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Morita equivalence (Definition)

Morita equivalence

This entry presents both the definition of Morita equivalent algebras and the Morita equivalence theorem, with a brief proof included.
Definition 1.1   Let $A$ and $B$ be two associative, but not necessarily commutative, algebras. Such algebras $A$ and $B$ are called Morita equivalent, if there is an equivalence of categories between $A$-mod and $B$-mod.
Theorem 1.1   Morita Equivalence Theorem Commutative algebras $A$ and $B$ are Morita equivalent if and only if they are isomorphic.

Proof. Following the above definition, isomorphic algebras are Morita equivalent. Let us assume that $A$ and $B$ are any two such Morita equivalent associative algebras. It follows then that

$\displaystyle A-mod \sim B-mod$
, and thus one also has that

$\displaystyle Z(A-mod) \simeq Z(B-mod).$
If $A$ and $B$ are both commutative, then by the Associative Algebra Lemma one also has that $A = Z_A$ and $B = Z_B.$



"Morita equivalence" is owned by bci1.

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See Also: B-mod category equivalence theorem, Yoneda lemma, Morita equivalence lemma for arbitrary algebras

Other names:  Morita equivalent algebras
Also defines:  equivalence of commutative algebras, Morita equivalent algebras
Keywords:  Morita equivalence

Attachments:
Morita equivalence lemma for arbitrary algebras (Example) by bci1

Cross-references: categories, theorem
There are 2 references to this object.

This is version 8 of Morita equivalence, born on 2009-06-15, modified 2009-06-15.
Object id is 806, canonical name is MoritaEquivalence.
Accessed 1310 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
Pending Errata and Addenda
None.
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