We study the geometry of $𝒟$-bundles—locally projective $𝒟$-modules—on algebraic curves, and apply them to the study of integrable hierarchies, specifically the multicomponent Kadomtsev–Petviashvili (KP) and spin Calogero–Moser (CM) hierarchies. We show that KP hierarchies have a geometric description as flows on moduli spaces of $𝒟$-bundles; in particular, we prove that the local structure of $𝒟$-bundles is captured by the full Sato Grassmannian. The rational, trigonometric, and elliptic solutions of KP...

In 1998 Victor Kac classified infinite-dimensional $\mathbb{Z}$-graded Lie superalgebras of finite depth. We construct new examples of infinite-dimensional Lie superalgebras with a $\mathbb{Z}$-gradation of infinite depth and finite growth and classify $\mathbb{Z}$-graded Lie superalgebras of infinite depth and finite growth under suitable hypotheses.

Si studiano gli omomorfismi reticolari ($ℒ$-omomorfismi) di algebre di Lie sopra anelli commutativi con unità. Le algebre di Lie sopra un campo e le $p$-algebre di Lie non ammettono $ℒ$-omomorfismi propri. Si assegnano condizioni necessarie e sufficienti affinchè un'algebra di Lie periodica o mista possieda un «$ℒ$-omomorfismo su una catena di lunghezza $n$.

The relative cohomology ${\mathrm{H}}_{\mathrm{diff}}^1(𝕂\left(1\right|3),\mathrm{𝔬𝔰𝔭}(2,3);{𝒟}_{\lambda ,\mu }\left(S^{1|3}\right))$ of the contact Lie superalgebra $𝕂\left(1\right|3)$ with coefficients in the space of differential operators ${𝒟}_{\lambda ,\mu }\left(S^{1|3}\right)$ acting on tensor densities on $S^{1|3}$, is calculated in N. Ben Fraj, I. Laraied, S. Omri (2013) and the generating $1$-cocycles are expressed in terms of the infinitesimal super-Schwarzian derivative $1$-cocycle $s\left(X_f\right)=D_1{D}_2{D}_3\left(f\right){\alpha }_3^{1/2}$, $X_f\in 𝕂\left(1\right|3)$ which is invariant with respect to the conformal subsuperalgebra $\mathrm{𝔬𝔰𝔭}(2,3)$ of $𝕂\left(1\right|3)$. In this work we study the supergroup case. We give an explicit construction of $1$-cocycles of the group...

We create a framework for odd Khovanov homology in the spirit of Bar-Natan's construction for the ordinary Khovanov homology. Namely, we express the cube of resolutions of a link diagram as a diagram in a certain 2-category of chronological cobordisms and show that it is 2-commutative: the composition of 2-morphisms along any 3-dimensional subcube is trivial. This allows us to create a chain complex whose homotopy type modulo certain relations is a link invariant. Both the original and the odd Khovanov...

We describe two constructions of a certain ${\mathbb{Z}}_4^3$-grading on the so-called Brown algebra (a simple structurable algebra of dimension $56$ and skew-dimension $1$) over an algebraically closed field of characteristic different from $2$. The Weyl group of this grading is computed. We also show how this grading gives rise to several interesting fine gradings on exceptional simple Lie algebras of types $E_6$, $E_7$ and $E_8$.

We show that a Poisson Lie group (G,π) is coboundary if and only if the natural action of G×G on M=G is a Poisson action for an appropriate Poisson structure on M (the structure turns out to be the well known ${\pi }_{+}$). We analyze the same condition in the context of Hopf algebras. A quantum analogue of the ${\pi }_{+}$ structure on SU(N) is described in terms of generators and relations as an example.

We construct a special class of fermionic Novikov superalgebras from linear functions. We show that they are Novikov superalgebras. Then we give a complete classification of them, among which there are some non-associative examples. This method leads to several new examples which have not been described in the literature.