In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces ; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication.

In the mathematical field of representation theory, a weight of an algebra A over a field F is an algebra homomorphism from A to F – a linear functional – or equivalently, a one dimensional representation of A over F. It is the algebra analogue of a multiplicative character of a group.

In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices ( or endomorphisms of a vector space ) in such a way that the Lie bracket is given by the commutator.

In the mathematical field of representation theory, a projective representation of a group G on a vector space V over a field F is a group homomorphism from G to the projective linear group

In the mathematical field of representation theory, a trivial representation is a representation of a group G on which all elements of G act as the identity mapping of V. A trivial representation of an associative or Lie algebra is a ( Lie ) algebra representation for which all elements of the algebra act as the zero linear map ( endomorphism ) which sends every element of V to the zero vector.

In the mathematical field of representation theory a real representation is usually a representation on a real vector space U, but it can also mean a representation on a complex vector space V with an invariant real structure, i. e., an antilinear equivariant map

In the mathematical field of representation theory, a representation of a Lie superalgebra is an action of Lie superalgebra L on a Z < sub > 2 </ sub >- graded vector space V, such that if A and B are any two pure elements of L and X and Y are any two pure elements of V, then

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