map would not be linear, since one would have for k ∈ K. One can rescue this attempt and restore linearity by imposing additional structure, by defining an algebra homomorphism Δ: A → A ⊗ A, and defining the tensor product representation as Such a homomorphism Δ is called a comultiplication if it satisfies certain axioms. The resulting structure is called a bialgebra. To be consistent with the definitions of the associative algebra, the coalgebra must be co-associative, and, if the algebra is unital, then the co-algebra must be co-unital as well. A Hopf algebra is a bialgebra with an additional piece of structure (the so-called antipode), which allows not only to define the tensor product of two representations, but also the Hom module of two representations (again, similarly to how it is done in the representation theory of groups). Motivation for a Lie algebra One can try to be more clever in defining a tensor product. Consider, for example, so that the action on the tensor product space is given by . This map is clearly linear in x, and so it does not have the problem of the earlier definition. However, it fails to preserve multiplication: . But, in general, this does not equal . This shows that this definition of a tensor product is too naive; the obvious fix is to define it such that it is antisymmetric, so that the middle two terms cancel. This leads to the concept of a Lie algebra. Non-unital algebras Some authors use the term "associative algebra" to refer to structures which do not necessarily have a multiplicative identity, and hence consider homomorphisms which are not necessarily unital. One example of a non-unital associative algebra is given by the set of all functions f: R → R' whose limit as x nears infinity is zero. Another example is the vector space of continuous periodic functions, together with the convolution product. See also Abstract algebra Algebraic structure Algebra over a field Sheaf of algebras, a sort of an algebra over a ringed space Notes References Nathan Jacobson, Structure of Rings James Byrnie Shaw (1907) A Synopsis of Linear Associative Algebra, link from Cornell University Historical Math Monographs. Ross Street (1998) Quantum Groups: an entrée to modern algebra'', an overview of index-free notation. Algebras Algebraic geometry