\(\Delta \implies \otimes\)

Comultiplication makes tensor product great again

In general, unlike group representations or Lie algebra representations, the tensor product of two representations of a generic associative algebra is not naturally a representation.

To define a representation on the tensor product \(V \otimes W\), the algebra \(A\) must possess additional structure—specifically, it must be a Bialgebra (or a Hopf Algebra).

Here is the breakdown of why this is the case and how the construction works.

1 The Problem

Suppose \((V, \rho_V)\) and \((W, \rho_W)\) are two representations (modules) of an associative algebra \(A\). To make \(V \otimes W\) a representation, we need a map: \[\rho_{V \otimes W} : A \to \text{End}(V \otimes W)\] that is an algebra homomorphism.

If we try to define the action “element-wise” (like we do in linear algebra), we run into a problem. For an element \(a \in A\), how should it act on \(v \otimes w\)? * If we try \(a(v \otimes w) = (av) \otimes (aw)\), this is generally not linear in \(a\). For example, \((a+b)(v \otimes w)\) would not equal \(a(v \otimes w) + b(v \otimes w)\). * If we try \(a(v \otimes w) = (av) \otimes w\), we are ignoring the \(W\) structure.

2 The Solution: The Comultiplication

To properly distribute the action of \(A\) across a tensor product, we need a way to “split” an element \(a \in A\) into two parts. This is provided by a comultiplication map: \[\Delta: A \to A \otimes A\] For \(V \otimes W\) to be an \(A\)-module, we define the action as: \[a \cdot (v \otimes w) = \Delta(a)(v \otimes w)\]

Using Sweedler notation, we write \(\Delta(a) = \sum a_{(1)} \otimes a_{(2)}\). Then the action is: \[a \cdot (v \otimes w) = \sum (a_{(1)} v) \otimes (a_{(2)} w)\]

3 Required Properties

For this tensor product to satisfy the axioms of a representation, \(A\) must be a Bialgebra, meaning: 1. \(\Delta\) is an algebra homomorphism: This ensures that \(ab \cdot (v \otimes w) = a \cdot (b \cdot (v \otimes w))\). 2. \(\Delta\) is coassociative: \((\Delta \otimes \text{id})\Delta = (\text{id} \otimes \Delta)\Delta\). This ensures that the tensor product of three representations \((V \otimes W) \otimes U\) is isomorphic to \(V \otimes (W \otimes U)\).

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4 Important Examples

Most “standard” algebras used in physics and representation theory are bialgebras, which is why we often take the tensor product for granted.

4.1 Group Algebras (\(k[G]\))

For a group \(G\), the algebra \(A = k[G]\) has a comultiplication defined by: \[\Delta(g) = g \otimes g \quad \text{for all } g \in G\] This leads to the familiar rule for group representations: \[g \cdot (v \otimes w) = (gv) \otimes (gw)\]

4.2 Universal Enveloping Algebras of Lie Algebras (\(U(\mathfrak{g})\))

For a Lie algebra \(\mathfrak{g}\), an element \(x \in \mathfrak{g}\) acts on the tensor product of two \(\mathfrak{g}\)-modules as: \[x \cdot (v \otimes w) = (xv) \otimes w + v \otimes (xw)\] This corresponds to the comultiplication: \[\Delta(x) = x \otimes 1 + 1 \otimes x\] This is essentially the Leibniz rule (the product rule for derivatives).

4.3 Quantum Groups (\(U_q(\mathfrak{g})\))

Quantum groups are non-commutative, non-cocommutative Hopf algebras. They have more complex comultiplication rules (e.g., \(\Delta(K) = K \otimes K\) and \(\Delta(E) = E \otimes 1 + K \otimes E\)), which allow for the study of “braided” tensor categories.

4.4 5. Summary Table

Structure	Algebra	Comultiplication \(\Delta(a)\)	Tensor Action \(a(v \otimes w)\)
Group	\(k[G]\)	\(g \otimes g\)	\((gv) \otimes (gw)\)
Lie Algebra	\(U(\mathfrak{g})\)	\(x \otimes 1 + 1 \otimes x\)	\((xv) \otimes w + v \otimes (xw)\)
General Bialgebra	\(A\)	\(\sum a_{(1)} \otimes a_{(2)}\)	\(\sum (a_{(1)}v) \otimes (a_{(2)}w)\)

4.5 Conclusion

If you are working with a plain associative algebra that has no defined comultiplication, you cannot tensor two representations. If you find yourself needing to tensor representations, you are likely working in the category of modules over a Bialgebra or Hopf Algebra.