This applet implements Derksen–Weyman–Zelevinsky's mutation of quivers with potentials (arXiv:0704.0649). This website is inspired by Keller's quiver mutation applet.
The code was generated with the assistance of Claude, based on instructions and materials provided by the author.
How does it work?
The part worth explaining is how the reduction (to go from the premutation to the mutation) of the quiver with potential is done.
Background.
Let \(Q = (Q_0,Q_1,h,t)\) be a quiver and let \(K\) be a field. Put \(R = K^{Q_0}\) and \(A = K^{Q_1}\). The complete path algebra of \(Q\) is \[R\langle\langle A\rangle\rangle = \prod_{d=0}^\infty A^d,\] where \(A^d = A^{\otimes_R d}\) is the \(d\)-fold tensor product, and hence corresponds to paths of length \(d\). Note that we compose paths from left to right. Thus, going through \(a\) and then \(b\) is read as \(ab = a\otimes b\). The arrow ideal \(\mathfrak{m}\) is the (two-sided) ideal of \(R\langle\langle A\rangle\rangle\) with powers \[\mathfrak{m}^n = \prod_{d=n}^\infty A^d.\]
Let \((A,S)\) be a quiver with potential.
Write \(S^{(n)}\) for the degree \(n\) part of \(S\) consisting of all cycles of length \(n\).
If \(S^{(2)} = 0\), then \((A,S)\) is already reduced. Assume \(S^{(2)} \neq 0\) and write \(S^{(2)} = \sum_{i=1}^M\lambda_ia_ib_i\), where \(0\neq\lambda_i\in K\) and \(a_ib_i\) is a 2-cycle. Write \(A_{x,y}\) for the subspace of \(A\) spanned by the arrows from \(x\) to \(y\). We fix a basis of this vector space.
Step 1. (Change of bases to get a potential like Equation (4.6) of [DWZ]). Consider two vertices \(x\neq y\) for which there is \(i\) such that \(t(a_i),h(a_i)\in \{x,y\}\). Put the coefficients \(\lambda_j\) in a matrix \(\Lambda\) which corresponds to the bilinear pairing \(\Lambda : A_{x,y} \times A_{y,x} \to K\). Note that the \(\lambda_j\)'s used are those attached to 2-cycles which pass through \(x\) and \(y\). Do a change of basis on \(A_{x,y}\) and on \(A_{y,x}\) so that \(\Lambda\), in these new bases, has at most one nonzero entry in each row and column, and every such entry is equal to \(1\). By fixing all other arrows, these changes of bases yield a right-equivalence of quivers with potentials. Write \(S'\) for the new potential. It is of the form \[S' = \sum_{i=1}^m (a_i'b_i' + a_i'u_i + v_ib_i') + S'',\] where each of the \(2m\) arrows \(a_1',b_1',\ldots,a_m',b_m'\) are distinct, each \(u_i\) and \(v_i\) is a linear combination of paths of length at least \(2\), and \(S''\) consists only of cycles of length at least \(3\), where none of them contains an \(a_i'\) or a \(b_i'\). This is Equation (4.6) of [DWZ]. If \(u_i = 0 = v_i\) for all \(i=1,\ldots,m\), then we have the desired decomposition of the Splitting Theorem (Theorem 4.6 in [DWZ]). Otherwise, we go to the next step.
Step 2. (Reduction like in Lemmas 4.7 and 4.8 of [DWZ]). We do like in the proof of Lemmas 4.7 and 4.8 of [DWZ], but stop either when \(u_i = 0 = v_i\) for all \(i=1,\ldots,m\), or when the length of \(u_i\) and \(v_i\) reach the maximal length of a potential term (currently set to 8). More precisely, given a potential (which we may write like \(S'\) above), we apply the automorphism \(\varphi : R\langle\langle A \rangle\rangle \to R\langle\langle A \rangle\rangle\) defined by \[\varphi(a_i') = a_i' - v_i, \quad \varphi(b_i') = b_i' - u_i,\; \text{ and }\; \varphi(c) = c,\] for all other arrows \(c\) of \(Q\). Note that there is not a unique way of picking each \(u_i\) and \(v_i\). Suppose we write the potential as \(S' = a_1' R_1 + R_2\), where \(a_1'\) doesn't appear in \(R_2\). Then, the algorithm takes \(u_1 = R_1 - b_1'\). In a similar fashion, we may write \(R_2 = b_1' R_3 + R_4\). Then, the algorithm picks \(v_1 = R_3\). We continue with \(a_2',b_2',\ldots,a_m',b_m'\) applied to \(R_4,\ldots\) This gives a way of defining \(\varphi\). Now, assume that \(u_i,v_i \in \mathfrak{m}^n\) for all \(i=1,\ldots,m\) (where \(n\) is between \(2\) and the maximal length of a potential term). Then, \[\varphi(S') = \sum_{i=1}^m (a_i'b_i' + a_i'u_i' + v_i'b_i') + S'''\] is such that \(u_i',v_i' \in \mathfrak{m}^{2n}\) for all \(i=1,\ldots,m\), which implies that the algorithm terminates.
Remark. Like mentioned in Remark 5 of [LF] and studied further in Section 2 of [L], it is quite often the case that the limiting process of [DWZ] is not needed and so the above algorithm yields the correct potential, with no approximations involved. However, when such a truncation of the potential is needed, the applet lets the user know.
References.
[DWZ] Harm Derksen, Jerzy Weyman, and Andrei Zelevinski. Quivers with potentials and their representations I: Mutations. Selecta Mathematica (2008).
[LF] Daniel Labardini-Fragoso. Quivers with potentials associated to triangulated surfaces. Proceedings of the London Mathematical Society (2009).
[L] Sefi Ladkani. Non-degenerate potentials on the quiver \(X_7\). Journal of Algebra (2025).