I am studying a category of representations of an infinite-dimensional Lie algebra, which is an abelian category. Unfortunately, the category does not have enough injectives/projectives. I would like to find some references to see how one computes the Ext-groups in this kind of situations.

More precisely, let $R$ be a unital ring (even a unital associative $\mathbb{K}$-algebra over a field $\mathbb{K}$). For an abelian subcategory $\mathcal{A}$ of the category of unitary left $R$-modules, possibly without enough injectives/projectives, how do I compute $\text{Ext}_\mathcal{A}^n\left(M,N\right)$ for each $M,N\in\mathcal{A}$ and $n\in\mathbb{N}$ (it is especially important to consider the case where $M$ and $N$ are simple $\mathcal{A}$-modules)? Does it help if $\mathcal{A}$ is assumed to be a full subcategory? Does it help if there exists duality in $\mathcal{A}$, i.e., an exact contravariant functor $(\_)^\vee:\mathcal{A}\to\mathcal{A}$ such that $\left(M^\vee\right)^\vee$ is naturally isomorphic to $M$ for all $M\in\mathcal{A}$? Does it help if $\mathcal{A}$ is the universal enveloping algebra of a Lie algebra (potentially infinite-dimensional, but countable-dimensional simple Lie algebras are of particular interest), especially if the base field is algebraically closed and of characteristic $0$?

I would like to establish the blocks of this category by studying extensions between two simple objects. I can find some full abelian subcategories of this category which have enough injectives/projectives, but not all simple objects are in such subcategories. So, at this moment, I think I really need to understand how Ext-groups are computed without the assumption that the (abelian) category has enough injectives/projectives.

For those who are curious how one defines Ext-groups without the use of injectives/projectives, please take a look at http://stacks.math.columbia.edu/tag/06XP. Even better, you can read "An Introduction to Homological Algebra" by Charles A. Weibel.

I am studying a category of representations of an infinite-dimensional Lie algebra, which is an abelian category. Unfortunately, the category does not have enough injectives/projectives. I would like to find some references to see how one computes the Ext-groups in this kind of situations.

More precisely, let $R$ be a unital ring (even a unital associative $\mathbb{K}$-algebra over a field $\mathbb{K}$). For an abelian subcategory $\mathcal{A}$ of the category of unitary left $R$-modules, possibly without enough injectives/projectives, how do I compute $\text{Ext}_\mathcal{A}^n\left(M,N\right)$ for each $M,N\in\mathcal{A}$ and $n\in\mathbb{N}$ (it is especially important to consider the case where $M$ and $N$ are simple $\mathcal{A}$-modules)? Does it help if $\mathcal{A}$ is assumed to be a full subcategory? Does it help if there exists duality in $\mathcal{A}$, i.e., an exact contravariant functor $(\_)^\vee:\mathcal{A}\to\mathcal{A}$ such that $\left(M^\vee\right)^\vee$ is naturally isomorphic to $M$ for all $M\in\mathcal{A}$? Does it help if $R$ is the universal enveloping algebra of a Lie algebra (potentially infinite-dimensional, but countable-dimensional simple Lie algebras are of particular interest), especially if the base field is algebraically closed and of characteristic $0$?

I would like to establish the blocks of this category by studying extensions between two simple objects. I can find some full abelian subcategories of this category which have enough injectives/projectives, but not all simple objects are in such subcategories. So, at this moment, I think I really need to understand how Ext-groups are computed without the assumption that the (abelian) category has enough injectives/projectives.

For those who are curious how one defines Ext-groups without the use of injectives/projectives, please take a look at http://stacks.math.columbia.edu/tag/06XP. Even better, you can read "An Introduction to Homological Algebra" by Charles A. Weibel.

I am studying a category of representations of an infinite-dimensional Lie algebra, which is an abelian category. Unfortunately, the category does not have enough injectives/projectives. I would like to find some references to see how one computes the Ext-groups in this kind of situations.

More precisely, let $R$ be a unital ring (even a unital associative $\mathbb{K}$-algebra over a field $\mathbb{K}$). For an abelian subcategory $\mathcal{A}$ of the category of unitary left $R$-modules, possibly without enough injectives/projectives, how do I compute $\text{Ext}_\mathcal{A}^n\left(M,N\right)$ for each $M,N\in\mathcal{A}$ and $n\in\mathbb{N}$ (it is especially important to consider the case where $M$ and $N$ are simple $\mathcal{A}$-modules)? Does it help if $\mathcal{A}$ is assumed to be a full subcategory? Does it help if there exists duality in $\mathcal{A}$, i.e., an exact contravariant functor $(\_)^\vee:\mathcal{A}\to\mathcal{A}$ such that $\left(M^\vee\right)^\vee$ is naturally isomorphic to $M$ for all $M\in\mathcal{A}$?

I would like to establish the blocks of this category by studying extensions between two simple objects. I can find some full abelian subcategories of this category which have enough injectives/projectives, but not all simple objects are in such subcategories. So, at this moment, I think I really need to understand how Ext-groups are computed without the assumption that the (abelian) category has enough injectives/projectives.

For those who are curious how one defines Ext-groups without the use of injectives/projectives, please take a look at http://stacks.math.columbia.edu/tag/06XP. Even better, you can read "An Introduction to Homological Algebra" by Charles A. Weibel.

I am studying a category of representations of an infinite-dimensional Lie algebra, which is an abelian category. Unfortunately, the category does not have enough injectives/projectives. I would like to find some references to see how one computes the Ext-groups in this kind of situations.

More precisely, let $R$ be a unital ring (even a unital associative $\mathbb{K}$-algebra over a field $\mathbb{K}$). For an abelian subcategory $\mathcal{A}$ of the category of unitary left $R$-modules, possibly without enough injectives/projectives, how do I compute $\text{Ext}_\mathcal{A}^n\left(M,N\right)$ for each $M,N\in\mathcal{A}$ and $n\in\mathbb{N}$ (it is especially important to consider the case where $M$ and $N$ are simple $\mathcal{A}$-modules)? Does it help if $\mathcal{A}$ is assumed to be a full subcategory? Does it help if there exists duality in $\mathcal{A}$, i.e., an exact contravariant functor $(\_)^\vee:\mathcal{A}\to\mathcal{A}$ such that $\left(M^\vee\right)^\vee$ is naturally isomorphic to $M$ for all $M\in\mathcal{A}$? Does it help if $\mathcal{A}$ is the universal enveloping algebra of a Lie algebra (potentially infinite-dimensional, but countable-dimensional simple Lie algebras are of particular interest), especially if the base field is algebraically closed and of characteristic $0$?

I would like to establish the blocks of this category by studying extensions between two simple objects. I can find some full abelian subcategories of this category which have enough injectives/projectives, but not all simple objects are in such subcategories. So, at this moment, I think I really need to understand how Ext-groups are computed without the assumption that the (abelian) category has enough injectives/projectives.

For those who are curious how one defines Ext-groups without the use of injectives/projectives, please take a look at http://stacks.math.columbia.edu/tag/06XP. Even better, you can read "An Introduction to Homological Algebra" by Charles A. Weibel.