spaces. More abstractly, many Liealgebraan are finite-dimensional and over fields of characteristic zero.

Structure and classification: Finite-dimensional Liealgebraan over C decompose into a direct sum of a solvable radical and a semisimple subalgebra (Levi decomposition). Semisimple Liealgebraan are classified by their root systems and Dynkin diagrams; this yields a complete list of simple algebras such as A_n, B_n, C_n, D_n, and the exceptional types.

Representations: Liealgebraan act on vector spaces via representations; the universal enveloping algebra U(g) allows the construction of modules and the study of weights, roots, and highest-weight representations.

Applications: Liealgebraan underpin the theory of Lie groups, differential geometry, and particle physics, where they describe continuous symmetries and conservation laws.