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I have seen the notion of the adjoint representation of a lie algebra $L$ ad the map: $\text{ad}:L \to \text{End}(L)$ defined by:

$$ \text{ad:}x \mapsto [x,-]$$

My question is, is there a similar name for the "take an element to the multiplication by that element" map in an associative algebra $A$? I.e. the map $m:A \to \text{End}(A)$ defined by: $$ m: a \mapsto a \times -$$

$m$ is therefore an algebra homomorphism, and hence should be a representation, right? I am not well versed in representation theory, so I may have gotten some concepts mixed up.

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    I would call that the (left) regular representation. – 2017-01-19

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It is called the (left) regular representation.