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The highest weight of a representation is the weight $\Lambda$ of a nonzero weight vector annihilated by every positive-root operator. Every other weight is obtained from $\Lambda$ by subtracting a nonnegative integer combination of simple roots. A finite-dimensional irreducible representation of a complex semisimple Lie algebra is determined by its dominant integral highest weight.

A singular vector is a nonzero weight vector annihilated by the positive nilpotent subalgebra ${\mathfrak{n}}^{+}$.

A weight basis of a Lie algebra representation is a basis consisting of weight vectors for a chosen Cartan subalgebra. When the Cartan action is diagonalizable, choose a basis separately in every weight space and combine them. This makes weight multiplicities directly visible and gives diagonal matrices for Cartan elements.

A weight of a representation is a simultaneous eigenvalue functional for a commuting Cartan subalgebra. In general algebra representations it can be expressed as a one-dimensional character of the acting commutative algebra. A weight vector is an eigenvector carrying that functional; a weight space is the corresponding eigenspace, and weight multiplicity is its dimension.

The weight space $V_{\mu }$ consists of all weight vectors of weight $\mu$, together with zero.

This is the simultaneous eigenspace belonging to a Weight.