Emanuela Petracci
A class of representations of a Lie superalgebra (over a commutative superring) in its symmetric algebra is studied. As an application we get a direct and natural proof of a strong form of the Poincare'-Birkhoff-Witt theorem, extending this theorem to a class of nilpotent Lie superalgebras. Other applications are presented. Our results are new already for Lie algebras.
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This work explores the deformation theory of algebraic structures in a very general setting. These structures include commutative, associative algebras, Lie algebras, and the infinity versions of these structures, the strongly homotopy associative and Lie algebras. In all these cases the algebra structure is determined by an element of a certain graded Lie algebra which plays the role of a differential on this algebra. We work out the deformation theory in terms of the Lie algebra of coderivations of an appropriate coalgebra structure and construct a universal infinitesimal deformation as well…
Graded Lie structure on cohomology of some exact monoidal categories
本文在正合幺半范畴中建立了李括号拓扑定义与代数定义的联系，证明了拓扑定义可产生Gerstenhaber代数结构，并将A∞-余导子与同伦提升技术推广到该范畴。
Given a $n$-dimensional Lie algebra $g$ over a field $k \supset \mathbb Q$, together with its vector space basis $X^0_1,..., X^0_n$, we give a formula, depending only on the structure constants, representing the infinitesimal generators, $X_i = X^0_i t$ in $g\otimes_k k [[t]]$, where $t$ is a formal variable, as a formal power series in $t$ with coefficients in the Weyl algebra $A_n$. Actually, the theorem is proved for Lie algebras over arbitrary rings $k\supset Q$. We provide three different proofs, each of which is expected to be useful for generalizations. The first proof is obtained by di…
We show that a set of monic polynomials in the free Lie superalgebra is a Gröbner-Shirshov basis for a Lie superalgebra if and only if it is a Gröbner-Shirshov basis for its universal enveloping algebra. We investigate the structure of Gröbner-Shirshov bases for Kac-Moody superalgebras and give explicit constructions of Gröbner-Shirshov bases for classical Lie superalgebras.
For two different natural definitions of Casimir operators for simple Lie algebras we show that their eigenvalues in the adjoint representation can be expressed polynomially in the universal Vogel's parameters $α, β, γ$ and give explicit formulae for the generating functions of these eigenvalues.
This paper presents new generators for the center of the universal enveloping algebra of the symplectic Lie algebra. These generators are expressed in terms of the column-permanent, and it is easy to calculate their eigenvalues on irreducible representations. We can regard these generators as the counterpart of central elements of the universal enveloping algebra of the orthogonal Lie algebra given in terms of the column-determinant by A. Wachi. The earliest prototype of all these central elements is the Capelli determinants in the universal enveloping algebra of the general linear Lie algebra…
New commutative subalgebras of the maximal Gel'fand-Kirillov dimension in the universal enveloping algebras of classical Lie algebras gl(n) and so(n) are constructed. In the case of sp(n) Gel'fand-Tsetlin algebra is extended to a maximally commutative one.
We describe a new method to determine faithful representations of small dimension for a finite dimensional nilpotent Lie algebra. We give various applications of this method. In particular we find a new upper bound on the minimal dimension of a faithful module for the Lie algebras being counter examples to a well known conjecture of J. Milnor.
Let $U$ be a finite dimentional vector space over $\mathbb R$ or $\mathbb C$, and let $ρ:G\to GL(U)$ be a representation of a connected Lie group $G$. A linear subspace $V\subset U$ is called universal if every orbit of $G$ meets $V$. We study universal subspaces for Lie groups, especially compact Lie groups. Jinpeng and Doković approached universality for compact groups through a certain topological obstruction. They showed that the non-vanishing of the obstruction class is sufficient for the universality of $V$, and asked whether it is also necessary under certain conditions. In this article…
Universal Capelli identities and quantum immanants for the queer Lie superalgebra
本文通过构造 queer 林格超代数的量子immanants，证明了通用Capelli恒等式，并展示了其与阶乘Schur Q多项式的一致性。
Let $\ggg=\ggg_\bz+\ggg_\bo$ be a basic classical Lie superalgebra over an algebraically closed field $k$ of characteristic $p>2$, and $G$ be an algebraic supergroup satisfying $\Lie(G)=\ggg$, with the purely even subgroup $G_\ev$ which is a reductive group. The center $\cz:=\cz(\ggg)$ of the universal enveloping algebra of $\ggg$ easily turns out to be a domain. In this paper, we prove that the quotient field of $\cz$ coincides with that of the subalgebra generated by the $G_{\ev}$-invariant ring $\cz^{G_\ev}$ of $\cz$ and the $p$-center $\cz_0$ of $U(\ggg_\bz)$.
A Lie algebra $\mathfrak{g}_\mathbb{Q}$ over $\mathbb{Q}$ is said to be $\mathbb{R}$-universal if every homomorphism from $\mathfrak{g}_\mathbb{Q}$ to $\mathfrak{gl}(n,\mathbb{R})$ is conjugate to a homomorphism into $\mathfrak{gl}(n,\mathbb{Q})$ (for every $n$). By using Galois cohomology, we provide a short proof of the known fact that every real semisimple Lie algebra has an $\mathbb{R}$-universal $\mathbb{Q}$-form. We also provide a classification of the $\mathbb{R}$-universal Lie algebras that are semisimple.
For a representation of a connected compact Lie group G in a finite dimensional real vector space U and a subspace V of U, invariant under a maximal torus of G, we obtain a sufficient condition for V to meet all G-orbits in U, which is also necessary in certain cases. The proof makes use of the cohomology of flag manifolds and the invariant theory of Weyl groups. Then we apply our condition to the conjugation representations of U(n), Sp(n), and SO(n) in the space of $n\times n$ matrices over C, H, and R, respectively. In particular, we obtain an interesting generalization of Schur's triangular…
We construct a universal continuous invariant bilinear form for the Lie algebra of compactly supported sections of a Lie algebra bundle in a topological sense. Moreover we construct a universal continuous central extension of a current algebra that is the tensor product of a finite-dimensional Lie algebra g and a certain topological algebra A. In particular taking A as the compactly supported smooth functions on a sigma-compact manifold M, we obtain a more detailed justification for a recent result of Janssens and Wockel concerning a universal extension for the Lie algebra of compactly support…
The notion of a non-associative universal enveloping algebra for a Lie triple system arises when Lie triple systems are considered as Bol algebras (more generally, Sabinin algebras). In this paper a new construction for these universal enveloping algebras is given, and their properties are studied. It is shown that universal enveloping algebras of Lie triple systems have surprisingly few ideals. It is conjectured, and the conjecture is verified on several examples, that the only proper ideal of the universal enveloping algebra of a simple Lie triple system is the augmentation ideal.
We construct the universal enveloping algebra of a Hom-Lie algebra and endow it with a Hom-Hopf algebra structure. We discuss group-like elements that we see as a Hom-group integrating the initial Hom-Lie algebra.
The goal of this paper is to explicitly describe in terms of generators and relations the universal central extension of the infinite dimensional Lie algebra, $\mathfrak{g} \otimes \mathbb{C}[t,t^{-1},u]$ with finite dimensional simple Lie algebra $\mathfrak{g}$ and $u^m=p(t)$.
We introduce the notion of Lie algebras with plus-minus pairs as well as regular plus-minus pairs. These notions deal with certain factorizations in universal enveloping algebras. We show that many important Lie algebras have such pairs and we classify, and give a full treatment of, the three dimensional Lie algebras with plus-minus pairs.
We present a universal formula for the dimension of the Cartan powers of the adjoint representation of a complex simple Lie algebra (i.e., a universal formula for the Hilbert functions of homogeneous complex contact manifolds), as well as several other universal formulas. These formulas generalize formulas of Vogel and Deligne and are given in terms of rational functions where both the numerator and denominator decompose into products of linear factors with integer coefficients. We also discuss some consequences of the formulas including a relation with Scorza varieties.
Structure Constant Formulas for the Universal Enveloping Algebras of the Nilpotent Lie Algebras of Dimension Five and Less
基于六维及以下 Lie 代数的分类，研究五维及以下 nilpotent Lie 代数的 universal enveloping algebras 的结构常数公式，推导并证明了相关公式。
We define global and local Weyl modules for $q \otimes A$, where $q$ is the queer Lie superalgebra and $A$ is an associative commutative unital $\mathbb{C}-$algebra. We prove that global Weyl modules are universal highest weight objects in certain category upto parity reversing functor $Π$. Then with the assumption that $A$ is finitely generated and with a special technical condition which simple root system of $q$ satisfy it is shown that the local Weyl modules are finite dimensional. Further they are universal highest map-weight objects in certain category upto $Π$. Finally we prove a tensor…
In this paper we study universal central extensions and non-abelian tensor product of hom-Lie-Rinehart algebras. We discuss about universal $α$- central extensions, and, lifting of automorphisms and $α$-derivations to central extensions for hom-Lie-Rinehart algebras. This is in turn provide such lifting of automorphisms and $α^k$-derivations to the central extensions for hom-Lie algebras.
In this paper we initiate a study of the relation between weight modules for simple Lie algebras and unitary representations of the corresponding simply-connected Lie groups. In particular we consider in detail from this point of view the universal covering group of SU(1,1), including new results on the discrete part of tensor products of irreducible representations. As a consequence of these results, we show that the set of smooth vectors of the tensor product intersects trivially some of the representations in the discrete spectrum.
In this paper we define integral forms for the universal enveloping superalgebras of the map superalgebras of Cartan type complex Lie superalgebras, and exhibit explicit integral bases for these integral forms.
Lie antialgebras is a class of supercommutative algebras recently appeared in symplectic geometry. We define the notion of enveloping algebra of a Lie antialgebra and study its properties. We show that every Lie antialgebra is canonically related to a Lie superalgebra and prove that its enveloping algebra is a quotient of the enveloping algebra of the corresponding Lie superalgebra.
Three kinds of universal central extension are considered for a perfect Lie algebra. More precisely, one can consider such a Lie algebra as a Lie triple system, or a Leibniz algebra and construct appropriate central extensions. We show that Lie triple system universal central extension is isomorphic either to Lie or Leibniz algebra universal central extension considered as a Lie triple system in appropriate way.
We investigate color Lie rings over finite group algebras and their universal enveloping algebras. We exhibit these universal enveloping algebras as PBW deformations of skew group algebras: Every color Lie ring over a finite group algebra with a particular Yetter-Drinfeld structure has universal enveloping algebra that is a quantum Drinfeld orbifold algebra. Conversely, every quantum Drinfeld orbifold algebra of a particular type arising from the action of an abelian group is the universal enveloping algebra of some color Lie ring over the group algebra. One consequence is that these quantum D…
We study the quotient Q_i(A) of a free algebra A by the ideal M_i(A) generated by relation that the i-th commutator of any elements is zero. In particular, we completely describe such quotient for i=4 (for i<=3 this was done previously by Feigin and Shoikhet). We also study properties of the ideals M_i(A), e.g. when M_i(A)M_j(A) is contained in M_{i+j-1}(A) (by a result of Gupta and Levin, it is always contained in M_{i+j-2}(A)).
We describe a type of Lie color algebra, which we call generic, whose universal enveloping algebra is a domain with finite global dimension. Moreover, it is an iterated Ore extension. We provide an application and show Grobner basis methods can be used to study universal enveloping algebras of factors of generic Lie color algebras.
Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$. We study a monoidal category $\mathbb{T}_α$ which is universal among all symmetric $\mathbb{K}$-linear monoidal categories generated by two objects $A$ and $B$ such that $A$ has a, possibly transfinite, filtration. We construct $\mathbb{T}_α$ as a category of representations of the Lie algebra $\mathfrak{gl}^M(V_*,V)$ consisting of endomorphisms of a fixed diagonalizable pairing $V_*\otimes V\to \mathbb{K}$ of vector spaces $V_*$ and $V$ of dimension $α$. Here $α$ is an arbitrary cardinal number. We describe explicitly th…
Real Nullstellensatz for 2-step nilpotent Lie algebras
本研究提出适用于2步幂零李代数的非交换实零点定理，将经典交换实零点定理推广至非交换框架，用泛包络*-代数的可积*-表示赋值替代多项式点赋值，刻画了对应同态公共核为实理想。
For n even, we prove Pozhidaev's conjecture on the existence of associative enveloping algebras for simple n-Lie algebras. More generally, for n even and any (n+1)-dimensional n-Lie algebra L, we construct a universal associative enveloping algebra U(L) and show that the natural map from L to U(L) is injective. We use noncommutative Grobner bases to present U(L) as a quotient of the free associative algebra on a basis of L and to obtain a monomial basis of U(L). In the last section, we provide computational evidence that the construction of U(L) is much more difficult for n odd.
The universal central extensions and their extension kernels of the matrix Lie superalgebra sl(m, n, A), the Steinberg Lie superalgebra st(m, n, A) in category {\bf SLeib} of Leibniz superalgebras are determined under a weak assumption (compared with \cite{MP}) using the first Hochschild homology and the first cyclic homology group.
We explicitly compute the first and second cohomology groups of the classical Lie superalgebras $sl_{m|n}$ and $osp_{2|2n}$ with coefficients in the finite dimensional irreducible modules and the Kac modules. We also show that the second cohomology groups of these Lie superalgebras with coefficients in the respective universal enveloping algebras (under the adjoint action) vanish. The latter result in particular implies that the universal enveloping algebras $U(sl_{m|n})$ and $U(osp_{2|2n})$ do not admit any non-trivial formal deformations of Gerstenhaber type.
Diagrammatic technique for Vogel's universality
本文通过Vogel的Λ-代数图示技术，展示了其在Lie代数中的普遍计算能力，并通过多个实例讨论了该方法的应用。
Functors between representation categories. Universal modules
该研究针对李代数的表示范畴关联问题，通过构造泛代数模与李代数模的张量积的李模结构，导出两类范畴间的函子，在有限维假设下证明函子有左伴随，进而构造出泛模。
We prove that the cohomology ring of a finite-dimensional restricted Lie superalgebra over a field of characteristic $p > 2$ is a finitely-generated algebra. Our proof makes essential use of the explicit projective resolution of the trivial module constructed by J. Peter May for any graded restricted Lie algebra. We then prove that the cohomological finite generation problem for finite supergroup schemes over fields of odd characteristic reduces to the existence of certain conjectured universal extension classes for the general linear supergroup $GL(m|n)$ that are similar to the universal exte…
A Lie group corresponding to the free Lie algebra and its universality
本文研究自由李代数完备化对应的李群，证明其稠密子群可唯一积分任意同态，并具有普适性，与Pestov的Banach--Lie群结果相关。
For q generic or a primitive l-th root of unity, q-Witt algebras are described by means of q-divided power algebras. The structure of the universal q-central extension of the q-Witt algebra, the q-Virasoro algebra, is also determined. q-Lie algebras are investigated and the q-PBW theorem for the universal enveloping algebras of q-Lie algebras is proved. A realization of a class of representations of the q-Witt algebras is given. Based on it, the q-holomorph structure for the q-Witt algebras is constructed, which interprets the realization in the context of representation theory.
Finite-dimensional Lie subalgebras of the universal enveloping algebra of $\mathfrak{sl}(2,\mathbb{C})$
本文完整分类了特殊线性李代数sl(2,C)通用包络代数中的所有非零有限维复数李子代数，证明幂零子代数必为阿贝尔、不可解子代数为约化且半单部分同构于sl(2,C)，非幂零可解子代数为阿贝尔中心与可解族r(d,m)的直和，并讨论了实斜埃尔米特情形及其与外尔代数实现的关系。
We describe properties of the nonstandard q-deformation $U'_q({\rm so}_n)$ of the universal enveloping algebra $U({\rm so}_n)$ of the Lie algebra ${\rm so}_n$ which does not coincide with the Drinfeld--Jimbo quantum algebra $U_q({\rm so}_n)$. Irreducible representations of this algebras for q a root of unity q^p=1 are given. These representations act on p^N-dimensional linear space (where N is a number of positive roots of the Lie algebra ${\rm so}_n$) and are given by $r={\rm dim} {\rm so}_n$ complex parameters.
We prove a version of the Poincaré-Birkhoff-Witt Theorem for profinite pronilpotent Lie algebras in which their symmetric and universal enveloping algebras are replaced with appropriate formal analogues and discuss some immediate corollaries of this result.
Let d \subset d' be finite-dimensional Lie algebras, H = U(d), H'=U(d') the corresponding universal enveloping algebras endowed with the cocommutative Hopf algebra structure. We show that if L is a primitive Lie pseudoalgebra over H then all finite irreducible L' = Cur_H^{H'} L-modules are of the form Cur_H^{H'} V, where V is an irreducible L-module, with a single class of exceptions. Indeed, when L = H(d, χ, ω), we introduce non current L'-modules that are obtained by modifying the current pseudoaction with an extra term depending on an element t \in \d' \setminus d, which must satisfy some t…
Equivariant deformation theory for nilpotent slices in symplectic Lie algebras
针对辛李代数中双块若尔当形幂零轨道的Slodowy切片非通用泊松形变问题，证明其幂零Slodowy簇与D型同构，C型切片为$\u2124_2$-等变通用泊松形变且有非交换对应。
Here we extend the algebro-geometric approach to free probability, started in~\cite{FMcK4,F14}, to general (non)-commutative probability theories. We show that any universal convolution product of moments of independent (non)-commutative random variables defined on a graded connected dual semi-group is given by a pro-unipotent group scheme. We show that moment-cumulant formulae have a natural interpretation within the theory of homogeneous Lie groups, which we generalise for the present purpose, and are given by the log and exp map, respectively. Finally, we briefly discuss the universal role …
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