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In the previous answers, almost everyone stressed on the reasons why have the representations of monoids not been studied widely, which probably answers your question.

But I would like to mention a couple of interesting cases where the structure of monoid algebras are fully understood (over $\mathbb{C}$ at least).

1.Brauer Algebras, which are introduced by Richard Brauer.

2.Temperley-lieb Algebras, which are quotients of Hecke algebras.

3.Partition Algebra, which are invented and studied by Paul Martin.

All of the above examples are part of a wider branch which is called Diagram Algebras.

In the previous answers, almost everyone stressed on the reasons why have the representations of monoids not been studied widely, which probably answers your question.

But I would like to mention a couple of interesting cases where the structure of monoid algebras are fully understood (over $\mathbb{C}$ at least).

1.Brauer Algebras, which are introduced by Richard Brauer.

2.Temperley-lieb Algebras, which are quotients of Hecke algebras.

3.Partition Algebra, which are introduced and studied by Paul Martin.

All of the above examples are part of a wider branch which is called Diagram Algebras.

In the previous answers, almost everyone stressed on the reasons why have the representations of monoids not been studied widely, which probably answers your question.

But I would like to mention a couple interesting cases were the structure of monoid algebras are fully understood (over the field of complexes).

1.Brauer Algebras, which are introduced by Richard Brauer.

2.Temperley-lieb Algebras, which are quotients of Hecke algebras.

3.Partition Algebra, which are invented and studied by Paul Martin.

All of the above examples are part of a wider branch which is called Diagram Algebras.

In the previous answers, almost everyone stressed on the reasons why have the representations of monoids not been studied widely, which probably answers your question.

But I would like to mention a couple of interesting cases where the structure of monoid algebras are fully understood (over $\mathbb{C}$ at least).

1.Brauer Algebras, which are introduced by Richard Brauer.

2.Temperley-lieb Algebras, which are quotients of Hecke algebras.

3.Partition Algebra, which are invented and studied by Paul Martin.

All of the above examples are part of a wider branch which is called Diagram Algebras.

In the previous answers, almost everyone stressed on the reasons why have the representations of monoids not been studied widely, which probably answers your question.

But I would like to mention a couple interesting cases were the structure of monoid algebras are fully understood (over the field of complexes).

1.Brauer Algebras, which are introduced by Richard Brauer.

2.Temperley-lieb Algebras, which are quotients of Hecke algebras.

3.Partition Algebra, which are invented and studied by Paul Martin.

Both of the above examples are part of a wider branch which is called Diagram Algebras.

In the previous answers, almost everyone stressed on the reasons why have the representations of monoids not been studied widely, which probably answers your question.

But I would like to mention a couple interesting cases were the structure of monoid algebras are fully understood (over the field of complexes).

1.Brauer Algebras, which are introduced by Richard Brauer.

2.Temperley-lieb Algebras, which are quotients of Hecke algebras.

3.Partition Algebra, which are invented and studied by Paul Martin.

All of the above examples are part of a wider branch which is called Diagram Algebras.