underlying vector spaces, then the tensor product of the representations would have V1 ⊗ V2 as the underlying vector space, with the action of uniquely determined by the assumption that for all and . In the language of homomorphisms, this means that we define by the formula . In the physics literature, the tensor product with the identity operator is often suppressed in the notation, with the formula written as , where it is understood that acts on the first factor in the tensor product and acts on the second factor in the tensor product. In the context of representations of the Lie algebra su(2), the tensor product of representations goes under the name "addition of angular momentum." In this context, might, for example, be the orbital angular momentum while is the spin angular momentum. Dual representations Let be a Lie algebra and be a representation of . Let be the dual space, that is, the space of linear functionals on . Then we can define a representation by the formula where for any operator , the transpose operator is defined as the "composition with " operator: The minus sign in the definition of is needed to ensure that is actually a representation of , in light of the identity If we work in a basis, then the transpose in the above definition can be interpreted as the ordinary matrix transpose. Representation on linear maps Let be -modules, a Lie algebra. Then becomes a -module by setting . In particular, ; that is to say, the -module homomorphisms from to are simply the elements of that are invariant under the just-defined action of on . If we take to be the base field, we recover the action of on given in the previous subsection. Representation theory of semisimple Lie algebras See Representation theory of semisimple Lie algebras. Enveloping algebras To each Lie algebra over a field k, one can associate a certain ring called the universal enveloping algebra of and denoted . The universal property of the universal enveloping algebra guarantees that every representation of gives rise to a representation of . Conversely, the PBW theorem tells us that sits inside , so that every representation of can be restricted to . Thus, there is a one-to-one correspondence between representations of and those of . The universal enveloping algebra plays an important role in the representation theory of semisimple Lie