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Algebra, Modules, and Tensor Symmetry Interfaces

1 · Prerequisites

2 · Summary

This page connects the standard algebra, module, representation, and tensor language. A unital representation of an associative algebra $A$ on a $k$ -vector space $V$ is a unital $k$ -linear algebra action on $V$ ; the two items here define that interface and prove its equivalence with a $k$ -bilinear unital left $A$ -module action. General module and tensor-product constructions remain on tensor-products-of-modules.

For a Lie algebra $g$ , the canonical map to Universal enveloping algebra turns a Lie representation into a unital $U (g)$ -module by Lie representations are U(g)-modules. Its tensor-product action is $x (v \otimes w) = x v \otimes w + v \otimes x w$ by Direct-sum, dual, Hom, and tensor representations, consistent with the primitive coproduct on $U (g)$ in Hopf-algebra structure on U(g). A Lie-algebra module alone does not specify an integrated representation of a chosen global Lie group.

The category of modules over a bialgebra becomes monoidal because its coproduct defines the tensor action and its counit defines the unit object (Monoidal category, The module category of a bialgebra and its tensor action, Bialgebra module tensor actions and their coherence). A monoidal category has associators and unitors satisfying the pentagon and triangle; a braiding is additional structure. If $H$ is a Hopf algebra over $k$ and $R \in H \otimes_{k} H$ is an algebraic quasitriangular structure (a finite sum of elementary tensors), then $R$ supplies a braiding on the category of all left $H$ -modules by A quasitriangular structure gives and is recovered from a module braiding. This braiding is symmetric exactly when $R$ is triangular, as defined in Braiding on the module category and triangular structures. Completed universal $R$ -matrices require a separate convergence and completion argument.

The Lie-representation to enveloping-module passage is summarized below. The diagram records the map from the Lie algebra, the given representation, and its unique extension to the enveloping algebra; the local theorem proves the extension and its uniqueness.

\mathfrak g \arrow[r,"\iota_{\mathfrak g}"] \arrow[dr,"\rho"'] & U(\mathfrak g) \arrow[d,"\overline{\rho}"] \\
& \operatorname{End}_k(V)

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

Representations of unital associative algebras

Definition

Let $k$ be a field, let $A$ be a unital associative $k$ -algebra (Algebras over a commutative ring, central structure maps, and algebra homomorphisms), and let $V$ be a $k$ -vector space. Write ${End}_{k} (V) = L_{k} (V, V)$ for the $k$ -linear endomorphisms of $V$ (The space $L (V, W)$ of linear maps with pointwise addition and scalar multiplication); its multiplication is composition and its identity is ${id}_{V}$ (The endomorphism ring ${End}_{R} (M)$ under addition and composition, Module endomorphisms form a ring under pointwise addition and composition). An (unital) representation of $A$ on $V$ is a $k$ -linear map $ρ : A ⟶ {End}_{k} (V)$ such that $ρ (1_{A}) = {id}_{V}$ and $ρ (a b) = ρ (a) \circ ρ (b) (a, b \in A) .$ The associated action is $a \cdot v : = ρ (a) (v)$ . The zero vector space is allowed; in that case ${id}_{V} = 0$ .

Remarks

This definition specifies an action of $A$ on one vector space. It does not by itself specify an action on $V \otimes_{k} W$ . For the module interpretation and the extra coproduct data used for bialgebra modules, see Associative algebra representations are unital modules and The module category of a bialgebra and its tensor action.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

Associative algebra representations are unital modules

Statement

Let $k$ be a field, $A$ a unital associative $k$ -algebra, and $V$ a $k$ -vector space. There is a bijection between unital representations $ρ : A \to {End}_{k} (V)$ of Representations of unital associative algebras and $k$ -bilinear left $A$ -module actions $A \times V \to V$ , $(a, v) \mapsto a \cdot v$ , for which $(a b) \cdot v = a \cdot (b \cdot v)$ and $1_{A} \cdot v = v$ . The maps are $a \cdot v = ρ (a) (v)$ and $ρ (a) (v) = a \cdot v$ . Here $k$ -bilinear means linear in each variable with respect to the given $k$ -vector structures; the module axioms are those of Unital left and right modules over a ring; unqualified module means left module.

Facts & Assumptions

Given: A field $k$ , a unital associative $k$ -algebra $A$ , a $k$ -vector space $V$ , and the representation and action data in the Statement.

[F1]

Representations of unital associative algebras: a representation is a $k$ -linear map $ρ : A \to {End}_{k} (V)$ preserving the unit and multiplication.

[F2]

Unital left and right modules over a ring; unqualified module means left module: a unital left module action is additive in both variables, associative, and satisfies $1_{A} v = v$ .

[F3]

Vector space over a field: scalar multiplication on each $k$ -vector space is compatible with addition and with multiplication in $k$ .

[F4]

The space $L (V, W)$ of linear maps with pointwise addition and scalar multiplication: ${End}_{k} (V)$ has pointwise vector-space operations on its $k$ -linear maps.

[F5]

Algebras over a commutative ring, central structure maps, and algebra homomorphisms: the multiplication and unit of a $k$ -algebra are compatible with its specified $k$ -algebra structure.

Proof

1.1givenF1F3F4F5construct

Given a representation $ρ$ , define $a \cdot v : = ρ (a) (v)$ . For fixed $a$ , this is $k$ -linear in $v$ because $ρ (a) \in {End}_{k} (V)$ ; for fixed $v$ , it is $k$ -linear in $a$ because $ρ$ is $k$ -linear, $A$ has its specified $k$ -vector structure, and evaluation at $v$ respects the pointwise vector-space operations of [F4]. Thus the action is $k$ -bilinear.

1.2givenF2F3F4F5construct

Conversely, given a $k$ -bilinear unital left $A$ -module action, define $ρ (a) (v) : = a \cdot v$ . Bilinearity in $v$ makes each $ρ (a)$ a $k$ -linear endomorphism, and bilinearity in $a$ makes $ρ$ $k$ -linear.

2.1givenF1F2step 1.1step 1.2algebra

For the action constructed in step 1.1, multiplicativity and preservation of the identity in [F1] give $(a b) \cdot v = a \cdot (b \cdot v)$ and $1_{A} \cdot v = v$ . For the map constructed in step 1.2, the unital left-module laws in [F2] give $ρ (a b) = ρ (a) \circ ρ (b)$ and $ρ (1_{A}) = {id}_{V}$ , which are the representation conditions in [F1].

3.1step 1.1step 1.2step 2.1algebra∎

Starting with $ρ$ , the first construction returns $a \cdot v = ρ (a) (v)$ ; starting with an action, the second returns $ρ (a) (v) = a \cdot v$ . Equality for every $a, v$ proves that the constructions are inverse and hence give the stated bijection.

5 · Examples, counterexamples and false statements

None yet.