Graded algebra

Let $(𝑀, +, 0)$ be a monoid. An algebra $(𝐴, \cdot)$ over $𝕂$ is said to be $𝑀$ -graded iff it is an $𝑀$ -graded vector space $𝐴 = ∐ 𝛼 \in 𝑀 𝐴 𝛼$ such that #m/def/ralg

𝐴 𝛼 \cdot 𝐴 𝛽 \subseteq 𝐴 𝛼 + 𝛽

for any $𝛼, 𝛽 \in 𝑀$ . If $(𝐴, \cdot)$ is a $𝕂$ -monoid, this definition is equivalent to that of a graded ring, and hence $1 \in 𝐴 + 0$ .

Category of graded algebras

Many of our typical algebra constructions carry over. These motivate $𝖦 𝗋 𝔄 𝖠 𝗅 𝗀 𝕂$ .

Graded algebra homomorphism
Graded subalgebra, Quotient graded algebra

Properties

If $𝑀 \leq 𝕂 +$ , the degree operator on $𝐴$ is a derivation.
If $𝐴$ is commutative, then $𝑀$ must be abelian/

Examples

Tensor algebra

Graded algebra

Category of graded algebras

Properties

Examples

See also