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[parent] Morita equivalence lemma for arbitrary algebras (Example)

Morita equivalence lemma for arbitrary algebras

Let us consider first an example of Morita equivalence; thus, for an integer $n \geq 1$, let $Mat_n(A)$ be the algebra of $n \times n$-matrices with entries in an algebra $A$. The following is a typical example of Morita equivalence that involves noncommutative algebras.

Theorem 1.1   Morita equivalence Lemma for arbitrary algebras

For any algebra $A$ and any integer $n \geq 1$, the algebras $A$ and $Mat_n(A)$ are Morita equivalent.

Important Notes:

  • Even if $A$ is a commutative algebra, the algebra $Mat_n(A)$ is of course not commutative for any $n > 1$ because the matrix multiplication is generally non-commutative.
  • In general, the algebra $A$ cannot be recovered from its corresponding abelian category $A$-mod. Therefore, in order for a concept in noncommutative geometry to have or retain an intrinsic meaning, such a concept must be Morita invariant that is, to remain within the same Morita equivalence class. This raises the important question: what properties of an algebra are Morita invariant ? The answer to this question is provided by the “Uniqueness Morita Theorem”.



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See Also: Morita equivalence, non-commutative geometry

Also defines:  Morita invariant, non-commutative algebra
Keywords:  Morita equivalence, non-commtutative geometry

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Cross-references: noncommutative geometry, concept, abelian category, non-commutative, matrix multiplication, noncommutative, Morita equivalence

This is version 5 of Morita equivalence lemma for arbitrary algebras, born on 2009-06-15, modified 2009-06-15.
Object id is 807, canonical name is MoritaEquivalenceLemmaForArbitraryAlgebras.
Accessed 700 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
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