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Let $A$ be a finite dimensional algebra. A NCR of an algebra $A$ is a faithful module $M$ such that $End_A(M)$ has finite global dimension. Call a faithful module PNCR (=pseudo-NCR) in case $End_A(M)$ has Cartan determinant equal to one.

While checking whether $M$ is PNCR takes mostly only seconds with a computer, checking whether $M$ is NCR is usually impossible with a computer as it takes too long.

Call $A$ great in case $M$ is a NCR if and only if $M$ is a PNCR. For example $A=K[x]/(x^n)$ is great.

Question 1: Are there other nontrivial great algebras?

 

Question 2: Are Brauer tree algebras great? (The Brauer tree algebras with 2 simples and multiplicity one are also great, but higher dimensional cases take too long with the computer aleady).

Let $A$ be a finite dimensional algebra. A NCR of an algebra $A$ is a faithful module $M$ such that $End_A(M)$ has finite global dimension. Call a faithful module PNCR (=pseudo-NCR) in case $End_A(M)$ has Cartan determinant equal to one.

While checking whether $M$ is PNCR takes mostly only seconds with a computer, checking whether $M$ is NCR is usually impossible with a computer as it takes too long.

Call $A$ great in case $M$ is a NCR if and only if $M$ is a PNCR. For example $A=K[x]/(x^n)$ is great.

Question 1: Are there other nontrivial great algebras?

Question 2: Are Brauer tree algebras great? (The Brauer tree algebras with 2 simples and multiplicity one are also great, but higher dimensional cases take too long with the computer aleady).

Let $A$ be a finite dimensional algebra. A NCR of an algebra $A$ is a faithful module $M$ such that $End_A(M)$ has finite global dimension. Call a faithful module PNCR (=pseudo-NCR) in case $End_A(M)$ has Cartan determinant equal to one.

While checking whether $M$ is PNCR takes mostly only seconds with a computer, checking whether $M$ is NCR is usually impossible with a computer as it takes too long.

Call $A$ great in case $M$ is a NCR if and only if $M$ is a PNCR. For example $A=K[x]/(x^n)$ is great.

Question 1: Are there other nontrivial great algebras?

Question 2: Are Brauer tree algebras great?

Let $A$ be a finite dimensional algebra. A NCR of an algebra $A$ is a faithful module $M$ such that $End_A(M)$ has finite global dimension. Call a faithful module PNCR (=pseudo-NCR) in case $End_A(M)$ has Cartan determinant equal to one.

While checking whether $M$ is PNCR takes mostly only seconds with a computer, checking whether $M$ is NCR is usually impossible with a computer as it takes too long.

Call $A$ great in case $M$ is a NCR if and only if $M$ is a PNCR. For example $A=K[x]/(x^n)$ is great.

Question 1: Are there other nontrivial great algebras?

Question 2: Are Brauer tree algebras great? (The Brauer tree algebras with 2 simples and multiplicity one are also great, but higher dimensional cases take too long with the computer aleady).

Let $A$ be a finite dimensional algebra. A NCR of an algebra $A$ is a faithful module $M$ such that $End_A(M)$ has finite global dimension. Call a faithful module PNCR (=pseudo-NCR) in case $End_A(M)$ has Cartan matrix equal to one.

While checking whether $M$ is PNCR takes mostly only seconds with a computer, checking whether $M$ is NCR is usually impossible with a computer as it takes too long.

Call $A$ great in case $M$ is a NCR if and only if $M$ is a PNCR. For example $A=K[x]/(x^n)$ is great.

Question 1: Are there other nontrivial great algebras?

Question 2: Are Brauer tree algebras great?

Let $A$ be a finite dimensional algebra. A NCR of an algebra $A$ is a faithful module $M$ such that $End_A(M)$ has finite global dimension. Call a faithful module PNCR (=pseudo-NCR) in case $End_A(M)$ has Cartan determinant equal to one.

While checking whether $M$ is PNCR takes mostly only seconds with a computer, checking whether $M$ is NCR is usually impossible with a computer as it takes too long.

Call $A$ great in case $M$ is a NCR if and only if $M$ is a PNCR. For example $A=K[x]/(x^n)$ is great.

Question 1: Are there other nontrivial great algebras?

Question 2: Are Brauer tree algebras great?