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Vector bundles on plane cubic curves and the classical Yang–Baxter equation

Igor Burban, Thilo Henrich (2015)

Journal of the European Mathematical Society

In this article, we develop a geometric method to construct solutions of the classical Yang–Baxter equation, attaching a family of classical r -matrices to the Weierstrass family of plane cubic curves and a pair of coprime positive integers. It turns out that all elliptic r -matrices arise in this way from smooth cubic curves. For the cuspidal cubic curve, we prove that the obtained solutions are rational and compute them explicitly. We also describe them in terms of Stolin’s classication and prove...

Vector form brackets in Lie algebroids

Albert Nijenhuis (1996)

Archivum Mathematicum

A brief exposition of Lie algebroids, followed by a discussion of vector forms and their brackets in this context - and a formula for these brackets in “deformed” Lie algebroids.

Veränderungen über einen Satz von Timmesfeld – I. Quadratic Actions

Adrien Deloro (2013)

Confluentes Mathematici

We classify quadratic SL 2 ( 𝕂 ) - and 𝔰𝔩 2 ( 𝕂 ) -modules by crude computation, generalising in the first case a Theorem proved independently by F.G. Timmesfeld and S. Smith. The paper is the first of a series dealing with linearisation results for abstract modules of algebraic groups and associated Lie rings.

Vertex algebras and the formal loop space

Mikhail Kapranov, Eric Vasserot (2004)

Publications Mathématiques de l'IHÉS

We construct a certain algebro-geometric version ℒ ( X ) of the free loop space for a complex algebraic variety X. This is an ind-scheme containing the scheme ℒ 0 ( X ) of formal arcs in X as studied by Kontsevich and Denef-Loeser. We describe the chiral de Rham complex of Malikov, Schechtman and Vaintrob in terms of the space of formal distributions on ℒ ( X ) supported in ℒ 0 ( X ) . We also show that ℒ ( X ) possesses a factorization structure: a certain non-linear version of a vertex algebra structure. This explains the heuristic...

Ward identities from recursion formulas for correlation functions in conformal field theory

Alexander Zuevsky (2015)

Archivum Mathematicum

A conformal block formulation for the Zhu recursion procedure in conformal field theory which allows to find n -point functions in terms of the lower correlations functions is introduced. Then the Zhu reduction operators acting on a tensor product of VOA modules are defined. By means of these operators we show that the Zhu reduction procedure generates explicit forms of Ward identities for conformal blocks of vertex operator algebras. Explicit examples of Ward identities for the Heisenberg and free...

Weak c*-Hopf algebras: the coassociative symmetry of non-integral dimensions

Gabriella Böhm, Kornél Szlachányi (1997)

Banach Center Publications

By allowing the coproduct to be non-unital and weakening the counit and antipode axioms of a C*-Hopf algebra too, we obtain a selfdual set of axioms describing a coassociative quantum group, that we call a weak C*-Hopf algebra, which is sufficiently general to describe the symmetries of essentially arbitrary fusion rules. It is the same structure that can be obtained by replacing the multiplicative unitary of Baaj and Skandalis with a partial isometry. The algebraic properties, the existence of...

Weak multiplier Hopf algebras. Preliminaries, motivation and basic examples

Alfons Van Daele, Shuanhong Wang (2012)

Banach Center Publications

Let G be a finite group. Consider the algebra A of all complex functions on G (with pointwise product). Define a coproduct Δ on A by Δ(f)(p,q) = f(pq) where f ∈ A and p,q ∈ G. Then (A,Δ) is a Hopf algebra. If G is only a groupoid, so that the product of two elements is not always defined, one still can consider A and define Δ(f)(p,q) as above when pq is defined. If we let Δ(f)(p,q) = 0 otherwise, we still get a coproduct on A, but Δ(1) will no longer be the identity in A ⊗ A. The pair (A,Δ)...

Weak polynomial identities and their applications

Vesselin Drensky (2021)

Communications in Mathematics

Let R be an associative algebra over a field K generated by a vector subspace V . The polynomial f ( x 1 , ... , x n ) of the free associative algebra K 〈 x 1 , x 2 , ... 〉 is a weak polynomial identity for the pair ( R , V ) if it vanishes in R when evaluated on V . We survey results on weak polynomial identities and on their applications to polynomial identities and central polynomials of associative and close to them nonassociative algebras and on the finite basis problem. We also present results on weak polynomial identities of degree three....

Weak Polynomial Identities for M1,1(E)

Di Vincenzo, Onofrio, La Scala, Roberto (2001)

Serdica Mathematical Journal

* Partially supported by Universita` di Bari: progetto “Strutture algebriche, geometriche e descrizione degli invarianti ad esse associate”.We compute the cocharacter sequence and generators of the ideal of the weak polynomial identities of the superalgebra M1,1 (E).

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