Algebra theory MOC

Module over an associative algebra

Let be a K-monoid over . A (left) -module is a -vector space equipped with a bilinear map

such that

1. for
2. for ,

which is a curried version of a unital algebra homomorphism

We also call this a representation of carried by .

Properties and further terminology

• automatically carries a Lie algebra representation of the commutator algebra of and any Lie subalgebra.
• A Submodule of is an invariant subspace under the action of .
• A module is irreducible iff it has no proper nontrivial submodules.
• A module is indecomposable iff it cannot be decomposed into the direct sum of two nonzero submodules.
• A module isomorphism is an Equivalence of group representations.
• The Regular representation shows that is a module over itself.

Explanation

Since a K-monoid over a field is itself a ring, it is possible to form a module over . The action of on and on induces an action of on , thus the module inherits the -linear structure of the underlying ring . Therefore is a vector space over .

Proof

Let be the identity element of the associative algebra . Then a distributive and linear field action is given by

since for any and :

satisfying ^V4;

satisfying ^V5;

satisfying ^V6; and

satisfying ^V7.

Such a module coïncides exactly with the notion of a Group representation of the algebra over .

See also

• Module over a Lie algebra
• Module over a group

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