Tits quadratic form

Let $Q = ( Q _ { 0 } , Q _ { 1 } )$ be a finite quiver (see [a8]), that is, an oriented graph with vertex set $Q_0$ and set $Q _ { 1 }$ of arrows (oriented edges; cf. also Graph, oriented; Quiver). Following P. Gabriel [a8], [a9], the Tits quadratic form $q_{Q} : \mathbf{Z} ^ { Q _ { 0 } } \rightarrow \mathbf{Z} $ of $Q$ is defined by the formula

\begin{equation*} q_Q ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ { i , j \in Q _ { 0 } } d _ { i j } x _ { i } x _ { j }, \end{equation*}

where $x = ( x _ { i } ) _ { i \in Q _ { 0 } } \in \mathbf{Z} ^ { Q _ { 0 } }$ and $d _ { i j}$ is the number of arrows from $i$ to $j$ in $Q _ { 1 }$.

There are important applications of the Tits form in representation theory. One easily proves that if $Q$ is connected, then $q_Q$ is positive definite if and only if $Q$ (viewed as a non-oriented graph) is any of the Dynkin diagrams $\mathbf{A} _ { n }$, ${\bf D} _ { n }$, ${\bf E} _ { 6 }$, $\mathbf{E} _ { 7 }$, or $\mathbf{E} _ { 8 }$ (cf. also Dynkin diagram). On the other hand, the Gabriel theorem [a8] asserts that this is the case if and only if $Q$ has only finitely many isomorphism classes of indecomposable $K$-linear representations, where $K$ is an algebraically closed field (see also [a2]). Let $\operatorname{rep}_K( Q )$ be the Abelian category of finite-dimensional $K$-linear representations of $Q$ formed by the systems $\mathbf{X} = ( X _ { i } , \phi _ { \beta } ) _ { j \in Q _ { 0 } , \beta \in Q _ { 1 }}$ of finite-dimensional vector $K$-spaces $X_j$, connected by $K$-linear mappings $\phi _ { \beta } : X _ { i } \rightarrow X _ { j }$ corresponding to arrows $\beta : i \rightarrow j$ of $Q$. By a theorem of L.A. Nazarova [a12], given a connected quiver $Q$ the category $\operatorname{rep}_K( Q )$ is of tame representation type (see [a7], [a10], [a19] and Quiver) if and only if $q_Q$ is positive semi-definite, or equivalently, if and only if $Q$ (viewed as a non-oriented graph) is any of the extended Dynkin diagrams $\tilde { A }_{ n }$, $\tilde { \mathbf{D} } _ { n }$, $\widetilde{\bf E} _ { 6 }$, $\tilde{\mathbf{E}} _ { 7 }$, or $\tilde{\bf E} _ { 8 }$ (see [a1], [a10], [a19]; and [a4] for a generalization).

Let $K _ { 0 } ( Q ) = K _ { 0 } ( \operatorname { rep } _ { K } ( Q ) )$ be the Grothendieck group of the category $\operatorname{rep}_K( Q )$. By the Jordan–Hölder theorem, the correspondence $\mathbf{X} \mapsto \underline{\operatorname { dim }} \mathbf{X} = ( \operatorname { dim } _ { K } X _ { j } ) _ { j \in Q _ { 0 } }$ defines a group isomorphism $\underline{\operatorname { dim }} : K _ { 0 } ( Q ) \rightarrow \mathbf{Z} ^ { Q _ { 0 } }$. One shows that the Tits form $q_Q$ coincides with the Euler characteristic $\chi _ { Q } : K _ { 0 } ( Q ) \rightarrow \mathbf{Z}$, $[ \mathbf{X} ] \mapsto \chi _ { Q } ( [ \mathbf{X} ] ) = \operatorname { dim } _ { K } \operatorname { End } _ { Q } ( \mathbf{X} ) - \operatorname { dim } _ { K } \operatorname { Ext } _ { Q } ^ { 1 } ( \mathbf{X} , \mathbf{X} )$, along the isomorphism $\underline{\operatorname { dim }} : K _ { 0 } ( Q ) \rightarrow \mathbf{Z} ^ { Q _ { 0 } }$, that is, $q_{Q} ( \underline { \operatorname { dim } } \mathbf{X} ) = \chi _ { Q } ( [ \mathbf{X} ] )$ for any $\mathbf{X}$ in $\operatorname{rep}_K( Q )$ (see [a10], [a17]).

The Tits quadratic form $q_Q$ is related with an algebraic geometry context defined as follows (see [a9], [a10], [a19]).

For any vector $v = ( v _ { j } ) _ { j \in Q _ { 0 } } \in \mathbf{N} ^ { Q _ { 0 } }$, consider the affine irreducible $K$-variety $\mathcal{A} _ { Q } ( v ) = \prod _ { i ,\, j \in Q _ { 0 } } \prod _ { ( \beta : j \rightarrow i ) \in Q _ { 1 } } \mathbf{M} _ { v _ { i } \times v _ { j } } ( K ) _ { \beta }$ of $K$-representations of $Q$ of the dimension type $v$ (in the Zariski topology), where $\mathbf{M} _ { v _ { i } \times v _ { j } } ( K ) _ { \beta } = \mathbf{M} _ { v _ { i } \times v _ { j } } ( K )$ is the space of $( v _ { i } \times v _ { j } )$-matrices for any arrow $\beta : j \rightarrow i$ of $Q$. Consider the algebraic group ${\cal G} \operatorname{l} _ { Q } ( d ) = \prod _ { j \in Q _ { 0 } } \operatorname{Gl} ( v _ { j } , K )$ and the algebraic group action $* : \mathcal{G} \text{l} _ { Q } ( d ) \times \mathcal{A} _ { Q } ( d ) \rightarrow \mathcal{A} _ { Q } ( d )$ defined by the formula $( h _ { j } ) ^ { * } ( M _ { i j } ^ { \beta } ) = ( h _ { i } ^ { - 1 } M _ { i j } ^ { \beta } h _ { j } )$, where $\beta : j \rightarrow i$ is an arrow of $Q$, $M _ { i j } ^ { \beta } \in \mathbf{M} _ { v _ { j } \times v _ { i } } ( K ) _ { \beta }$, $h _ { j } \in \operatorname{Gl} ( v _ { j } , K )$, and $h _ { i } \in \operatorname{Gl} ( v _ { i } , K )$. An important role in applications is played by the Tits-type equality $q_Q ( v ) = \operatorname { dim } {\cal G}\operatorname{l} _ { Q } ( v ) - \operatorname { dim } {\cal A} _ { Q } ( v )$, $v \in \mathbf N ^ { Q _ 0}$, where $\operatorname{dim}$ denotes the dimension of the algebraic variety (see [a8]).

Following the above ideas, Yu.A. Drozd [a5] introduced and successfully applied a Tits quadratic form in the study of finite representation type of the Krull–Schmidt category $\operatorname{Mat}_I$ of matrix $K$-representations of partially ordered sets $( I , \preceq )$ with a unique maximal element (see [a10], [a19]). In [a6] and [a7] he also studied bimodule matrix problems and the representation type of boxes $\mathcal{B}$ by means of an associated Tits quadratic form $q_{\cal B} : {\bf Z} ^ { n } \rightarrow {\bf Z}$ (see also [a18]). In particular, he showed [a6] that if $\mathcal{B}$ is of tame representation type, then $q_{\mathcal{B}}$ is weakly non-negative, that is, $q _ { \mathcal B } ( v ) \geq 0$ for all $v \in {\bf N} ^ { n }$.

K. Bongartz [a3] associated with any finite-dimensional basic $K$-algebra $R$ a Tits quadratic form as follows. Let $\{ e _ { 1 } , \ldots , e _ { n } \}$ be a complete set of primitive pairwise non-isomorphic orthogonal idempotents of the algebra $R$. Fix a finite quiver $Q = ( Q _ { 0 } , Q _ { 1 } )$ with $Q _ { 0 } = \{ 1 , \ldots , n \}$ and a $K$-algebra isomorphism $R \simeq K Q / I$, where $K Q$ is the path $K$-algebra of the quiver $Q$ (see [a1], [a10], [a19]) and $I$ is an ideal of $R$ contained in the square of the Jacobson radical $\operatorname{rad} R$ of $R$ and containing a power of $\operatorname{rad} R$. Assume that $Q$ has no oriented cycles (and hence the global dimension of $R$ is finite). The Tits quadratic form $q_R : {\bf Z} ^ { n } \rightarrow \bf Z$ of $R$ is defined by the formula

\begin{equation*} q_{ R} ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ {( \beta : i \rightarrow j ) \in Q _ { 1 } } x _ { i } x _ { j } + \sum _ { ( \beta : i \rightarrow j ) \in Q _ { 1 } } r _ { i ,\, j } x _ { i } x _ { j }, \end{equation*}

where $r_{i, j} = | L \cap e _ { j } I e _ { i } |$, for a minimal set $L$ of generators of $I$ contained in $\sum _ { i , j \in Q _ { 0 } } e _ { j } I { e }_i$. One checks that $r_{i,\,j} = \operatorname { dim } _ { K } \operatorname { Ext } _ { R } ^ { 2 } ( S _ { j } , S _ { i } )$, where $S _ { t }$ is the simple $R$-module associated to the vertex $t \in Q_0$. Then the definition of $q_{ R}$ depends only on $R$, and when $R$ is of global dimension at most two, the form $q_{ R}$ coincides with the Euler characteristic $\chi _ { R } : K _ { 0 } ( \operatorname { mod } R ) \rightarrow \mathbf Z$, $[ X ] \mapsto \chi _ { R } ( [ X ] ) = \sum _ { m = 0 } ^ { \infty } ( - 1 ) ^ { m } \operatorname { dim } _ { K } \operatorname { Ext } _ { R } ^ { m } ( X , X )$, under a group isomorphism $\underline{\operatorname { dim }} : K _ { 0 } ( \operatorname { mod } R ) \rightarrow \mathbf{Z} ^ { Q _ { 0 } }$, where $K_0 \pmod{R}$ is the Grothendieck group of the category $\operatorname{mod} R$ of finite-dimensional right $R$-modules (see [a17]). Note that $q_R = q_Q$ if $R = K Q$.

By applying a Tits-type equality as above, Bongartz [a3] proved that if $R$ is of finite representation type, then $q_{ R}$ is weakly positive, that is, $q _ { R } ( v ) > 0$ for all non-zero vectors $v \in {\bf N} ^ { n }$. The converse implication does not hold in general, but it has been established if the Auslander–Reiten quiver of $R$ (see Riedtmann classification) has a post-projective component (see [a10]), by applying an idea of Drozd [a5]. J.A. de la Peña [a14] proved that if $R$ is of tame representation type, then $q_{ R}$ is weakly non-negative. The converse implication does not hold in general, but it has been proved under a suitable assumption on $R$ (see [a13] and [a16] for a discussion of this problem and relations between the Tits quadratic form and the Euler quadratic form of $R$).

Let $( I , \preceq )$ be a partially ordered set with partial order relation $\preceq$ and let $\operatorname { max}I$ be the set of all maximal elements of $( I , \preceq )$. Following [a5] and [a15], D. Simson [a20] defined the Tits quadratic form $q_l : \mathbf{Z} ^ { l } \rightarrow \mathbf{Z}$ of $( I , \preceq )$ by the formula

\begin{equation*} q _I( x ) = \sum _ { i \in I } x _ { i } ^ { 2 } + \sum _ { \substack {i \prec j} \\{j\in I\backslash \operatorname {max} I} } x _ { i } x _ { j } - \sum _ { p \in \operatorname { max } I } ( \sum _ { i \prec p } x _ { i } ) x _ { p } \end{equation*}

and applied it in the study of prinjective $KI$-modules, that is, finite-dimensional right modules $X$ over the incidence $K$-algebra $K I = K ( I , \preceq )$ of $( I , \preceq )$ such that there is an exact sequence $0 \rightarrow P _ { 1 } \rightarrow P _ { 0 } \rightarrow X \rightarrow 0$, where $P_0$ is a projective $KI$-module and $P _ { 1 }$ is a direct sum of simple projectives. The additive Krull–Schmidt category $\operatorname { prin } K I$ of prinjective $KI$-modules is equivalent to the category of matrix $K$-representations of $( I , \preceq )$ [a20]. Under an identification $K_{0} ( \operatorname { prin } K I ) \simeq \mathbf{Z} ^ { I }$, the Tits form $q_{l}$ is equal to the Euler characteristic $\chi _ { K I } : K _ { 0 } ( \operatorname { prin } K I ) \rightarrow \bf Z$. A Tits-type equality is also valid for $q_{l}$ [a15]. It has been proved in [a20] that $q_{l}$ is weakly positive if and only if $\operatorname { prin } K I$ has only a finite number of iso-classes of indecomposable modules. By [a15], if $\operatorname { prin } K I$ is of tame representation type, then $q_{l}$ is weakly non-negative. The converse implication does not hold in general, but it has been proved under some assumption on $( I , \preceq )$ (see [a11]).

A Tits quadratic form $q _ { \Lambda } : \mathbf{Z} ^ { n } \rightarrow \mathbf{Z}$ for a class of classical $D$-orders $\Lambda$, where $D$ is a complete discrete valuation domain, has been defined in [a21]. Criteria for the finite lattice type and tame lattice type of $\Lambda$ are given in [a21] by means of $q _ { \Lambda }$.

For a class of $K$-co-algebras $C$, a Tits quadratic form $q _ { C } : \mathbf{Z} ^ { ( l _ { C } ) } \rightarrow \mathbf{Z}$ is defined in [a22], and the co-module types of $C$ are studied by means of $q_{C}$, where $I _ { C }$ is a complete set of pairwise non-isomorphic simple left $C$-co-modules and ${\bf Z} ^ { ( I _ { C } ) }$ is a free Abelian group of rank $| I _ { C } |$.

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