Related papers: The PBW theorem for the affine Yangians
We construct algebra homomorphisms from affine Yangians to the current algebras of rectangular $W$-algebras both in type A. The construction is given via the coproduct and the evaluation map for the affine Yangians. As a consequence, we…
We determine the PBW deformations of the wreath product of a symmetric group with a deformed preprojective algebra of an affine Dynkin quiver. In particular, we show that there is precisely one parameter which does not come from deformation…
We consider the cohomological Hall algebra Y of a Lagrangian substack of the moduli stack of representations of the preprojective algebra of an arbitrary quiver Q, and its actions on the cohomology of quiver varieties. We conjecture that Y…
We construct twisted noncommutative gauge theories on twistor space and show that they are equivalent to four-dimensional twist-noncommutative gauge theories. In particular, we study twists of the Poincar\'e algebra. We explain how such a…
Take the matrix Lie superalgebra $gl_{N|N}$ with the standard generators $E_{ij}$ where $i,j=-N,...,-1,1,...,N$. Define an involutive automorphism of $gl_{N|N}$ by sending $E_{ij}$ to $E_{-i,-j}$. Then the corresponding twisted subalgebra…
A well-known and old result of Hazewinkel and Koszul states that the cohomology of a finite-dimensional Lie algebra is isomorphic, up to a suitable shift, to its twisted homology, a Lie-theoretical version of Poincare duality. This paper…
Inspired by the commutator and anticommutator algebras derived from algebras graded by groups, we introduce noncommutatively graded algebras. We generalize various classical graded results to the noncommutatively graded situation concerning…
Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra with associated Yangian $Y_\hbar\mathfrak{g}$ and Yangian double $\mathrm{D}Y_\hbar\mathfrak{g}$. An elementary result of fundamental importance to the theory of Yangians is that, for…
This short note is an announcement of results. We continue the study of Yangian-type algebras initiated in the paper arXiv:2208.04809. These algebras share a number of properties of the Yangians of type A but are more massive. We refine and…
Let ${\mathbf U}^-_q$ be the negative part of the quantum enveloping algebra associated to a simply laced Kac-Moody Lie algebra ${\mathfrak g}$, and $\underline{\mathbf U}^-_q$ the algebra corresponding to the fixed point subalgebra of…
Many rings and algebras arising in quantum mechanics can be interpreted as skew PBW (Poincar\'e-Birkhoff-Witt) extensions. Indeed, Weyl algebras, enveloping algebras of finite-dimensional Lie algebras (and its quantization), Artamonov…
We review the main ideas underlying the emerging theory of Yangians -- the new type of hidden symmetry in string-inspired models. Their classification by quivers is a far-going generalization of simple Lie algebras classification by Dynkin…
A deformation $U$, of a graded $K$-algebra $A$ is said to be of PBW type if $grU$ is $A$. It has been shown for Koszul and $N$-Koszul algebras that the deformation is PBW if and only if the relations of $U$ satisfy a Jacobi type condition.…
We construct a model of the affine nil-Hecke algebra as a subalgebra of the Nichols-Woronowicz algebra associated to a Yetter-Drinfeld module over the affine Weyl group. We also discuss the Peterson isomorphism between the homology of the…
We investigate the differential smoothness of a certain family of skew Poincar\'e-Birkhoff-Witt extensions.
The Brylinski-Kostant filtration on a representation of a finite-dimensional semisimple Lie algebra has interpretations in terms of the algebra, geometry and combinatorics of the representation. Its extension to affine Lie algebras was…
We define a family of homomorphisms on a collection of convolution algebras associated with quiver varieties, which gives a kind of coproduct on the Yangian associated with a symmetric Kac-Moody Lie algebra. We study its property using…
We develop versions of the Poincar\'e-Birkhoff-Witt and Cartier-Milnor-Moore theorems in the setting of braided Hopf algebras. To do so, we introduce new analogues of a Lie algebra in the setting of a braided monoidal category, using the…
We construct an universal enveloping algebra associated to the ternary extension of Lie (super)algebras called Lie algebra of order three. A Poincar\'e-Birkhoff-Witt theorem is proven is this context. It this then shown that this universal…
In this paper, we describe recent work towards the mirror P=W conjecture, which relates the weight filtration on a cohomology of a log Calabi--Yau manifold to the perverse Leray filtration on the cohomology of the homological mirror dual…