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Oksana Yakimova

Publications and source records attributed to Oksana Yakimova.

At least 19 recordsLinked to original sources
arXiv2026

Near-derivations and their applications to Lie algebras

E.B. Vinberg's theory of quasi-derivations of algebras is extended to a broader framework of near-derivations. This deepens connections between Poisson geometry and Lie theory. Although basic results apply to arbitrary algebras, our substantial applications concern the Poisson algebra $(\mathcal S(\mathfrak q),\{\ ,\,\})$ of a Lie algebra $\mathfrak q$. We develop a method for obtaining quasi-derivations via the use of squares of derivations, which allows us to provide quasi-derivations of the simple Lie algebras. It is shown that (1) a near-derivation $D$ of $(\mathcal S(\mathfrak q),\{\ ,\,\})$ yields a pencil of compatible Poisson brackets on $\mathfrak q^*$ and (2) using $D$ one may naturally construct a Poisson-commutative subalgebra of $\mathcal S(\mathfrak q)$. A special attention is given to near-derivations of $(\mathcal S(\mathfrak q),\{\ ,\,\})$ induced from near-derivations of $\mathfrak q$. This provides some old and new families of compatible Poisson brackets. We also compare properties of near-derivations of $\mathfrak q$ and Nijenhuis operators in $\mathfrak{gl}(\mathfrak q)$.

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arXiv2026

Horospherical splittings of $\mathfrak g$ and related Poisson commutative subalgebras of $\mathcal S(\mathfrak g)$

Let a Lie algebra $\mathfrak q$ be a linear sum of two complementary subalgebras $\mathfrak h$ and $\mathfrak r$. We continue our investigations initiated in (J. London Math. Soc. 103 (2021), 1577-1595), where compatible Poisson brackets associated with splitting $\mathfrak q=\mathfrak h\oplus\mathfrak r$ and related Poisson-commutative subalgebras of the symmetric algebra $\mathcal S(\mathfrak q)$ are studied. In this article, we further develop the general theory and study in more details splittings of the reductive Lie algebras such that both $\mathfrak h$ and $\mathfrak r$ are solvable horospherical subalgebras. We also derive some results of the Adler-Kostant-Symes theory using our approach.

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arXiv2025

Invariants of twisted current algebras and related Poisson-commutative subalgebras

Let q be a finite-dimensional Lie algebra and $θ$ an automorphism of q of order m. We extend $θ$ to an automorphism of the loop algebra of q and consider the fixed-point subalgebra $q[t,t^{-1}]^θ$. Using a splitting of $q[t,t^{-1}]^θ$, we construct $θ$-twisted Poisson-commutative versions of the Feigin--Frenkel centre and the universal Gaudin subalgebra introduced by Ilin and Rybnikov in 2021.

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arXiv2025

Contact Lie algebras, generic stabilisers, and affine seaweeds

Let $\mathfrak q=Lie Q$ be an algebraic Lie algebra of index 1, i.e., a generic $Q$-orbit on $\mathfrak q^*$ has codimension 1. We show that the following conditions are equivalent: $\mathfrak q$ is contact; a generic $Q$-orbit on $\mathfrak q^*$ is not conical; there is a generic stabiliser for the coadjoint action of $\mathfrak q$. In addition, if $\mathfrak q$ is contact, then the subalgebra $S(\mathfrak q)_{\sf si}\subset S(\mathfrak q)$ generated by symmetric semi-invariants of $\mathfrak q$ is a polynomial ring. We study also affine seaweed Lie algebras of type ${\sf A}$ and find some contact as well as non-contact examples among them.

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arXiv2025

A combinatorial approach to the index of seaweed subalgebras of Kac--Moody algebras

In 2000, Dergachev and Kirillov introduced subalgebras of "seaweed type" in $\mathfrak{gl}_n$ and computed their index using certain graphs. Then seaweed subalgebras $\mathfrak q\subset\mathfrak g$ were defined by Panyushev for any reductive $\mathfrak g$. A few years later Joseph generalised this notion to the setting of (untwisted) affine Kac--Moody algebras $\widehat{\mathfrak g}$. Furthermore, he proved that the index of such a seaweed can be computed by the same formula that had been known for $\mathfrak g$. In this paper, we construct graphs that help to understand the index of a seaweed $\mathfrak q\subset\widehat{\mathfrak g}$, where $\widehat{\mathfrak g}$ is of affine type A or C.

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arXiv2024

Some remarks on periodic gradings

Let $\mathfrak q$ be a finite-dimensional Lie algebra, $\vartheta\in Aut(\mathfrak q)$ a finite order automorphism, and $\mathfrak q_0$ the subalgebra of fixed points of $\vartheta$. Using $\vartheta$ one can construct a pencil $\mathcal P$ of compatible Poisson brackets on $S(\mathfrak q)$, and thereby a `large' Poisson-commutative subalgebra $Z(\mathfrak q,\vartheta)$ consisting of $\mathfrak q_0$-invariants in $S(\mathfrak q)$. We study one particular bracket $\{\,\,,\,\}_{\infty}\in\mathcal P$ and the related Poisson centre ${\mathcal Z}_\infty$. It is shown that ${\mathcal Z}_\infty$ is a polynomial ring, if $\mathfrak q$ is reductive.

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arXiv2022

Automorphisms of finite order, periodic contractions, and Poisson-commutative subalgebras of $\mathcal S(\mathfrak g)$

Let $\mathfrak g$ be a semisimple Lie algebra, $\vartheta\in {\sf Aut}(\mathfrak g)$ a finite order automorphism, and $\mathfrak g_0$ the subalgebra of fixed points of $\vartheta$. Recently, we noticed that using $\vartheta$ one can construct a pencil of compatible Poisson brackets on $\mathcal S(\mathfrak g)$, and thereby a `large' Poisson-commutative subalgebra $\mathcal Z(\mathfrak g,\vartheta)$ of $\mathcal S(\mathfrak g)^{\mathfrak g_0}$. In this article, we study invariant-theoretic properties of $(\mathfrak g,\vartheta)$ that ensure good properties of $\mathcal Z(\mathfrak g,\vartheta)$. Associated with $\vartheta$ one has a natural Lie algebra contraction $\mathfrak g_{(0)}$ of $\mathfrak g$ and the notion of a good generating system (=g.g.s.) in $\mathcal S(\mathfrak g)^\mathfrak g$. We prove that in many cases the equality ${\mathsf{ind\,}}\mathfrak g_{(0)}={\mathsf{ind\,}}\mathfrak g$ holds and $\mathcal S(\mathfrak g)^\mathfrak g$ has a g.g.s. According to V.G. Kac's classification of finite order automorphisms (1969), $\vartheta$ can be represented by a Kac diagram, $\mathcal K(\vartheta)$, and our results often use this presentation. The most surprising observation is that $\mathfrak g_{(0)}$ depends only on the set of nodes in $\mathcal K(\vartheta)$ with nonzero labels, and that if $\vartheta$ is inner and a certain label is nonzero, then $\mathfrak g_{(0)}$ is isomorphic to a parabolic contraction of $\mathfrak g$.

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arXiv2021

A bi-Hamiltonian nature of the Gaudin algebras

Let $\mathfrak q$ be a Lie algebra over a field $\mathbb K$ and $p,\tilde p\in\mathbb K[t]$ two different normalised polynomials of degree at least 2. As vector spaces both quotient Lie algebras $\mathfrak q[t]/(p)$ and $\mathfrak q[t]/(\tilde p)$ can be identified with $W=\mathfrak q{\cdot}1\oplus\mathfrak q\bar t\oplus\ldots\oplus\mathfrak q\bar t^{n-1}$. If $\mathrm{deg}\,(p-\tilde p)$ is at most 1, then the Lie brackets $[\,\,,\,]_p$, $[\,\,,\,]_{\tilde p}$ induced on $W$ by $p$ and $\tilde p$, respectively, are compatible. By a general method, known as the Lenard-Magri scheme, we construct a subalgebra $Z=Z(p,\tilde p)\subset {\mathcal S}(W)^{\mathfrak q{\cdot}1}$ such that $\{Z,Z\}_p=\{Z,Z\}_{\tilde p}=0$. If ${\mathrm{tr.deg\,}}{\mathcal S}(\mathfrak q)^{\mathfrak q}=\mathrm{ind}\,\mathfrak q$ and $\mathfrak q$ has the codim-$2$ property, then ${\mathrm{tr.deg\,}} Z$ takes the maximal possible value, which is $((n-1)\dim\mathfrak q)/2+((n+1)\mathrm{ind}\,\mathfrak q)/2$. If $\mathfrak q=\mathfrak g$ is semisimple, then $Z$ contains the Hamiltonians of a suitably chosen Gaudin model. Therefore, in a non-reductive case, we obtain a completely integrable generalisation of Gaudin models.

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arXiv2020

Compatible Poisson brackets associated with 2-splittings and Poisson commutative subalgebras of $\mathcal S(\mathfrak g)$

Let ${\mathcal S}(\mathfrak g)$ be the symmetric algebra of a reductive Lie algebra $\mathfrak g$ equipped with the standard Poisson structure. If ${\mathcal C}\subset\mathcal S(\mathfrak g)$ is a Poisson-commutative subalgebra, then ${\rm trdeg\,}{\mathcal C}\le\boldsymbol{b}(\mathfrak g)$, where $\boldsymbol{b}(\mathfrak g)=(\dim\mathfrak g+{\rm rk}\mathfrak g)/2$. We present a method for constructing the Poisson-commutative subalgebra $\mathcal Z_{\langle\mathfrak h,\mathfrak r\rangle}$ of transcendence degree $\boldsymbol{b}(\mathfrak g)$ via a vector space decomposition $\mathfrak g=\mathfrak h\oplus\mathfrak r$ into a sum of two spherical subalgebras. There are some natural examples, where the algebra $\mathcal Z_{\langle\mathfrak h,\mathfrak r\rangle}$ appears to be polynomial. The most interesting case is related to the pair $(\mathfrak b,\mathfrak u_-)$, where $\mathfrak b$ is a Borel subalgebra of $\mathfrak g$. Here we prove that ${\mathcal Z}_{\langle\mathbb b,\mathbb u_-\rangle}$ is maximal Poisson-commutative and is complete on every regular coadjoint orbit in $\mathfrak g^*$. Other series of examples are related to decompositions associated with involutions of $\mathfrak g$.

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arXiv2020

Symmetrisation and the Feigin-Frenkel centre

For simple Lie algebras of types B, C, and D, we provide new explicit formulas for the generators of the Feigin-Frenkel centre. These formulas make use of the symmetrisation map as well as some well-chosen symmetric invariants of $\mathfrak g$. There are some general results on the role of the symmetrisation map in the explicit description of the FF-centre. The advantage of our method is that it reduces questions about elements of the FF-centre to questions on the structure of the symmetric invariants in a type-free way. As an illustration, we deal with type G$_2$ by hand.

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arXiv2019

Monomial bases and branching rules

Following a question of Vinberg, a general method to construct monomial bases for finite-dimensional irreducible representations of a reductive Lie algebra was developed in a series of papers by Feigin, Fourier, and Littelmann. Relying on this method, we construct monomial bases of multiplicity spaces associated with the restriction of the representation to a reductive subalgebra. As an application, we produce monomial bases for representations of the general linear and symplectic Lie algebras associated with natural chains of subalgebras. We also show that our basis in type A is related to both the Gelfand-Tsetlin basis and the Littelmann basis via triangular transition matrices which implies that the triangularity property extends to the matrix connecting the Gelfand-Tsetlin and canonical bases. A similar relationship holds between our basis in type C and a suitably modified version of the basis constructed earlier by the first author.

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arXiv2019

Quantisation and nilpotent limits of Mishchenko-Fomenko subalgebras

For any simple Lie algebra $\mathfrak{g}$ and an element $μ\in\mathfrak{g}^*$, the corresponding commutative subalgebra $\mathcal{A}_μ$ of $\mathcal{U}(\mathfrak{g})$ is defined as a homomorphic image of the Feigin-Frenkel centre associated with $\mathfrak{g}$. It is known that when $μ$ is regular this subalgebra solves Vinberg's quantisation problem, as the graded image of $\mathcal{A}_μ$ coincides with the Mishchenko-Fomenko subalgebra $\overline{\mathcal{A}}_μ$ of $\mathcal{S}(\mathfrak{g})$. By a conjecture of Feigin, Frenkel and Toledano Laredo, this property extends to an arbitrary element $μ$. We give sufficient conditions which imply the property for certain choices of $μ$. In particular, this proves the conjecture in type C and gives a new proof in type A. We show that the algebra $\mathcal{A}_μ$ is free in both cases and produce its generators in an explicit form. Moreover, we prove that in all classical types generators of $\mathcal{A}_μ$ can be obtained via the canonical symmetrisation map from certain generators of $\overline{\mathcal{A}}_μ$. The symmetrisation map is also used to produce free generators of nilpotent limits of the algebras $\mathcal{A}_μ$ and give a positive solution of Vinberg's problem for these limit subalgebras.

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arXiv2018

Poisson-commutative subalgebras of $S(\mathfrak g)$ associated with involutions

The symmetric algebra $S(\mathfrak g)$ of a reductive Lie algebra $\mathfrak g$ is equipped with the standard Poisson structure, i.e., the Lie-Poisson bracket. Poisson-commutative subalgebras of $S(\mathfrak g)$ attract a great deal of attention, because of their relationship to integrable systems and, more recently, to geometric representation theory. The transcendence degree of a Poisson-commutative subalgebra ${\mathcal C}\subset S(\mathfrak g)$ is bounded by the "magic number" $\boldsymbol{b}(\mathfrak g)$ of $\mathfrak g$. The "argument shift method" of Mishchenko-Fomenko was basically the only known source of $\mathcal C$ with ${\rm trdeg\,}{\mathcal C}=\boldsymbol{b}(\mathfrak g)$. We introduce an essentially different construction related to symmetric decompositions $\mathfrak g=\mathfrak g_0\oplus\mathfrak g_1$. Poisson-commutative subalgebras $\mathcal Z,\tilde{\mathcal Z}\subset S(\mathfrak g)^{\mathfrak g_0}$ of the maximal possible transcendence degree are presented. If the $\mathbb Z_2$-contraction $\mathfrak g_0\ltimes\mathfrak g_1^{\sf ab}$ has a polynomial ring of symmetric invariants, then $\tilde{\mathcal Z}$ is a polynomial maximal Poisson-commutative subalgebra of $S(\mathfrak g)^{\mathfrak g_0}$, and its free generators are explicitly described.

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arXiv2018

A combinatorial identity for the Jacobian of $t$-shifted invariants

Let $\mathfrak g$ be a simple Lie algebra. There are classical formulas for the Jacobians of the generating invariants of the Weyl group of $\mathfrak g$ and of the images under the Harich-Chandra projection of the generators of $ZU(\mathfrak g)$. We present a modification of these formulas related to Takiff Lie algebras.

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arXiv2018

Nilpotent subspaces and nilpotent orbits

Let $G$ be a semisimple algebraic group with Lie algebra $\mathfrak g$. For a nilpotent $G$-orbit $\mathcal O\subset\mathfrak g$, let $d_\mathcal O$ denote the maximal dimension of a subspace $V\subset \mathfrak g$ that is contained in the closure of $\mathcal O$. In this note, we prove that $d_\mathcal O \le \frac{1}{2}\dim\mathcal O$ and this upper bound is attained if and only if $\mathcal O$ is a Richardson orbit. Furthermore, if $V$ is $B$-stable and $\dim V= \frac{1}{2}\dim\mathcal O$, then $V$ is the nilradical of a polarisation of $\mathcal O$. Every nilpotent orbit closure has a distinguished $B$-stable subspace constructed via an $\mathfrak{sl}_2$-triple, which is called the Dynkin ideal. We then characterise the nilpotent orbits $\mathcal O$ such that the Dynkin ideal (1) has the minimal dimension among all $B$-stable subspaces $\mathfrak c$ such that $\mathfrak c\cap\mathcal O$ is dense in $\mathfrak c$, or (2) is the only $B$-stable subspace $\mathfrak c$ such that $\mathfrak c\cap\mathcal O$ is dense in $\mathfrak c$.

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arXiv2017

Some semi-direct products with free algebras of symmetric invariants

Let $\mathfrak g$ be a complex reductive Lie algebra and $V$ the underling vector space of a finite-dimensional representation of $\mathfrak g$. Then one can consider a new Lie algebra $\mathfrak q=\mathfrak g{\ltimes} V$, which is a semi-direct product of $\mathfrak g$ and an Abelian ideal $V$. We outline several results on the algebra $\mathbb C[\mathfrak q^*]^{\mathfrak q}$ of symmetries invariants of $\mathfrak q$ and describe all semi-direct products related to the defining representation of $\mathfrak{sl}_n$ with $\mathbb C[\mathfrak q^*]^{\mathfrak q}$ being a free algebra.

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arXiv2017

Takiff algebras with polynomial rings of symmetric invariants

Extending results of Rais-Tauvel, Macedo-Savage, and Arakawa-Premet, we prove that under mild restrictions on the Lie algebra $\mathfrak q$ having the polynomial ring of symmetric invariants, the m-th Takiff algebra of $\mathfrak q$, $\mathfrak q\langle m\rangle$, also has a polynomial ring of symmetric invariants.

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