Algebra

A symbol representing an unknown or changeable value.

An expression consisting of terms with variables and coefficients combined by addition.

An equation where the highest power of the variable is 1.

Expressing a polynomial as a product of two or more polynomials.

An equation where the highest power of the variable is 2.

An expression showing the relationship between two values using inequality symbols.

A set of two or more equations that must be satisfied simultaneously.

A function where the base is a positive constant and the exponent is the variable.

The inverse of exponential function, indicating how many times a base must be multiplied to get a number.

A sequence where the difference between consecutive terms is constant.

A sequence where the ratio between consecutive terms is constant.

The sum of terms in a sequence, which can be finite or infinite.

A mathematical object defined as a multilinear map. A (p,q)-tensor has p covariant and q contravariant components.

An operation constructing a new vector space from two vector spaces. Characterized by the universal property of bilinear maps.

Transformation rules for components under coordinate changes. Covariant components transform like basis, contravariant oppositely.

A group with smooth manifold structure. Group operations are differentiable, representing continuous symmetries.

The tangent space at the identity of a Lie group, equipped with Lie bracket. Encodes infinitesimal structure of the group.

A map from Lie algebra to Lie group. Generates finite transformations from infinitesimal generators.

Fundamental Lie groups represented as matrices: GL(n), SL(n), O(n), SO(n), U(n), SU(n), Sp(n).

A representation of a Lie group acting on its own Lie algebra. Essential for studying Lie algebra structure.

A symmetric bilinear form on a Lie algebra. Essential for determining semisimplicity and classification.

A finite set of vectors encoding the structure of semisimple Lie algebras. Classified by Dynkin diagrams.

A Lie algebra with no solvable ideals. Decomposes as direct sum of simple Lie algebras; Killing form is non-degenerate.