# 1. Definition

A Nakayama algebra is a finite-dimensional algebra over a field $\mathbb{F}$ where every indecomposable projective or injective module is uniserial, see [ARS97], [AF92] and [SY11].

Let $Q$ be a finite quiver with path algebra $\mathbb{F} Q$, and let $I$ be a two-sided ideal in $\mathbb{F} Q$. Let $J$ denote the ideal generated by all arrows in $\mathbb{F} Q$. Then $I$ is called admissible in case $J^m \subseteq I \subseteq J^2$ for some $m \geq 2$. We will restrict in this survey on Nakayama algebras given by quiver and admissible relations (note that over algebraically closed fields this is no loss of generality when doing homological algebra, as any finite dimensional algebra is Morita equivalent to a quiver algebra). In this language, Nakayama algebras can be characterised as algebras $\mathbb{F}Q/I$ for admissible ideals $I$ and a finite quiver $Q$ that is either an oriented linear quiver or an oriented cyclic quiver.

Let $A$ be an Nakayama algebra with $n$ simple modules and let $e_i$ denote the idempotent corresponding to the vertex $i$ in the quiver. The Kupisch series of $A$ is the sequence $[c_0,c_1,...,c_{n-1}]$, where $c_i \geq 1$ denotes the vector space dimension of the indecomposable projective module $e_iA$.

# 2. Nakayama algebras with linear quiver and Dyck paths

The area sequence $[a_1,\dots,a_n]$ of a Dyck path of semilength $n$ is given by setting $a_i$ to be the number of full boxes between the path and the main diagonal, see St000012.

Sending a Nakayama algebra with $n+1$ simple modules and Kupisch sequence $[c_0,\dots,c_n]$ to unique Dyck path with area sequence $[c_{n-1}-2,\dots,c_1-2,c_0-2]$ is a bijection between Nakayama algebras on $n+1$ simple modules and with linear quiver and Dyck paths of semilength $n$.

All statistics on Dyck paths in the database that describe properties of Nakayama algebras with linear quiver use this bijective identification.

# 3. Properties

Let $A$ be a Nakayama algebra over the field $\mathbb{F}$ with $n$ simple modules and Kupisch series $[c_0,\dots,c_{n-1}]$ and let $M$ and $N$ be $A$-modules and let $S$ be a simple $A$-module. Denote by $D:=Hom_{\mathbb{F}}(-,\mathbb{F})$ the natural duality.

• The Jacobson radical of a general algebra $A$ is defined as the intersection of all maximal right ideals of $A$.

• The Loewy length of a general algebra $A$ is defined as the smallest integer $n$ such that for the Jacobson radical $J$ we have $J^n=0$.

• Every indecomposable module over $A$ is uniserial and can be written in the form $e_i A / e_i J^k$ for $i$ a point in the quiver and $1 \leq k \leq c_i$. The modules $e_i A / e_i J^1$ are exactly the simple modules.

• A module isomorphic to $D(Ae_i)$ is called indecomposable injective.

• The projective cover of $M$ is the unique map (up to isomorphism) $P \rightarrow M$ such that $P$ is projective of minimal vector space dimension. Dually the injective envelope of $M$ is by definition the map $M \rightarrow I$ such that $I$ is injective of minimal vector space dimension. One often just speaks of $P$ as the projective cover for short and also as $I$ being the injective envelope.

• The first syzygy module $\Omega^{1}(M)$ is the kernel of the projective cover $P \rightarrow M$ of $M$. Inductively, one defines for $n \geq 0$ the $n$-th syzygy module of $M$ as $\Omega^n(M) := \Omega^1(\Omega^{n-1}(M))$ with $\Omega^0(M)=M$.

• The projective dimension of $M$ is defined as the smallest integer $n \geq 0$ such that $\Omega^n(M)$ is projective and as infinite in case no such $n$ exists. The injective dimension of $M$ is defined as the projective dimension of $D(M)$.

• Define $Ext_A^1(M,N)$ as $Ext_A^1(M,N) := D(\overline{Hom}_A(N,\tau(M)))$, where $\tau(M)$ denotes the Auslander-Reiten translate of $M$ and $\overline{Hom}_A(X,Y)$ denotes the space of homomorphisms between two $A$-modules $X$ and $Y$ modulo the space of homomorphisms between $X$ and $Y$ that factor over an injective $A$-module.

• For $n \geq 1$, define $Ext_A^n(M,N) := Ext_A^1(\Omega^{n-1}(M),N)$ with $Ext_A^0(M,N) := Hom_A(M,N)$.

• The global dimension of $A$ is defined as the maximal projective dimension of a simple module.

• For Nakayama algebras with a linear quiver, the global dimension equals the maximal projective dimension of an indecomposable injective module.
• Let $P_i$ be the projective cover of $\Omega^i(M)$. Then $M$ is said to have codominant dimension $n$ in case $n$ is the smallest integer such that $P_n$ is non-injective.

• The dominant dimension of $M$ is the codominant dimension of $D(M)$.

• The dominant dimension of $A$ is the dominant dimension of the regular $A$-module.

• Let $P_i$ be the projective cover of $\Omega^i(D(A))$. $A$ is said to be $k$-Gorenstein in case the injective dimension of $P_i$ is at most $i$ for all $i=0,1,...,k$.

• The maximal $k$ such that $A$ is $k$-Gorenstein is called the $k$-Gorenstein degree of $A$.

• $M$ is called $r$-torsionfree in case $Ext_A^i(D(A),\tau(M))=0$ for all $i=1,2,..,r$.

• 1-torsionfree modules are called torsionless and 2-torsionfree modules are called reflexive.

• The torsionfree index of a module $M$ is defined as the maximal number $r$ such that $M$ is $r$-torsionfree.

• $S$ is called $k$-regular for a $k \geq 1$ in case $S$ has projective dimension $k$ and $Ext_A^k(S,A)$ is one-dimensional while $Ext_A^i(S,A)=0$ for all $i=0,1,...,k-1$.

• The module $Hom_A(Hom_A(M,A),A)$ is called the double dual of $M$.

• The grade of $M$ is defined as $\inf \{ i \geq 0 | Ext_A^i(M,A) \neq 0 \}$.

• The Cartan matrix $C_A$ of $A$ is defined as the matrix with entries $(dim_{\mathbb{F}}(e_i A e_j))_{i,j}$.

• The Coxeter matrix of $A$ with finite global dimension is defined as $-C_A^{-1}C_A^T$.

• The Coxeter polynomial of $A$ is defined as the characteristic polynomial of the Coxeter matrix of $A$.

• $M$ is called a tilting module in case it has projective dimension equal to one, $Ext_A^1(M,M)=0$ and the number of indecomposable summands equals the number of simple modules.

• The socle of $M$ is the submodule of $M$ given by the sum of all simple submodules.

# 4. References

[AF92]   F. W. Anderson and R. Kent, Rings and Categories of Modules, Graduate Texts in Mathematics Vol. 13, Springer (1992).

[ARS97]   M. Auslander, I. Reiten and S. Smalø, Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics 36. Cambridge University Press (1997), xiv+425pp.

[SY11]   A. Skowronski and K. Yamagata, Frobenius Algebras I: Basic Representation Theory, EMS Textbooks in Mathematics (2011).