Definition for "Lie Algebra" in the Quantum Dictionary

- Modern Physics -

Quantum Mechanics Dictionary

A mathematical structure used to represent quantum states and the operations performed on them. It specifies how states are expressed, related, combined, and decomposed into components that support calculation. It does not by itself determine physical predictions without a mapping to observables and measurement rules.

Natural & Formal Sciences Dictionary

A vector space over a field equipped with a bilinear, antisymmetric bracket that satisfies the Jacobi identity, encoding the infinitesimal structure of continuous symmetry groups.

- Mathematics & Logic -

Pure Mathematics Dictionary

A Lie algebra is a vector space g over a field together with a bilinear bracket [·,·]: g×g → g that is antisymmetric ([x,y] = −[y,x]) and satisfies the Jacobi identity [x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0, encoding infinitesimal symmetries.

- Pure Mathematics -

Algebra Dictionary

A vector space over a specified field equipped with a bilinear bracket operation [·,·] that is alternating (skew-symmetric in characteristic ≠2) and satisfies the Jacobi identity; the bracket encodes infinitesimal commutator structure.