We say that a Lie ring R is called a unique addition Lie ring, or briefly a UA-Lie ring, if any commutator-preserving bijection on R preserves the addition as well. We prove that any semisimple Lie algebra and any its parabolic subalgebra is a UA-Lie ring. Also we describe wide classes of solvable UA-Lie rings.
In recent joint work with D. Nakano, an analogue 𝒩 of the nilpotent cone was constructed for classical simple Lie superalgebras, and 𝒩 was shown to consist of only finitely many nilpotent orbits. In this talk, we determine several geometric properties such as the dimension and irreducibility of 𝒩 for the Lie superalgebra 𝔤𝔩(m|n), and we give a more detailed description of the geometry and structure of the nilpotent orbits in this case. We also demonstrate connections between our nilpotent orbit representatives and certain signed Young diagrams appearing in the work of Kraft and Procesi.
Rota-Baxter operators on Lie algebras were first studied by Belavin, Drinfeld and Semenov-Tian-Shansky as operator forms of the classical Yang-Baxter equation. Integrating the Rota-Baxter operators on Lie algebras, we introduce the notion of Rota-Baxter operators on Lie groups and more generally on groups. Then the factorization theorem can be achieved directly on groups. We introduce the notion of post-Lie groups, whose differentiations are post-Lie algebras. A Rota-Baxter operator on a group naturally induces a post-group. Post-groups are also closely related to operads, braces, Lie-Butcher groups and various structures.
Pre-Calabi-Yau algebras were introduced in the last decade by M. Kontsevich, A. Takeda and Y. Vlassopoulos using the necklace bracket. This notion is equivalent to a cyclic A∞-algebra for the natural bilinear form in the finite-dimensional case. Moreover, W-K. Yeung showed that double Poisson DG structures provide an example of pre-Calabi-Yau structures. In 2020, D. Fernandez and E. Herscovich proved that given a morphism of double Poisson DG algebras from A to B, one can produce a cyclic A∞-algebra and A∞-morphisms between the latter and the cyclic A∞-algebras associated to A and B. I will explain how to generalize this result to pre-Calabi-Yau algebras by doing an explicit construction of a (cyclic) A∞-algebra and A∞-morphisms given a pre-Calabi-Yau morphism.
The celebrated problem of deformation quantization discusses deformations of Poisson algebras into associative algebras, a question that is, in the end, motivated by quantum mechanics. I shall discuss this question and some of its generalisations from the purely algebraic point of view using the theory of operads. In particular, I shall show how to prove that there are, in a strict mathematical sense, only two meaningful deformation problems for Poisson algebras, namely deforming them in the class of all Poisson algebras or all associative algebras, and there is only one meaningful deformation problem for the so called almost Poisson algebras (also sometimes known as generic Poisson algebras), namely deforming them in the class of all almost Poisson algebras. For instance, this explains the existing body of work in the mathematical physics literature asserting that some classes of non-associative star products cannot be alternative, are always flexible etc.
The notion of a double Poisson bracket on an associative algebra was introduced by M. Van den Bergh in order to induce a (usual) Poisson bracket on the representation spaces of this algebra. I will start by reviewing the basics of this theory and its relation to other interesting operations, such as Leibniz brackets and H0-Poisson structures. I will then explain some recent results and generalisations related to double Poisson brackets.
A set-theoretic solution of the Yang-Baxter equation is a mapping r : X×X → X×X satisfying (r×1)(1×r)(r×1) = (1×r)(r×1)(1×r). A solution r : (x, y) → (σx(y), τy (x)) is called non-degenerate if the mappings σx and τy are permutations, for all x, y ∈ X. A solution is called involutive if r2 = 1. If (X, r) is a non-degenerate involutive solution (X, r) then the relation ∼ defined by x ∼ y ≡ σx = σy is a congruence. A solution is of multipermutation level 2 if |(X/ ∼)/ ∼ | = 1. In our talk, we focus on these solutions and we present several constructions and properties.
I will talk about the development of the theory of polynomial identities initiated by important questions such as Burnside's asking if every finitely generated torsion group is finite. The field was enriched by contributions of many great mathematicians. Most notably Lie rings methods were developed and used by Zelmanov in the 1990s to give a positive solution to the restricted Burnside problem which awarded him the Fields medal. It has been of great interest to expand the theory to other varieties of algebraic structures. In particular, I will review when a group algebra or enveloping algebra satisfy a polynomial identity.
Let g be a 2n-dimensional solvable Lie algebra. A complex structure on g is an endomorphism J that satisfies J2=-Id and NJ(X,Y)=0, for every X,Y ∈ g, being NJ(X,Y):=[X,Y]+J[JX,Y]+J[X,JY]-[JX,JY]. Suppose that g simultaneously admits a complex structure J and a symplectic structure ω. Although J and ω are initially two unrelated structures, one can ask for an additional condition involving both of them. In this sense, the pair (J,ω) is said to be a complex symplectic structure if J is symmetric with respect to ω, in the sense that ω(JX,Y)=ω(X,JY), for every X,Y ∈ g. In this talk, we will present some methods to find certain types of solvable Lie algebras (such as nilpotent or almost Abelian) admitting complex symplectic structures.
In classification problems over the real field ℝ first Galois cohomology sets play an important role, as they often make it possible to classify the orbits of a real Lie group. In this talk we outline an algorithm to compute the first Galois cohomology set H1(G,ℝ) of a complex reductive algebraic group G defined over the real field ℝ. The algorithm is in a large part based on computations in the Lie algebra of G. This is joint work with Mikhail Borovoi.
Étale affine representations of Lie algebras and algebraic groups arise in the context of affine geometry on Lie groups, operad theory, deformation theory and Yang-Baxter equations. For reductive groups, every étale affine representation is equivalent to a linear representation and we obtain a special case of a prehomogeneous representation. Such representations have been classified by Sato and Kimura in some cases. The induced representation on the Lie algebra level gives rise to a pre-Lie algebra structure on the Lie algebra 𝔤 of G. For a Lie group G, a pre-Lie algebra structure on 𝔤 corresponds to a left-invariant affine structure on G. This refers to a well-known question by John Milnor from 1977 on the existence of complete left-invariant affine structures on solvable Lie groups. We present results on the existence of étale affine representations of reductive groups and Lie algebras and discuss a related conjecture of V. Popov concerning flattenable groups and linearizable subgroups of the affine Cremona group.
In 1985, K. Saito introduced elliptic root systems as root systems belonging to a real vector space F equipped with a symmetric bilinear form I with signature (l, 2, 0). Such root systems are studied in view of simply elliptic singularities which are surface singularities with a regular elliptic curve in its resolution. K. Saito had classified elliptic root systems R with its one-dimensional subspace G of the radical of I, in the case when R/G ⊂ F/G is a reduced affine root system. In our joint work with A. Fialowski and Y. Saito, we have completed its classification; we classified the pair (R,G) whose quotient R/G ⊂ F/G is a non-reduced affine root system. In this talk, we give an overview of elliptic root systems and describe some of the new root systems we have found.
We call a Lie algebra 𝔤 di-semisimple if it can be written as a vector space sum 𝔤 = 𝔰1 + 𝔰2, where 𝔰1 and 𝔰2 are semisimple subalgebras of 𝔤 and we say that 𝔤 is strongly di-semisimple if g can be written as a direct vector space sum of semisimple subalgebras. We will show that complex strongly di-semisimple Lie algebras have to be semisimple themselves. We will then use this result to show that if a pair of complex Lie algebras (𝔤,𝔫) with 𝔤 semisimple admits a so called post-Lie algebra structure, then 𝔫 must be isomorphic to 𝔤.
I will start by reviewing the notions of vertex algebra, Poisson vertex algebra, and Lie conformal algebra, and their relations to each other. Then I will present a unified approach to all these algebras as Lie algebras in certain pseudo-tensor categories, or equivalently, as morphisms from the Lie operad to certain operads. As an application, I will introduce a cohomology theory of vertex algebras similarly to Lie algebra cohomology, and will show how it relates to the cohomology of Poisson vertex algebras and of Lie conformal algebras.
Murray Gerstenhaber constructed a graded Lie structure (Gerstenhaber bracket) on Hochschild cohomology, which makes Hochschild cohomology a Lie algebra. However, it is not easy to calculate bracket structure with the original definition. There is an alternative technique to compute Gerstenhaber bracket on Hochschild cohomology, introduced by Chris Negron and Sarah Witherspoon. It is also known that Hopf algebra cohomology has a bracket and the bracket is trivial when a Hopf algebra is quasi-triangular. We use a similar technique to the technique given by Negron and Witherspoon to calculate the Lie structure on Hochschild cohomology of the Taft algebra Tp for any integer p>2 which is a nonquasi-triangular Hopf algebra. Then, we find the corresponding bracket on Hopf algebra cohomology of Tp. We show that the bracket is indeed zero on Hopf algebra cohomology of Tp, as in all known quasi-triangular Hopf algebras. This example is the first known bracket computation for a nonquasi triangular algebra.
Murray Gerstenhaber constructed Lie structure (Gerstenhaber bracket) on Hochschild cohomology, which makes Hochschild cohomology a graded Lie algebra. Later, it is shown that Hopf algebra cohomology also has a Lie structure. We will introduce a general formula for the bracket on Hopf algebra cohomology of any Hopf algebra with bijective antipode on the bar resolution that is reminiscent of Gerstenhaber’s original formula for Hochschild cohomology.
Different versions of the Burnside Problem ask what one can say about finitely generated periodic groups under additional assumptions. For associative algebras, Kurosh type problems ask similar questions about properties of finitely generated nil (more generally, algebraic) algebras. Similarly, one considers finitely generated restricted Lie algebras with a nil p-mapping. Now we study an oscillating intermediate growth in nil restricted Lie algebras.
Namely, for any field of positive characteristic, we construct a family of 3-generated restricted Lie algebras of intermediate oscillating growth. We call them Phoenix algebras, because of the following. a) For infinitely many periods of time the algebra is 'almost dying' by having a quasi-linear growth, namely the lower Gelfand-Kirillov dimension is 1, more precisely, the growth is of type n (ln ⋯ ln n)κ (ln q times), where q ∈ ℕ, κ > 0 are constants. b) On the other hand, for infinitely many n the growth function has a rather fast intermediate behaviour of type exp(n/(ln n)λ), λ being a constant determined by characteristic, for such periods the algebra is 'resuscitating'. c) Moreover, the growth function is bounded and oscillating between these two types of behaviour. d) These restricted Lie algebras have a nil p-mapping.
We also construct nil Lie superalgebras and nil Jordan superalgebras of similar oscillating intermediary growth over arbitrary field.
Dixmier and Moeglin showed that if L is a finite-dimensional complex Lie algebra then the primitive ideals of the enveloping algebra U(L) are the prime ideals of Spec(U(L)) that are locally closed in the Zariski topology. In addition, they proved that a prime ideal P of U(L) is primitive if and only if the Goldie ring of quotients of U(L)/P has the property that its centre is just the base field of the complex numbers. Algebras that share this characterization of primitive ideals are said to satisfy the Dixmier-Moeglin equivalence. We give an overview of this property and mention some recent work on proving this equivalence holds for certain classes of twisted homogenous coordinate rings and classes of Hopf algebras of small Gelfand-Kirillov dimension.
For a superalgebra A, and even subalgebra a, one may define an associated diagrammatic monoidal supercategory Web(A,a), which generalizes a number of symmetric web category constructions. In this talk, I will define and discuss Web(A,a)), focusing on two interesting applications: Firstly, Web(A,a) is equipped with an asymptotically faithful functor to the category of 𝔤𝔩n(A)-modules generated by symmetric powers of the natural module, and may be used to establish Howe dualities between 𝔤𝔩n(A) and 𝔤𝔩m(A) in some cases. Secondly, Web(A,a) yields a diagrammatic presentation for the ‘Schurification' TAa(n,d). For various choices of A/a, these Schurifications have proven connections to RoCK blocks of Hecke algebras, and conjectural connections to RoCK blocks of Schur algebras and Sergeev superalgebras.
Let n be a positive integer and let Q = (qij) be a multipicatively antisymmetric n × n matrix over a field 𝕂, that is qii = 1 for 1 ≤ i ≤ n and, for 1 ≤ i,j ≤ n, qij ≠ 0 and qji=qij-1. The quantized (co-ordinate ring of) quantum n-space R=𝒪Q(𝕂n) is the 𝕂-algebra generated by x1,x2, . . .,xn subject to the relations xixj = qijxjxi for 1 ≤ i < j ≤ n.
Although the space of derivations Der(R) is well understood through work of Alev and Chamarie in 1982, less is known about the space Derσ(R) of σ-derivations of R. The only case in the literature where the σ-derivations of R are determined appears to be when n = 2 and, for some λ ∈ 𝕂*, σ(x1)=λx1 and σ(x_2)=λ-1 x2. This case appears in a 2018 paper by Almulhem and Brzeziński that was motivated by differential geometry. This talk will discuss the classification of the σ-derivations of R for all n when σ is toric, that is each xi is an eigenvector for σ, with a view to applications to iterated Ore extensions of 𝕂. Any such classification must include the inner σ-derivations of R, that is those for which there exists a ∈ R such that δa(r)=ar-σ(r)a for all r ∈ R.
The methods are based on two of the classical methods of non-commutative algebra, namely localization and grading, in this case by ℤn. Localization at the set {x1d1x2d2 . . . xndn} yields the quantum n-torus T=𝒪Q((𝕂*)n) to which σ and all σ-derivations extend. A σ-derivation δ of T is homogeneous, of weight (d1,d2, . . ., dn), if δ(xi) ∈ 𝕂x1d1x2d2 . . .,xidi+1. . . xndn for 1 ≤ i ≤ n and every σ-derivation of T is a unique linear combination of homogeneous σ-derivations. It turns out that if δ is a homogeneous σ-derivation of T then either the automorphism σ is inner or the σ-derivation δ is inner and the Ore extension T[x ; σ,δ] can, by a change of variables, be expressed as an Ore extension of either automorphism type or derivation type. This dichotomy influences the space Derσ(R) which can be identified with { δ ∈ Derσ(T) : δ(R) ⊆ R }. The most obvious σ-derivations included here are the homogeneous σ-derivations of weight (d1,d2, . . ., dn) where each di ≥ 0, but more interesting are those for which one di = -1. There are two types of these, depending on whether σ or δ is inner on T. In the latter case we are in a common situation where a σ-derivation of a ring R is not inner on R but becomes inner on the localization of R at the powers of a normal element of R, giving rise to a distinguished normal or central element of the Ore extension R[x ; σ,δ].
