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Bivariant K-theory of generalized Weyl algebras
(European Mathematical Society Publishing House, 2020-01-01)
We compute the isomorphism class in \mathfrak{KK}^{\mathrm {alg}} of all noncommutative generalized Weyl algebras A=\mathbb C[h](\sigma, P) ,where \sigma(h)=qh+h_0 is an automorphism of \mathcal C[h] , except when q\neq 1 is a root of unity. In particular, we compute the isomorphism class in \mathfrak{KK}^{\mathrm {alg}} of the quantum Weyl algebra, the primitive factors B_{\lambda} of U(\mathfrak{sl}_2) and the quantum weighted projective lines \mathcal{O}(\mathbb{WP}_q(k, l)) .
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Bi-traceable graphs, the intersection of three longest paths and Hippchen's conjecture
(Cornell University, 2021-01-19)
Let $P,Q$ be longest paths in a simple graph. We analyze the possible connections between the components of $P\cup Q\setminus (V(P)\cap V(Q))$ and introduce the notion of a bi-traceable graph. We use the results for all the possible configurations of the intersection points when $\#V(P)\cap V(Q)\le 5$ in order to prove that if the intersection of three longest paths $P,Q,R$ is empty, then $\#(V(P)\cap V(Q))\ge 6$. We also prove Hippchen's conjecture for $k\le 6$: If a graph $G$ is $k$-connected for $k\le 6$, and $P$ and $Q$ are longest paths in $G$, then $\#(V(P)\cap V(Q))\ge 6$.
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Twisted tensor products of K3 with K3
(Bellwether Publishing, Ltd., 2021-01-01)
Jorge A. Guccione and Juan J. Guccione were supported by UBACyT 20020150100153BA (UBA) and PIP 11220110100800CO (CONICET). This work was supported by CONCYTEC-FONDECYT within the framework of the contest ?Proyectos de Investigaci?n B?sica 2020-01? [contract number 120-2020-FONDECYT].
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On the classification of graded twisted planes
(Springer Science+Business Media, 2022-05-04)
We use a representation of a graded twisted tensor product of K[x] with K[y] in L(Kℕ0) in order to obtain a nearly complete classification of these graded twisted tensor products via infinite matrices. There is one particular example and three main cases: quadratic algebras classified in Conner and Goetz (J. Noncommut. Geom. 15(1), 41–78, 2021), a family called A(n, d, a) with the n + 1-extension property for n ≥ 2, and a third case, not fully classified, which contains a family B(a, L) parameterized by quasi-balanced sequences.
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On two conjectures about the intersection of longest paths and cycles
(Elsevier B.V., 2024-11-01)
A conjecture attributed to Smith states that every two longest cycles in a k-connected graph intersect in at least k vertices. In this paper, we show that every two longest cycles in a k-connected graph on n vertices intersect in at least min⁡{n,8k−n−16} vertices, which confirms Smith's conjecture when k≥(n+16)/7. An analog conjecture for paths instead of cycles was stated by Hippchen. By a simple reduction, we relate both conjectures, showing that Hippchen's conjecture is valid when either k≤7 or k≥(n+9)/7.
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The Grobner basis and solution set of a polynomial system related to the Jacobian conjecture
(Pontificia Universidad Católica del Perú, 2025-11-05)
We compute the Groebner basis of a system of polynomial equations related to the Jacobian conjecture, and describe completely the solution set.
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Solutions of the braid equation attached to power series
(Taylor and Francis Ltd., 2024-01-01)
Let K be an algebraically closed characteristic 0 field and let V be the K-vector space with basis (Formula presented.). In this paper we use the theory developed in [17] to associate with each power series (Formula presented.) satisfying (Formula presented.), an involutive solution (Formula presented.), of the braid equation, and we begin the study of this correspondence. For (Formula presented.), let Vn be the subspace of V generated by (Formula presented.). Each one of the solution s(f) induces by restriction a solution (Formula presented.) on (Formula presented.). We also begin the study of these solutions.
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Set-theoretic type solutions of the braid equation
(Academic Press Inc., 2024-04-15)
In this paper we begin the study of set-theoretic type solution of the braid equation. Our theory includes set-theoretical solutions as basic examples. More precisely, the linear solution associated to a set-theoretic solution on a set X can be regarded as coming from the coalgebra kX, where k is a field and the elements of X are grouplike. We introduce and study a broader class of linear solutions associated in a similar way to more general coalgebras. We show that the relationships between set-theoretical solutions, q-cycle sets, q-braces, skew-braces, matched pairs of groups and invertible 1-cocycles remain valid in our setting.
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Extensions of Linear Cycle Sets and Cohomology
(Springer Science and Business Media Deutschland GmbH, 2023-03-01)
We generalize the cohomology theory for linear cycle sets introduced by Lebed and Vendramin. Our cohomology classifies extensions of linear cycle sets by trivial ideals, whereas the cohomology of Lebed and Vendramin only deals with central ideals I (which are automatically trivial). Therefore our theory gives an analog to the theory of extensions of braces by trivial ideals constructed by Bachiller, but from a cohomological point of view. We also study the general notions of extensions of linear cycle sets and the equivalence of extensions.
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Hopf Q -braces structures on rank one pointed Hopf algebras
(World Scientific, 2025-01-01)
In this paper we determine all the Hopf [Formula: see text]-brace structures on rank one pointed Hopf algebras and compute the socle of each one of them. We also identify which among them are Hopf skew-braces. Then we determine when two Hopf [Formula: see text]-brace structures on rank one pointed Hopf algebras are isomorphic, and, finally, we compute all the weak braiding operators on these Hopf algebras.
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