arXiv subjects

Lukas Tappeiner

Publications and source records attributed to Lukas Tappeiner.

3 recordsLinked to original sources

arXiv2025

GKLO representations of twisted Yangians in type $\mathsf{AI}$ and quantizations of symmetric quotients of the affine Grassmannian

We construct an analogue of Gerasimov-Kharchev-Lebedev-Oblezin (GKLO) representations for twisted Yangians of type $\mathsf{AI}$, using the recently found current presentation of these algebras due to Lu, Wang and Zhang. These new representations allow us to define interesting truncations of twisted Yangians, which, in the spirit of Ciccoli-Drinfeld-Gavarini quantum duality, reflect the Poisson geometry of homogeneous spaces. As our main result, we prove that a truncated twisted Yangian quantizes a scheme supported on quotients of transverse slices in the affine Grassmannian.

math.RT↗

arXiv2025

Shifted twisted Yangians and finite $W$-algebras of classical type

We introduce parabolic presentations of twisted Yangians of types AI and AII, interpolating between the R-matrix presentation and the Drinfeld presentation. Then we formulate and provide parabolic presentations for the shifted twisted Yangians. We define quotient algebras known as truncated shifted twisted Yangians and equip them with baby comultiplications, generalizing the work of Brundan and Kleshchev. PBW bases for all (truncated) shifted twisted Yangians of type AI and AII are established along the way. Applying the theory of universal equivariant quantizations of conic symplectic singularities we show that the truncated twisted shifted Yangian is isomorphic to the finite $W$-algebra which quantizes a suitable Slodowy slice. This provides a presentation of the finite $W$-algebra associated with every even nilpotent element in type {\sf B} and {\sf C}, as well as every nilpotent element with two Jordan blocks in type {\sf D}. Finally we make a conjecture which would supply presentations in the remaining even cases in type {\sf D}.

math.QA↗

arXiv2025

Shifted twisted Yangians and Slodowy slices in classical Lie algebras

In this paper we introduce the shifted twisted Yangian of type {\sf AI}, following the work of Lu--Wang--Zhang, and we study their semiclassical limits, a class of Poisson algebras. We demonstrate that they coincide with the Dirac reductions of the semiclassical shifted Yangian for $\mathfrak{gl}_n$. We deduce that these shifted twisted Yangians admit truncations which are isomorphic to Slodowy slices for many non-rectangular nilpotent elements in types {\sf B}, {\sf C}, {\sf D}. As a direct consequence we obtain parabolic presentations of the semiclassical shifted twisted Yangian, analogous to those introduced by Brundan--Kleshchev for the Yangian of type {\sf A}. Finally we give Poisson presentations of Slodowy slices for all even nilpotent elements in types {\sf B}, {\sf C}, {\sf D}, generalising the recent work of the second author.

math.RT↗