A Hopf algebra quantizing a necklace Lie algebra canonically associated to a quiver

By Travis Schedler. 2004

64 citations recorded by the source corpus.

V. Ginzburg, and independently R. Bocklandt and L. Le Bruyn, defined an infinite-dimensional "necklace" Lie algebra canonically associated to any quiver. Following suggestions of V. Turaev, P. Etingof, and Ginzburg, we define a cobracket and prove that it defines a Lie bialgebra structure. We then present a Hopf algebra quantizing this Lie bialgebra, and prove that it is a Hopf algebra satisfying the PBW property. We present representations into spaces of differential operators on representations of the quiver, which quantize the trace representations of the Lie algebra given by Ginzburg.

Subfields: Algebra and Number Theory, Mathematical Physics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/bba826e1fa58c39bbe06e3af720404c951bbb051

Citations in the corpus

• Calabi-Yau Frobenius algebras (2007, unknown): “an singularity” algebra HH0(Π˜ Q,ΠQ˜) at degrees ≤h−2, and composing with the kernel map ker(L։Y) →L. The asserted relation to the Kleinian singularity C2/Γ associated to Qis that Z˜ Q ∼=e i0ΠQ˜ei0 ([Sch05], Theorem 10.1.1), and the latter is just isomorphic to C[x,y]Γ after tensoring by C(or, by an”
• Calabi-Yau algebras (2006, unknown): “,Φ) cyc. This way, the super-commutator quotient D(F,Φ) cyc acquires a natural structure of DG Lie super-algebra, with Lie bracket {−,−}and differential d. Further, the construction due to T. Schedler [Sc] gives an odd Lie cobracket ν: D(F,Φ) cyc →Sym2 2 D(F,Φ) cyc 2 , f→pr Xn i=1 ∂ f ∂xi∂θi − ∂ f ∂θ”