<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsUnitForm</code></td><td class="tdright">( category )</td></tr></table></div>
<p>The category for unit forms, which we identify with symmetric integral matrices with 2 along the diagonal.</p>
<p>Returns: the bilinear form associated to a unit form <var class="Arg">B</var>.</p>
<p>The bilinear form associated to the unitform <var class="Arg">B</var> given by a matrix <code class="code">B</code> is defined for two vectors <code class="code">x</code> and <code class="code">y</code> as: <span class="SimpleMath">\(x*B*y^T\)</span>.</p>
<p>Returns: true is the unitform <var class="Arg">B</var> is weakly non-negative, otherwise false.</p>
<p>The unit form <var class="Arg">B</var> is weakly non-negative is <span class="SimpleMath">\(B(x,y) \geq 0\)</span> for all <span class="SimpleMath">\(x\neq 0\)</span> in <span class="SimpleMath">\(\mathbb{Z}^n\)</span>, where <span class="SimpleMath">\(n\)</span> is the dimension of the square matrix associated to <var class="Arg">B</var>.</p>
<p>Returns: true is the unitform <var class="Arg">B</var> is weakly positive, otherwise false.</p>
<p>The unit form <var class="Arg">B</var> is weakly positive if <span class="SimpleMath">\(B(x,y) > 0\)</span> for all <span class="SimpleMath">\(x\neq 0\)</span> in <span class="SimpleMath">\(\mathbb{Z}^n\)</span>, where <span class="SimpleMath">\(n\)</span> is the dimension of the square matrix associated to <var class="Arg">B</var>.</p>
<p>Returns: the positive roots of a unit form, if the unit form is weakly positive. If they have not been computed, an error message will be returned saying "no method found!".</p>
<p>This attribute will be attached to <var class="Arg">B</var> when <code class="code">IsWeaklyPositiveUnitForm</code> is applied to <var class="Arg">B</var> and it is weakly positive.</p>
<p>Returns: the quadratic form associated to a unit form <var class="Arg">B</var>.</p>
<p>The quadratic form associated to the unitform <var class="Arg">B</var> given by a matrix <code class="code">B</code> is defined for a vector <code class="code">x</code> as: <span class="SimpleMath">\(\frac{1}{2}x*B*x^T\)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TitsUnitFormOfAlgebra</code>( <var class="Arg">A</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">A</var> -- a finite dimensional (quotient of a) path algebra (by an admissible ideal).<br /></p>
<p>Returns: the Tits unit form associated to the algebra <var class="Arg">A</var>.</p>
<p>This function returns the Tits unitform associated to a finite dimensional quotient of a path algebra by an admissible ideal or path algebra, given that the underlying quiver has no loops or minimal relations that starts and ends in the same vertex. That is, then it returns a symmetric matrix <span class="SimpleMath">\(B\)</span> such that for <span class="SimpleMath">\(x = (x_1,...,x_n) (1/2)*(x_1,...,x_n)B(x_1,...,x_n)^T = \sum_{i=1}^n x_i^2 - \sum_{i,j} \dim_k \Ext^1(S_i,S_j)x_ix_j + \sum_{i,j} \dim_k \Ext^2(S_i,S_j)x_ix_j\)</span>, where <span class="SimpleMath">\(n\)</span> is the number of vertices in <span class="SimpleMath">\(Q\)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EulerBilinearFormOfAlgebra</code>( <var class="Arg">A</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">A</var> -- a finite dimensional (quotient of a) path algebra (by an admissible ideal).<br /></p>
<p>Returns: the Euler (non-symmetric) bilinear form associated to the algebra <var class="Arg">A</var>.</p>
<p>This function returns the Euler (non-symmetric) bilinear form associated to a finite dimensional (basic) quotient of a path algebra <var class="Arg">A</var>. That is, it returns a bilinear form (function) defined by<br /> <span class="SimpleMath">\(f(x,y) = x*\textrm{CartanMatrix}(A)^{(-1)}*y\)</span><br /> It makes sense only in case <var class="Arg">A</var> is of finite global dimension.</p>
<p>Returns: the unit form in the category <code class="func">IsUnitForm</code> (<a href="chap12_mj.html#X79E906FD7B3DF010"><span class="RefLink">12.2-1</span></a>) associated to the matrix <var class="Arg">B</var>.</p>
<p>The function checks if <var class="Arg">B</var> is a symmetric integral matrix with 2 along the diagonal, and returns an error message otherwise. In addition it sets the attributes, <code class="func">BilinearFormOfUnitForm</code> (<a href="chap12_mj.html#X81A3BD6C8027B193"><span class="RefLink">12.2-2</span></a>), <code class="func">QuadraticFormOfUnitForm</code> (<a href="chap12_mj.html#X799A82E58690F9C5"><span class="RefLink">12.2-6</span></a>) and <code class="func">SymmetricMatrixOfUnitForm</code> (<a href="chap12_mj.html#X7837555780590360"><span class="RefLink">12.2-7</span></a>).</p>
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