Twisting of properads

An operator I on a real Lie algebra is called a complex structure operator if and the -eigenspace is a Lie subalgebra in the complexification of . A hypercomplex structure on a Lie algebra is a triple of complex structures and K on satisfying the quaternionic relations. We call a hypercomplex nilpotent Lie algebra -solvable if there exists a sequence of -invariant subalgebrassuch that . We give examples of -solvable hypercomplex structures on a nilpotent Lie algebra and ...

Added: December 3, 2023

The orbit method for locally nilpotent infinite-dimensional Lie algebras

Ignatyev Mikhail, Petukhov A., Journal of Algebra 2021 Vol. 585 P. 501–557

Let $\mathfrak{n}$ be a locally nilpotent infinite-dimensional Lie algebra over $\mathbb{C}$. Let $\mathrm{U}(\mathfrak{n})$ and $\mathrm{S}(\mathfrak{n})$ be its respective universal enveloping algebra and symmetric algebra. Consider the Jacobson topology on the primitive spectrum of $\mathrm{U}(\mathfrak{n})$, and the Poisson topology on the primitive Poisson spectrum of $\mathrm{S}(\mathfrak{n})$. We provide a homeomorphism between the corresponding topological spaces (at the ...

Added: October 8, 2023

Transposed Poisson structure on Galilean and solvable Lie algebras

Lopatkin V., Kaygorodov I., Zhang Z., Journal of Geometry and Physics 2023 No. 187 P. 1–20

Transposed Poisson structures on complex Galilean type Lie algebras and superalgebras are described. It is proven that all principal Galilean Lie algebras do not have non-trivial 12-derivations and as it follows they do not admit non-trivial transposed Poisson structures. Also, we proved that each complex finite-dimensional solvable Lie algebra admits a non-trivial transposed Poisson structure and a non-trivial Hom-Lie structure. ...

Added: May 5, 2023

On homology of Lie algebras over commutative rings

Pavutnitskiy F., Ivanov S., Zaikovskii A. et al., Journal of Algebra 2021 Vol. 586 P. 99–139

We study five different types of the homology of a Lie algebra over a commutative ring which are naturally isomorphic over fields. We show that they are not isomorphic over commutative rings, even over , and study connections between them. In particular, we show that they are naturally isomorphic in the case of a Lie ...

Added: October 7, 2021

PBW Property for Associative Universal Enveloping Algebras Over an Operad

Khoroshkin A., International Mathematics Research Notices 2022 Vol. 2022 No. 4 P. 3106–3143

Given a symmetric operad $\mathcal{P}$ and a $\mathcal{P}$-algebra $V$, the associative universal enveloping algebra ${\mathsf{U}_{\mathcal{P}}}$ is an associative algebra whose category of modules is isomorphic to the abelian category of $V$-modules. We study the notion of PBW property for universal enveloping algebras over an operad. In case $\mathcal{P}$ is Koszul a criterion ...

Added: February 13, 2021

Extensions of vertex algebras. Constructions and applications

Feigin B. L., Russian Mathematical Surveys 2017 Vol. 72 No. 4 P. 707–763

This paper discusses the main known constructions of vertex operator algebras. The starting point is the lattice algebra. Screenings distinguish subalgebras of lattice algebras. Moreover, one can construct extensions of vertex algebras. Combining these constructions gives most of the known examples. A large class of algebras with big centres is constructed. Such algebras have applications ...

Added: November 5, 2020

Quantum-classical duality for Gaudin magnets with boundary

Васильев М., Zabrodin A., Zotov A., Nuclear Physics B - Proceedings Supplements 2020 Vol. 952 No. 114931 P. 1–20

We establish a remarkable relationship between the quantum Gaudin models with boundary and the classical many-body integrable systems of Calogero-Moser type associated with the root systems of classical Lie algebras (B, C and D). We show that under identification of spectra of the Gaudin Hamiltonians HjG with particles velocities q˙j of the classical model all ...

Added: August 20, 2020

Representation theoretic realization of non-symmetric Macdonald polynomials at infinity

Feigin E., Kato S., Makedonskyi I., Journal fuer die reine und angewandte Mathematik 2020 Vol. 764 P. 181–216

We study the non-symmetric Macdonald polynomials specialized at infinity from various points of view. First, we define a family of modules of the Iwahori algebra whose characters are equal to the non-symmetric Macdonald polynomials specialized at infinity. Second, we show that these modules are isomorphic to the dual spaces of sections of certain sheaves on ...

Added: August 12, 2020

Large tensor products and Littlewood-Richardson coefficients

Feigin E., Journal of Lie Theory 2019 Vol. 29 No. 4 P. 927–940

The Littlewood-Richardson coefficients describe the decomposition of tensor products of irreducible representations of a simple Lie algebra into irreducibles. Assuming the number of factors is large, one gets a measure on the space of weights. This limiting measure was extensively studied by many authors. In particular, Kerov computed the corresponding density in a special case in ...

Added: December 9, 2019

On wild Lie algebras

Makedonskyi I., / Series arXiv "math". 2012.

We give a criterion of tameness and wildness for a finite-dimensional Lie algebra over an algebraically closed field. ...

Added: December 3, 2018

On finite dimensional Lie algebras of planar vector fields with rational coefficients

Makedonskyi I., Petravchuk A., / Series arXiv "math". 2013.

The Lie algebra of planar vector fields with coefficients from the field of rational functions over an algebraically closed field of characteristic zero is considered. We find all finite-dimensional Lie algebras that can be realized as subalgebras of this algebra. ...

Added: December 3, 2018

On a higher structure on operadic deformation complexes

Shoikhet B., Theory and Applications of Categories 2018 Vol. 33 No. 32 P. 988–1030

In this paper, we prove that there is a canonical homotopy (n+1)-algebra structure on the shifted operadic deformation complex Def(e_n → P)[−n] for any operadP and a map of operads f : e_n → P. This result generalizes a result of Tamarkin, who considered the case P = End_{O_p}(X). Another more computational proof of the same result was recently sketched by Calaque and ...

Added: November 11, 2018

Clifford algebras and their applications to Lie groups and spinors

Shirokov D., , in: Proceedings of the Nineteenth International Conference on Geometry, Integrability and QuantizationVol. 19.: Sofia: Avangard Prima, 2018. Ch. 1 P. 11–53.

We discuss some well-known facts about Clifford algebras: matrix representations, Cartan’s periodicity of 8, double coverings of orthogonal groups by spin groups, Dirac equation in different formalisms, spinors in <span data-mathml="nn dimensions, etc. We also present our point of view on some problems. Namely, we discuss the generalization of the Pauli theorem, the basic ideas of the ...

Added: January 31, 2018

On dimension growth of modular irreducible representations of semisimple Lie algebras

Bezrukavnikov R., Ivan Losev, / Series arXiv "math". 2017. No. 1708.01385.

Added: October 9, 2017

On Some Lie Groups Containing Spin Group in Clifford Algebra

Shirokov D., Journal of Geometry and Symmetry in Physics 2016 Vol. 42 P. 73–94

In this paper we consider some Lie groups in complexified Clifford algebras. Using relations between operations of conjugation in Clifford algebras and matrix operations we prove isomorphisms between these groups and classical matrix groups (symplectic, orthogonal, linear, unitary) in the cases of arbitrary dimension and arbitrary signature. Also we obtain isomorphisms of corresponding Lie algebras ...

Added: December 14, 2016