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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t1301401.png" /> be a finite quiver (see [[#References|[a8]]]), that is, an oriented graph with vertex set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t1301402.png" /> and set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t1301403.png" /> of arrows (oriented edges; cf. also [[Graph, oriented|Graph, oriented]]; [[Quiver|Quiver]]). Following P. Gabriel [[#References|[a8]]], [[#References|[a9]]], the Tits quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t1301404.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t1301405.png" /> is defined by the formula
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t1301406.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t1301407.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t1301408.png" /> is the number of arrows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t1301409.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014010.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014011.png" />.
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Out of 177 formulas, 174 were replaced by TEX code.-->
  
There are important applications of the Tits form in representation theory. One easily proves that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014012.png" /> is connected, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014013.png" /> is positive definite if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014014.png" /> (viewed as a non-oriented graph) is any of the Dynkin diagrams <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014018.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014019.png" /> (cf. also [[Dynkin diagram|Dynkin diagram]]). On the other hand, the Gabriel theorem [[#References|[a8]]] asserts that this is the case if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014020.png" /> has only finitely many isomorphism classes of indecomposable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014021.png" />-linear representations, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014022.png" /> is an [[Algebraically closed field|algebraically closed field]] (see also [[#References|[a2]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014023.png" /> be the [[Abelian category|Abelian category]] of finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014024.png" />-linear representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014025.png" /> formed by the systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014026.png" /> of finite-dimensional vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014027.png" />-spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014028.png" />, connected by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014029.png" />-linear mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014030.png" /> corresponding to arrows <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014031.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014032.png" />. By a theorem of L.A. Nazarova [[#References|[a12]]], given a connected quiver <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014033.png" /> the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014034.png" /> is of tame representation type (see [[#References|[a7]]], [[#References|[a10]]], [[#References|[a19]]] and [[Quiver|Quiver]]) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014035.png" /> is positive semi-definite, or equivalently, if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014036.png" /> (viewed as a non-oriented graph) is any of the extended Dynkin diagrams <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014040.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014041.png" /> (see [[#References|[a1]]], [[#References|[a10]]], [[#References|[a19]]]; and [[#References|[a4]]] for a generalization).
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Let $Q = ( Q _ { 0 } , Q _ { 1 } )$ be a finite quiver (see [[#References|[a8]]]), that is, an oriented graph with vertex set $Q_0$ and set $Q _ { 1 }$ of arrows (oriented edges; cf. also [[Graph, oriented|Graph, oriented]]; [[Quiver|Quiver]]). Following P. Gabriel [[#References|[a8]]], [[#References|[a9]]], the Tits quadratic form $q_{Q} : \mathbf{Z} ^ { Q _ { 0 } } \rightarrow \mathbf{Z} $ of $Q$ is defined by the formula
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014042.png" /> be the [[Grothendieck group|Grothendieck group]] of the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014043.png" />. By the [[Jordan–Hölder theorem|Jordan–Hölder theorem]], the correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014044.png" /> defines a group isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014045.png" />. One shows that the Tits form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014046.png" /> coincides with the [[Euler characteristic|Euler characteristic]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014048.png" />, along the isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014049.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014050.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014051.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014052.png" /> (see [[#References|[a10]]], [[#References|[a17]]]).
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\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*}
  
The Tits quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014053.png" /> is related with an [[Algebraic geometry|algebraic geometry]] context defined as follows (see [[#References|[a9]]], [[#References|[a10]]], [[#References|[a19]]]).
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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 $Q _ { 1 }$.
  
For any vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014054.png" />, consider the affine irreducible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014055.png" />-variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014056.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014057.png" />-representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014058.png" /> of the dimension type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014059.png" /> (in the [[Zariski topology|Zariski topology]]), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014060.png" /> is the space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014061.png" />-matrices for any arrow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014062.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014063.png" />. Consider the [[Algebraic group|algebraic group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014064.png" /> and the algebraic group action <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014065.png" /> defined by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014066.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014067.png" /> is an arrow of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014070.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014071.png" />. An important role in applications is played by the Tits-type equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014073.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014074.png" /> denotes the [[Dimension|dimension]] of the [[Algebraic variety|algebraic variety]] (see [[#References|[a8]]]).
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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|Dynkin diagram]]). On the other hand, the Gabriel theorem [[#References|[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|algebraically closed field]] (see also [[#References|[a2]]]). Let $\operatorname{rep}_K( Q )$ be the [[Abelian category|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 [[#References|[a12]]], given a connected quiver $Q$ the category $\operatorname{rep}_K( Q )$ is of tame representation type (see [[#References|[a7]]], [[#References|[a10]]], [[#References|[a19]]] and [[Quiver|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 [[#References|[a1]]], [[#References|[a10]]], [[#References|[a19]]]; and [[#References|[a4]]] for a generalization).
  
Following the above ideas, Yu.A. Drozd [[#References|[a5]]] introduced and successfully applied a Tits quadratic form in the study of finite representation type of the Krull–Schmidt category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014075.png" /> of matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014076.png" />-representations of partially ordered sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014077.png" /> with a unique maximal element (see [[#References|[a10]]], [[#References|[a19]]]). In [[#References|[a6]]] and [[#References|[a7]]] he also studied bimodule matrix problems and the representation type of boxes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014078.png" /> by means of an associated Tits quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014079.png" /> (see also [[#References|[a18]]]). In particular, he showed [[#References|[a6]]] that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014080.png" /> is of tame representation type, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014081.png" /> is weakly non-negative, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014082.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014083.png" />.
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Let $K _ { 0 } ( Q ) = K _ { 0 } ( \operatorname { rep } _ { K } ( Q ) )$ be the [[Grothendieck group|Grothendieck group]] of the category $\operatorname{rep}_K( Q )$. By the [[Jordan–Hölder theorem|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|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 [[#References|[a10]]], [[#References|[a17]]]).
  
K. Bongartz [[#References|[a3]]] associated with any finite-dimensional basic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014084.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014085.png" /> a Tits quadratic form as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014086.png" /> be a complete set of primitive pairwise non-isomorphic orthogonal idempotents of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014087.png" />. Fix a finite quiver <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014088.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014089.png" /> and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014090.png" />-algebra isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014091.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014092.png" /> is the path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014093.png" />-algebra of the quiver <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014094.png" /> (see [[#References|[a1]]], [[#References|[a10]]], [[#References|[a19]]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014095.png" /> is an ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014096.png" /> contained in the square of the [[Jacobson radical|Jacobson radical]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014097.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014098.png" /> and containing a power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014099.png" />. Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140100.png" /> has no oriented cycles (and hence the global dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140101.png" /> is finite). The Tits quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140102.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140103.png" /> is defined by the formula
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The Tits quadratic form $q_Q$ is related with an [[Algebraic geometry|algebraic geometry]] context defined as follows (see [[#References|[a9]]], [[#References|[a10]]], [[#References|[a19]]]).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140104.png" /></td> </tr></table>
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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|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|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|dimension]] of the [[Algebraic variety|algebraic variety]] (see [[#References|[a8]]]).
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140105.png" />, for a minimal set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140106.png" /> of generators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140107.png" /> contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140108.png" />. One checks that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140109.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140110.png" /> is the simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140111.png" />-module associated to the vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140112.png" />. Then the definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140113.png" /> depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140114.png" />, and when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140115.png" /> is of global dimension at most two, the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140116.png" /> coincides with the Euler characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140117.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140118.png" />, under a group isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140119.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140120.png" /> is the [[Grothendieck group|Grothendieck group]] of the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140121.png" /> of finite-dimensional right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140122.png" />-modules (see [[#References|[a17]]]). Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140123.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140124.png" />.
+
Following the above ideas, Yu.A. Drozd [[#References|[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 [[#References|[a10]]], [[#References|[a19]]]). In [[#References|[a6]]] and [[#References|[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 [[#References|[a18]]]). In particular, he showed [[#References|[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 $v \in {\bf N} ^ { n }$.
  
By applying a Tits-type equality as above, Bongartz [[#References|[a3]]] proved that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140125.png" /> is of finite representation type, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140126.png" /> is weakly positive, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140127.png" /> for all non-zero vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140128.png" />. The converse implication does not hold in general, but it has been established if the Auslander–Reiten quiver of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140129.png" /> (see [[Riedtmann classification|Riedtmann classification]]) has a post-projective component (see [[#References|[a10]]]), by applying an idea of Drozd [[#References|[a5]]]. J.A. de la Peña [[#References|[a14]]] proved that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140130.png" /> is of tame representation type, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140131.png" /> is weakly non-negative. The converse implication does not hold in general, but it has been proved under a suitable assumption on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140132.png" /> (see [[#References|[a13]]] and [[#References|[a16]]] for a discussion of this problem and relations between the Tits quadratic form and the Euler quadratic form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140133.png" />).
+
K. Bongartz [[#References|[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 [[#References|[a1]]], [[#References|[a10]]], [[#References|[a19]]]) and $I$ is an ideal of $R$ contained in the square of the [[Jacobson radical|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
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140134.png" /> be a partially ordered set with partial order relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140135.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140136.png" /> be the set of all maximal elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140137.png" />. Following [[#References|[a5]]] and [[#References|[a15]]], D. Simson [[#References|[a20]]] defined the Tits quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140138.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140139.png" /> 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*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140140.png" /></td> </tr></table>
+
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|Grothendieck group]] of the category $\operatorname{mod} R$ of finite-dimensional right $R$-modules (see [[#References|[a17]]]). Note that $q_R = q_Q$ if $R = K Q$.
  
and applied it in the study of prinjective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140142.png" />-modules, that is, finite-dimensional right modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140143.png" /> over the incidence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140144.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140145.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140146.png" /> such that there is an [[Exact sequence|exact sequence]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140147.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140148.png" /> is a projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140149.png" />-module and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140150.png" /> is a direct sum of simple projectives. The additive Krull–Schmidt category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140151.png" /> of prinjective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140152.png" />-modules is equivalent to the category of matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140153.png" />-representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140154.png" /> [[#References|[a20]]]. Under an identification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140155.png" />, the Tits form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140156.png" /> is equal to the Euler characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140157.png" />. A Tits-type equality is also valid for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140158.png" /> [[#References|[a15]]]. It has been proved in [[#References|[a20]]] that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140159.png" /> is weakly positive if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140160.png" /> has only a finite number of iso-classes of indecomposable modules. By [[#References|[a15]]], if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140161.png" /> is of tame representation type, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140162.png" /> is weakly non-negative. The converse implication does not hold in general, but it has been proved under some assumption on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140163.png" /> (see [[#References|[a11]]]).
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By applying a Tits-type equality as above, Bongartz [[#References|[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 [[#References|[a10]]]), by applying an idea of Drozd [[#References|[a5]]]. J.A. de la Peña [[#References|[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 [[#References|[a13]]] and [[#References|[a16]]] for a discussion of this problem and relations between the Tits quadratic form and the Euler quadratic form of $R$).
  
A Tits quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140164.png" /> for a class of classical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140165.png" />-orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140166.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140167.png" /> is a complete discrete valuation domain, has been defined in [[#References|[a21]]]. Criteria for the finite lattice type and tame lattice type of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140168.png" /> are given in [[#References|[a21]]] by means of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140169.png" />.
+
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 [[#References|[a5]]] and [[#References|[a15]]], D. Simson [[#References|[a20]]] defined the Tits quadratic form $q_l : \mathbf{Z} ^ { l } \rightarrow \mathbf{Z}$ of $( I , \preceq )$ by the formula
  
For a class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140170.png" />-co-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140171.png" />, a Tits quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140172.png" /> is defined in [[#References|[a22]]], and the co-module types of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140173.png" /> are studied by means of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140174.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140175.png" /> is a complete set of pairwise non-isomorphic simple left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140176.png" />-co-modules and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140177.png" /> is a free Abelian group of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140178.png" />.
+
\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|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 $( I , \preceq )$ [[#References|[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 $q_{l}$ [[#References|[a15]]]. It has been proved in [[#References|[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 [[#References|[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 [[#References|[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 [[#References|[a21]]]. Criteria for the finite lattice type and tame lattice type of $\Lambda$ are given in [[#References|[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 [[#References|[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 $| I _ { C } |$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> V.I. Auslander, I. Reiten, S. Smalø, "Representation theory of Artin algebras" , ''Studies Adv. Math.'' , '''36''' , Cambridge Univ. Press (1995) {{MR|1314422}} {{ZBL|0834.16001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> I.N. Bernstein, I.M. Gelfand, V.A. Ponomarev, "Coxeter functors and Gabriel's theorem" ''Russian Math. Surveys'' , '''28''' (1973) pp. 17–32 ''Uspekhi Mat. Nauk.'' , '''28''' (1973) pp. 19–33 {{MR|393065}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> K. Bongartz, "Algebras and quadratic forms" ''J. London Math. Soc.'' , '''28''' (1983) pp. 461–469 {{MR|0724715}} {{ZBL|0532.16020}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> V. Dlab, C.M. Ringel, "Indecomposable representations of graphs and algebras" , ''Memoirs'' , '''173''' , Amer. Math. Soc. (1976) {{MR|0447344}} {{ZBL|0332.16015}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> Yu.A. Drozd, "Coxeter transformations and representations of partially ordered sets" ''Funkts. Anal. Prilozhen.'' , '''8''' (1974) pp. 34–42 (In Russian) {{MR|0351924}} {{ZBL|0356.06003}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> Yu.A. Drozd, "On tame and wild matrix problems" , ''Matrix Problems'' , Akad. Nauk. Ukr. SSR., Inst. Mat. Kiev (1977) pp. 104–114 (In Russian) {{MR|498704}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> Yu.A. Drozd, "Tame and wild matrix problems" , ''Representations and Quadratic Forms'' (1979) pp. 39–74 (In Russian) {{MR|0600111}} {{ZBL|0454.16014}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> P. Gabriel, "Unzerlegbare Darstellungen 1" ''Manuscripta Math.'' , '''6''' (1972) pp. 71–103 (Also: Berichtigungen 6 (1972), 309) {{MR|332887}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> P. Gabriel, "Représentations indécomposables" , ''Séminaire Bourbaki (1973/74)'' , ''Lecture Notes in Mathematics'' , '''431''' , Springer (1975) pp. 143–169 {{MR|0485996}} {{ZBL|0335.17005}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> P. Gabriel, A.V. Roiter, "Representations of finite dimensional algebras" , ''Algebra VIII'' , ''Encycl. Math. Stud.'' , '''73''' , Springer (1992) {{MR|1239446}} {{MR|1239447}} {{ZBL|0839.16001}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> S. Kasjan, D. Simson, "Tame prinjective type and Tits form of two-peak posets II" ''J. Algebra'' , '''187''' (1997) pp. 71–96 {{MR|1425560}} {{ZBL|0944.16013}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> L.A. Nazarova, "Representations of quivers of infinite type" ''Izv. Akad. Nauk. SSSR'' , '''37''' (1973) pp. 752–791 (In Russian) {{MR|0338018}} {{ZBL|0298.15012}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> J.A. de la Peña, "Algebras with hypercritical Tits form" , ''Topics in Algebra'' , ''Banach Center Publ.'' , '''26: 1''' , PWN (1990) pp. 353–369 {{MR|}} {{ZBL|0731.16008}} </TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> J.A. de la Peña, "On the dimension of the module-varieties of tame and wild algebras" ''Commun. Algebra'' , '''19''' (1991) pp. 1795–1807 {{MR|}} {{ZBL|0818.16013}} </TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> J.A. de la Peña, D. Simson, "Prinjective modules, reflection functors, quadratic forms and Auslander–Reiten sequences" ''Trans. Amer. Math. Soc.'' , '''329''' (1992) pp. 733–753 {{MR|1025753}} {{ZBL|0789.16010}} </TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> J.A. de la Peña, A. Skowroński, "The Euler and Tits forms of a tame algebra" ''Math. Ann.'' , '''315''' (2000) pp. 37–59 {{MR|}} {{ZBL|0941.16010}} </TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top"> C.M. Ringel, "Tame algebras and integral quadratic forms" , ''Lecture Notes in Mathematics'' , '''1099''' , Springer (1984) {{MR|0774589}} {{ZBL|0546.16013}} </TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top"> A.V. Roiter, M.M. Kleiner, "Representations of differential graded categories" , ''Lecture Notes in Mathematics'' , '''488''' , Springer (1975) pp. 316–339 {{MR|0435145}} {{ZBL|0356.16011}} </TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top"> D. Simson, "Linear representations of partially ordered sets and vector space categories" , ''Algebra, Logic Appl.'' , '''4''' , Gordon &amp; Breach (1992) {{MR|1241646}} {{ZBL|0818.16009}} </TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top"> D. Simson, "Posets of finite prinjective type and a class of orders" ''J. Pure Appl. Algebra'' , '''90''' (1993) pp. 77–103 {{MR|1246276}} {{ZBL|0815.16006}} </TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top"> D. Simson, "Representation types, Tits reduced quadratic forms and orbit problems for lattices over orders" ''Contemp. Math.'' , '''229''' (1998) pp. 307–342 {{MR|1676228}} {{ZBL|0921.16007}} </TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top"> D. Simson, "Coalgebras, comodules, pseudocompact algebras and tame comodule type" ''Colloq. Math.'' , '''in press''' (2001) {{MR|1874368}} {{ZBL|1055.16038}} </TD></TR></table>
+
<table>
 +
<tr><td valign="top">[a1]</td> <td valign="top"> V.I. Auslander, I. Reiten, S. Smalø, "Representation theory of Artin algebras" , ''Studies Adv. Math.'' , '''36''' , Cambridge Univ. Press (1995) {{MR|1314422}} {{ZBL|0834.16001}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> I.N. Bernstein, I.M. Gelfand, V.A. Ponomarev, "Coxeter functors and Gabriel's theorem" ''Russian Math. Surveys'' , '''28''' (1973) pp. 17–32 ''Uspekhi Mat. Nauk.'' , '''28''' (1973) pp. 19–33 {{MR|393065}} {{ZBL|}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top"> K. Bongartz, "Algebras and quadratic forms" ''J. London Math. Soc.'' , '''28''' (1983) pp. 461–469 {{MR|0724715}} {{ZBL|0532.16020}} </td></tr><tr><td valign="top">[a4]</td> <td valign="top"> V. Dlab, C.M. Ringel, "Indecomposable representations of graphs and algebras" , ''Memoirs'' , '''173''' , Amer. Math. Soc. (1976) {{MR|0447344}} {{ZBL|0332.16015}} </td></tr><tr><td valign="top">[a5]</td> <td valign="top"> Yu.A. Drozd, "Coxeter transformations and representations of partially ordered sets" ''Funkts. Anal. Prilozhen.'' , '''8''' (1974) pp. 34–42 (In Russian) {{MR|0351924}} {{ZBL|0356.06003}} </td></tr><tr><td valign="top">[a6]</td> <td valign="top"> Yu.A. Drozd, "On tame and wild matrix problems" , ''Matrix Problems'' , Akad. Nauk. Ukr. SSR., Inst. Mat. Kiev (1977) pp. 104–114 (In Russian) {{MR|498704}} {{ZBL|}} </td></tr><tr><td valign="top">[a7]</td> <td valign="top"> Yu.A. Drozd, "Tame and wild matrix problems" , ''Representations and Quadratic Forms'' (1979) pp. 39–74 (In Russian) {{MR|0600111}} {{ZBL|0454.16014}} </td></tr><tr><td valign="top">[a8]</td> <td valign="top"> P. Gabriel, "Unzerlegbare Darstellungen 1" ''Manuscripta Math.'' , '''6''' (1972) pp. 71–103 (Also: Berichtigungen 6 (1972), 309) {{MR|332887}} {{ZBL|}} </td></tr><tr><td valign="top">[a9]</td> <td valign="top"> P. Gabriel, "Représentations indécomposables" , ''Séminaire Bourbaki (1973/74)'' , ''Lecture Notes in Mathematics'' , '''431''' , Springer (1975) pp. 143–169 {{MR|0485996}} {{ZBL|0335.17005}} </td></tr><tr><td valign="top">[a10]</td> <td valign="top"> P. Gabriel, A.V. Roiter, "Representations of finite dimensional algebras" , ''Algebra VIII'' , ''Encycl. Math. Stud.'' , '''73''' , Springer (1992) {{MR|1239446}} {{MR|1239447}} {{ZBL|0839.16001}} </td></tr><tr><td valign="top">[a11]</td> <td valign="top"> S. Kasjan, D. Simson, "Tame prinjective type and Tits form of two-peak posets II" ''J. Algebra'' , '''187''' (1997) pp. 71–96 {{MR|1425560}} {{ZBL|0944.16013}} </td></tr><tr><td valign="top">[a12]</td> <td valign="top"> L.A. Nazarova, "Representations of quivers of infinite type" ''Izv. Akad. Nauk. SSSR'' , '''37''' (1973) pp. 752–791 (In Russian) {{MR|0338018}} {{ZBL|0298.15012}} </td></tr><tr><td valign="top">[a13]</td> <td valign="top"> J.A. de la Peña, "Algebras with hypercritical Tits form" , ''Topics in Algebra'' , ''Banach Center Publ.'' , '''26: 1''' , PWN (1990) pp. 353–369 {{MR|}} {{ZBL|0731.16008}} </td></tr><tr><td valign="top">[a14]</td> <td valign="top"> J.A. de la Peña, "On the dimension of the module-varieties of tame and wild algebras" ''Commun. Algebra'' , '''19''' (1991) pp. 1795–1807 {{MR|}} {{ZBL|0818.16013}} </td></tr><tr><td valign="top">[a15]</td> <td valign="top"> J.A. de la Peña, D. Simson, "Prinjective modules, reflection functors, quadratic forms and Auslander–Reiten sequences" ''Trans. Amer. Math. Soc.'' , '''329''' (1992) pp. 733–753 {{MR|1025753}} {{ZBL|0789.16010}} </td></tr><tr><td valign="top">[a16]</td> <td valign="top"> J.A. de la Peña, A. Skowroński, "The Euler and Tits forms of a tame algebra" ''Math. Ann.'' , '''315''' (2000) pp. 37–59 {{MR|}} {{ZBL|0941.16010}} </td></tr><tr><td valign="top">[a17]</td> <td valign="top"> C.M. Ringel, "Tame algebras and integral quadratic forms" , ''Lecture Notes in Mathematics'' , '''1099''' , Springer (1984) {{MR|0774589}} {{ZBL|0546.16013}} </td></tr><tr><td valign="top">[a18]</td> <td valign="top"> A.V. Roiter, M.M. Kleiner, "Representations of differential graded categories" , ''Lecture Notes in Mathematics'' , '''488''' , Springer (1975) pp. 316–339 {{MR|0435145}} {{ZBL|0356.16011}} </td></tr><tr><td valign="top">[a19]</td> <td valign="top"> D. Simson, "Linear representations of partially ordered sets and vector space categories" , ''Algebra, Logic Appl.'' , '''4''' , Gordon &amp; Breach (1992) {{MR|1241646}} {{ZBL|0818.16009}} </td></tr><tr><td valign="top">[a20]</td> <td valign="top"> D. Simson, "Posets of finite prinjective type and a class of orders" ''J. Pure Appl. Algebra'' , '''90''' (1993) pp. 77–103 {{MR|1246276}} {{ZBL|0815.16006}} </td></tr><tr><td valign="top">[a21]</td> <td valign="top"> D. Simson, "Representation types, Tits reduced quadratic forms and orbit problems for lattices over orders" ''Contemp. Math.'' , '''229''' (1998) pp. 307–342 {{MR|1676228}} {{ZBL|0921.16007}} </td></tr><tr><td valign="top">[a22]</td> <td valign="top"> D. Simson, "Coalgebras, comodules, pseudocompact algebras and tame comodule type" ''Colloq. Math.'' , '''in press''' (2001) {{MR|1874368}} {{ZBL|1055.16038}} </td></tr>
 +
</table>

Latest revision as of 06:50, 15 February 2024

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 $| I _ { C } |$.

References

[a1] V.I. Auslander, I. Reiten, S. Smalø, "Representation theory of Artin algebras" , Studies Adv. Math. , 36 , Cambridge Univ. Press (1995) MR1314422 Zbl 0834.16001
[a2] I.N. Bernstein, I.M. Gelfand, V.A. Ponomarev, "Coxeter functors and Gabriel's theorem" Russian Math. Surveys , 28 (1973) pp. 17–32 Uspekhi Mat. Nauk. , 28 (1973) pp. 19–33 MR393065
[a3] K. Bongartz, "Algebras and quadratic forms" J. London Math. Soc. , 28 (1983) pp. 461–469 MR0724715 Zbl 0532.16020
[a4] V. Dlab, C.M. Ringel, "Indecomposable representations of graphs and algebras" , Memoirs , 173 , Amer. Math. Soc. (1976) MR0447344 Zbl 0332.16015
[a5] Yu.A. Drozd, "Coxeter transformations and representations of partially ordered sets" Funkts. Anal. Prilozhen. , 8 (1974) pp. 34–42 (In Russian) MR0351924 Zbl 0356.06003
[a6] Yu.A. Drozd, "On tame and wild matrix problems" , Matrix Problems , Akad. Nauk. Ukr. SSR., Inst. Mat. Kiev (1977) pp. 104–114 (In Russian) MR498704
[a7] Yu.A. Drozd, "Tame and wild matrix problems" , Representations and Quadratic Forms (1979) pp. 39–74 (In Russian) MR0600111 Zbl 0454.16014
[a8] P. Gabriel, "Unzerlegbare Darstellungen 1" Manuscripta Math. , 6 (1972) pp. 71–103 (Also: Berichtigungen 6 (1972), 309) MR332887
[a9] P. Gabriel, "Représentations indécomposables" , Séminaire Bourbaki (1973/74) , Lecture Notes in Mathematics , 431 , Springer (1975) pp. 143–169 MR0485996 Zbl 0335.17005
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How to Cite This Entry:
Tits quadratic form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tits_quadratic_form&oldid=21999
This article was adapted from an original article by Daniel Simson (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article