In the literature and in some posts here, there has been variation in the undefined use of the term "block" for a category of modules over a ring, or more abstractly an abelian category (all of which are categories of modules by Freyd-Mitchell). This raises the natural question:

What is meant by a "block" in an abelian category?

The concept originates to some extent in the modular representation theory of finite groups or their group algebras pioneered by Richard Brauer. Here a block is just an indecomposable two-sided ideal of the group algebra, corresponding to a primitive central idempotent. But in later developments the language of homological algebra plays a greater role than the group algebra or its center: the category of modules decomposes into a direct sum of subcategories, which are as small as possible relative to permitting no nontrivial extensions among their simple objects (irreducible representations). This approach seems to me to generalize best to other situations, where a center or central characters may be elusive and where the decomposition may be infinite, etc.

There is some inconsistency in the literature about allowing "blocks" which might be further decomposed into direct sums. In classical or Kac-Moody Lie theory this usually reflects the influence of infinitesimal/central characters. But full centers in Lie theory and related quantum groups may be unknown or unneeded. The approach Kac took to his analogue of the Weyl character formula for integrable modules of an affine Lie algebra relied just on a single Casimir-type operator (which works equally well in the classical finite-dimensional case).

In practice looser definitions of "block" than the homological one work well enough in many settings, but it creates some confusion when the word is used with no definition at all. Should there be a standard convention? In a recent comment to another post I paraphrased Humpty Dumpty ("my remote ancestor"), who actually said: "When I use a word it means just what I choose it to mean --- neither more nor less."

In the literature and in some posts here, there has been variation in the undefined use of the term "block" for a category of modules over a ring, or more abstractly an abelian category (all of which are categories of modules by Freyd-Mitchell). This raises the natural question:

What is meant by a "block" in an abelian category?

The concept originates to some extent in the modular representation theory of finite groups or their group algebras pioneered by Richard Brauer. Here a block is just an indecomposable two-sided ideal of the group algebra, corresponding to a primitive central idempotent. But in later developments the language of homological algebra plays a greater role than the group algebra or its center: the category of modules decomposes into a direct sum of subcategories, which are as small as possible relative to permitting no nontrivial extensions among their simple objects (irreducible representations). This approach generalizes well to other situations, where a center or central characters may be elusive and where the decomposition may be infinite, etc. By now "blocks" occur in many areas of representation theory influenced by classical Lie theory: algebraic groups, restricted enveloping algebras, quantum analogues, finite $W$-algebras, Cherednik algebras, Kac-Moody algebras and groups, Lie superalgebras.

There is some inconsistency in the literature about allowing "blocks" which might be further decomposed into direct sums. In classical or Kac-Moody Lie theory this usually reflects the special influence of infinitesimal/central characters. But full centers in Lie theory and related quantum groups may be unknown or unneeded, e.g., the approach Kac took to his analogue of the Weyl character formula for integrable modules of an affine Lie algebra relied just on a single Casimir-type operator (which works equally well in the classical finite-dimensional case). In Jantzen's book Representations of Algebraic Groups (AMS, 2003), the discussion of blocks for algebraic group schemes in II.7.1 is careful but not completely general.

In practice looser definitions of "block" than the homological one work well enough in many settings, but it creates some confusion when the word is used with no definition at all. Is there a single convention which reduces in familiar cases to older usage? In a recent comment to another post I paraphrased Humpty Dumpty ("my remote ancestor"), who actually said: "When I use a word it means just what I choose it to mean --- neither more nor less." But communication is better when the short and convenient word "block" starts out with a common meaning.

In the literature and in some posts here, there has been variation in the undefined use of the term "block" for a category of modules over a ring, or more abstractly an abelian category (all of which are categories of modules by Freyd-Mitchell). This raises the natural question:

What is meant by a "block" in an abelian category?

The concept originates to some extent in the modular representation theory of finite groups or their group algebras pioneered by Richard Brauer. Here a block is just an indecomposable two-sided ideal of the group algebra, corresponding to a primitive central idempotent. But in later developments the language of homological algebra plays a greater role than the group algebra or its center: the module category decomposes into a direct sum of subcategories, which are as small as possible relative to permitting no nontrivial extensions among their simple objects (irreducible representations). This approach seems to me to generalize best to other situations, where a center or central characters may be elusive and where the decomposition may be infinite, etc.

There is some inconsistency in the literature about allowing "blocks" which might be further decomposed into direct sums. In classical or Kac-Moody Lie theory this usually reflects the influence of infinitesimal/central characters. But full centers in Lie theory and related quantum groups may be unknown or unneeded. The approach Kac took to his analogue of the Weyl character formula for integrable modules of an affine Lie algebra relied just on a single Casimir-type operator (which works equally well in the classical finite-dimensional case).

In practice looser definitions of "block" than the homological one work well enough in many settings, but it creates some confusion when the word is used with no definition at all. Should there be a standard convention? In a recent comment to another post I paraphrased Humpty Dumpty ("my remote ancestor"), who actually said: "When I use a word it means just what I choose it to mean --- neither more nor less."

In the literature and in some posts here, there has been variation in the undefined use of the term "block" for a category of modules over a ring, or more abstractly an abelian category (all of which are categories of modules by Freyd-Mitchell). This raises the natural question:

What is meant by a "block" in an abelian category?

The concept originates to some extent in the modular representation theory of finite groups or their group algebras pioneered by Richard Brauer. Here a block is just an indecomposable two-sided ideal of the group algebra, corresponding to a primitive central idempotent. But in later developments the language of homological algebra plays a greater role than the group algebra or its center: the category of modules decomposes into a direct sum of subcategories, which are as small as possible relative to permitting no nontrivial extensions among their simple objects (irreducible representations). This approach seems to me to generalize best to other situations, where a center or central characters may be elusive and where the decomposition may be infinite, etc.

There is some inconsistency in the literature about allowing "blocks" which might be further decomposed into direct sums. In classical or Kac-Moody Lie theory this usually reflects the influence of infinitesimal/central characters. But full centers in Lie theory and related quantum groups may be unknown or unneeded. The approach Kac took to his analogue of the Weyl character formula for integrable modules of an affine Lie algebra relied just on a single Casimir-type operator (which works equally well in the classical finite-dimensional case).

In practice looser definitions of "block" than the homological one work well enough in many settings, but it creates some confusion when the word is used with no definition at all. Should there be a standard convention? In a recent comment to another post I paraphrased Humpty Dumpty ("my remote ancestor"), who actually said: "When I use a word it means just what I choose it to mean --- neither more nor less."

In the literature and in some posts here, there has been variation in the undefined use of the term "block" for module categories or other abelian categories having similar properties. This raises the natural question:

What is meant by a "block" in a module category?

The concept originates to some extent in the modular representation theory of finite groups or their group algebras pioneered by Richard Brauer. Here a block is just an indecomposable two-sided ideal of the group algebra, corresponding to a primitive central idempotent. But in later developments the language of homological algebra plays a greater role than the group algebra or its center: the module category decomposes into a direct sum of subcategories, which are as small as possible relative to permitting no nontrivial extensions among their simple objects (irreducible representations). This approach seems to me to generalize best to other situations, where a center or central characters may be elusive and where the decomposition may be infinite, etc.

There is some inconsistency in the literature about allowing "blocks" which might be further decomposed into direct sums. In classical or Kac-Moody Lie theory this usually reflects the influence of infinitesimal/central characters. But full centers in Lie theory and related quantum groups may be unknown or unneeded. The approach Kac took to his analogue of the Weyl character formula for integrable modules of an affine Lie algebra relied just on a single Casimir-type operator (which works equally well in the classical finite-dimensional case).

In practice looser definitions of "block" than the homological one work well enough in many settings, but it creates some confusion when the word is used with no definition at all. Should there be a standard convention? In a recent comment to another post I paraphrased Humpty Dumpty ("my remote ancestor"), who actually said: "When I use a word it means just what I choose it to mean --- neither more nor less."

In the literature and in some posts here, there has been variation in the undefined use of the term "block" for a category of modules over a ring, or more abstractly an abelian category (all of which are categories of modules by Freyd-Mitchell). This raises the natural question:

What is meant by a "block" in an abelian category?

The concept originates to some extent in the modular representation theory of finite groups or their group algebras pioneered by Richard Brauer. Here a block is just an indecomposable two-sided ideal of the group algebra, corresponding to a primitive central idempotent. But in later developments the language of homological algebra plays a greater role than the group algebra or its center: the module category decomposes into a direct sum of subcategories, which are as small as possible relative to permitting no nontrivial extensions among their simple objects (irreducible representations). This approach seems to me to generalize best to other situations, where a center or central characters may be elusive and where the decomposition may be infinite, etc.

There is some inconsistency in the literature about allowing "blocks" which might be further decomposed into direct sums. In classical or Kac-Moody Lie theory this usually reflects the influence of infinitesimal/central characters. But full centers in Lie theory and related quantum groups may be unknown or unneeded. The approach Kac took to his analogue of the Weyl character formula for integrable modules of an affine Lie algebra relied just on a single Casimir-type operator (which works equally well in the classical finite-dimensional case).

In practice looser definitions of "block" than the homological one work well enough in many settings, but it creates some confusion when the word is used with no definition at all. Should there be a standard convention? In a recent comment to another post I paraphrased Humpty Dumpty ("my remote ancestor"), who actually said: "When I use a word it means just what I choose it to mean --- neither more nor less."