Emil Sköldberg
The generating function of the Betti numbers of the Heisenberg Lie algebra over a field of characteristic 2 is calculated using discrete Morse theory.
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We provide a systematic approach to obtain formulas for characters and Kostant ${\mathfrak u}$-homology groups of the oscillator modules of the finite dimensional general linear and ortho-symplectic superalgebras, via Howe dualities for infinite dimensional Lie algebras. Specializing these Lie superalgebras to Lie algebras, we recover, in a new way, formulas for Kostant homology groups of unitarizable highest weight representations of Hermitian symmetric pairs. In addition, two new reductive dual pairs related to the above-mentioned ${\mathfrak u}$-homology computation are worked out.
In this note, we compute the homology with trivial coefficients of Lie algebras of generalized Jacobi matrices of type $B, C$ and $D$ over an associative unital $k$-algebra with $k$ being a field of characteristic $0$.
Using Howe duality we compute explicitly Kostant-type homology groups for a wide class of representations of the infinite-dimensional Lie superalgebra $\hat{\frak{gl}}_{\infty|\infty}$ and its classical subalgebras at positive integral levels. We also obtain Kostant-type homology formulas for the Lie algebra $ widehat{\frak{gl}}_\infty$ at negative integral levels. We further construct resolutions in terms of generalized Verma modules for these representations.
We revisit a recent paper of Fialowski and Iohara. They compute the homology of the Lie algebra $\mathfrak{gl}(\infty,R)$ for $R$ an associative unital algebra over a field of characteristic zero. We explain how to obtain essentially the same results by a completely different method.
The u-homology formulas for unitarizable modules at negative levels over classical Lie algebras of infinite rank of types gl(n), sp(2n) and so(2n) are obtained. As a consequence, we recover the Enright's formulas for three Hermitian symmetric pairs of classical types (SU(p; q); SU(p) X SU(q)), (Sp(2n);U(n)) and (SO*(2n);U(n)).
This paper aims to characterize the minimal dimensions and super-dimensions of faithful representations for the Heisenberg Lie superalgebras over an algebraically closed field of characteristic zero.
We study $\mathbb Z$-graded modules of nonzero level with arbitrary weight multiplicities over Heisenberg Lie algebras and the associated generalized loop modules over affine Kac-Moody Lie algebras. We construct new families of such irreducible modules over Heisenberg Lie algebras. Our main result establishes the irreducibility of the corresponding generalized loop modules providing an explicit construction of many new examples of irreducible modules for affine Lie algebras. In particular, to any function $ϕ:\mathbb N\rightarrow \{\pm\}$ we associate a $ϕ$-highest weight module over the Heisen…
The Heisenberg algebra of a vector space and Hochschild homology
本文针对非交换光滑真簇，在对称商层的总霍奇同调上构造海森伯代数的三个等价作用，还为普通交换簇给出基于对应关系的第四个几何描述。
Restricted Cohomology of Heisenberg Lie Algebras
本文在正特征代数闭域上，利用普通上同调计算了海森堡李代数的受限1-和2-上同调空间。
On the Cohomology of Restricted Heisenberg Lie Algebras
本文分类了特征为正的域上海森堡李代数的所有互不同构受限结构，并利用普通上同调计算其受限上同调，给出受限一维中心扩张的显式公式。
Irreducible modules for map Heisenberg-Virasoro Lie algebras
该研究聚焦映射海森堡-Virasoro李代数的不可约模，完成了其不可约Harish-Chandra模的完整分类，还分类了在Witt代数泛包络代数零次部分上限制为秩1自由模的非权不可约模。
We define a Fourier transform and a convolution product for functions and distributions on Heisenberg--Clifford Lie supergroups. The Fourier transform exchanges the convolution and a pointwise product, and is an intertwining operator for the left regular representation. We generalize various classical theorems, including the Paley--Wiener--Schwartz theorem, and define a convolution Banach algebra.
The Heisenberg-Virasoro Lie conformal superalgebra
本文引入 Heisenberg-Virasoro 李共形超代数并定义 Ramond 型超 Heisenberg-Virasoro 代数，构造了一类包含最高权模、Whittaker 模及高阶 Whittaker 模的单模，并给出同构于 Neveu-Schwarz 型的子代数。
We calculate the homology of three families of 2-step nilpotent Lie (super)algebras associated with the symplectic, orthogonal, and general linear groups. The symplectic case was considered by Getzler and the main motivation for this work was to complete the calculations started by him. In all three cases, these algebras can be realized as the nilpotent radical of a parabolic subalgebra of a simple Lie algebra, and our first approach relies on a theorem of Kostant, but is otherwise elementary and involves combinatorics of Weyl groups and partitions which may be of independent interest. Our sec…
We study the derived representation scheme DRep_n(A) parametrizing the n-dimensional representations of an associative algebra A over a field of characteristic zero. We show that the homology of DRep_n(A) is isomorphic to the Chevalley-Eilenberg homology of the current Lie coalgebra gl_n^*(C) defined over a Koszul dual coalgebra of A. We extend this isomorphism to representation schemes of Lie algebras: for a finite-dimensional reductive Lie algebra g, we define the derived affine scheme DRep_g(a) parametrizing the representations (in g) of a Lie algebra a; we show that the homology of DRep_g(…
For modular Lie superalgebras, new notions are introduced: Divided power homology and divided power cohomology. For illustration, we give presentations (in terms of analogs of Chevalley generators) of finite dimensional Lie (super)algebras with indecomposable Cartan matrix in characteristic 2 (and in other characteristics for completeness of the picture). We correct the currently available in the literature notions of Chevalley generators and Cartan matrix in the modular and super cases, and an auxiliary notion of the Dynkin diagram. In characteristic 2, the defining relations of simple classi…
Let $G$ be an irreducible Hermitian Lie group and $D=G/K$ its bounded symmetric domain in $\mathbb C^d$ of rank $r$. Each $γ$ of the Harish-Chandra strongly orthogonal roots $\{γ_1, \cdots, γ_r\}$ defines a Heisenberg parabolic subgroup $P=MAN$ of $G$. We study the principal series representations $\Ind_P^G(1\otimes e^ν\otimes 1)$ of $G$ induced from $P$. We find the complementary series, reduction points, and unitary subrepresentations in this family of representations.
Cyclic homology of Jordan superalgebras and related Lie superalgebras
本文研究Jordan超代数的循环同调与其Tits-Kantor-Koecher Lie超代数二阶上同调的关系，重点处理Kantor二重情形，并将结果用于任意系数环上Hamilton型与接触型Lie超代数的二阶上同调和泛中心扩张计算。
Heisenberg parabolically induced representations of Hermitian Lie groups, Part II: Next-to-minimal representations and branching rules
本文证明由海森堡型极大抛物子群诱导的球面表示族端点处的可酉化子表示是次极小表示，并研究了其在SO₀(2,n)和E₆₍₋₁₄₎中的分支规则。
We consider algebras over a field $k$ of characteristic zero. The article is concerned with the isomorphism of graded vectorspaces \[ H(\gl(A))\iso\wedge (HC(A)[-1]) \] between the Lie algebra homology of matrices and the free graded commutative algebra on the cyclic homology of the $k$-algebra $A$, shifted down one degree. For unital algebras this isomorphism is a classical result obtained by Loday and Quillen and independently by Tsygan. For $H$-unital algebras, it is known to hold too, as is that the proof follows from results of Hanlon's. However, to our knowledge, the proof is not immedia…
The Broadhurst-Kreimer (BK) conjecture describes the Hilbert series of a bigraded Lie algebra A related to the multizeta values. Brown proposed a conjectural description of the homology of this Lie algebra (homological conjecture (HC)), and showed it implies the BK conjecture. We show that a part of HC is equivalent to a presentation of A, and that the remaining part of HC is equivalent to a weaker statement. Finally, we prove that granted the first part of HC, the remaining part of HC is equivalent to either of the following equivalent statements: (a) the vanishing of the third homology group…
We classify connected Lie groups which are locally isomorphic to generalized Heisenberg groups. For a given generalized Heisenberg group $N$, there is a one-to-one correspondence between the set of isomorphism classes of connected Lie groups which are locally isomorphic to $N$ and a union of certain quotients of noncompact Riemannian symmetric spaces.
We introduce nilpotent k-ary Lie algebras including analogues of Heisenberg Lie algebras and free nilpotent Lie algebras. We study homology of k-ary nilpotent Lie algebras by using a modification of Chevalley-Eilenberg complex. For some classes of nilpotent k-ary Lie algebras and in particular Heisenberg k-ary Lie algebras we give explicit formulas for Betti numbers. Representation stability of free nilpotent k-ary Lie algebras is proven and lower bounds for Betti numbers are described by Schur modules. We also verify that toral rank conjecture holds for the classes we studied. Moreover, for 2…
Representations of non-finitely graded Heisenberg-Virasoro type Lie algebras
该研究构造并研究了与Heisenberg-Virasoro型李代数相关的非有限分次李代数$\u27e8HV\u27e9(a,b;\u03b5)$，通过组合技术完成其秩1自由$\u27e8U(\u27e8h\u27e9)\u27e9$-模的分类，明确了模的多样性、单性及同构类。
Lie algebra homology with coefficients tensor products of the adjoint representation in relative polynomial degree 2
本文计算了自由李代数在adjoint表示张量积系数下的相对次数2的上同调，证实了Gadish和Hainaut的猜想。
The trace (or zeroth Hochschild homology) of Khovanov's Heisenberg category is identified with a quotient of the algebra W_{1+\infty}. This induces an action of W_{1+\infty} on symmetric functions.
Complete Structural Analysis of $q$-Heisenberg Algebras: Homology, Rigidity, Automorphisms, and Deformations
本文研究了$q$-Heisenberg代数的结构性质，包括同调、刚性、自同构与变形，揭示其在量子变形和表示论中的重要贡献。
Let K be a compact Lie group. We compute the abelianization of the Lie algebra of equivariant vector fields on a smooth K-manifold X. We also compute the abelianization of the Lie algebra of strata preserving smooth vector fields on the quotient X/K.
Unitary and Nonunitary Representations of the Heisenberg-Weyl Lie Algebra
本文研究了海森堡-魏尔李代数的幺正与非幺正表示，分析了张量积和不可分解表示的构造与性质。
Presented is a structure theorem for the Leibniz Homology, HL_*, of an Abelian extension of a simple real Lie algebra g. As applications, results are stated for affine extensions of the classical Lie algebras sl_n(R), so_n(R), and sp_n(R). Furthermore, HL_*(h) is calculated when h is the Lie algebra of the Poincare group as well as the Lie algebra of the affine Lorentz group. The general theorem identifies all of these in terms of g-invariants.
For a finite dimensional Lie algebra $\g$ over a field $\k$ of characteristic zero, the $μ$-function (respectively $μ_{nil}$-function) is defined to be the minimal dimension of $V$ such that $\g$ admits a faithful representation (respectively a faithful nilrepresentation) on $V$. Let $\h_m$ be the Heisenberg Lie algebra of dimension $2m + 1$ and let $\mathfrak{a}_n$ be the abelian Lie algebra of dimension $n$. The aim of this paper is to compute $μ(\h_m \oplus \mathfrak{a}_n)$ and $μ_{nil}(\h_m \oplus \mathfrak{a}_n)$ for all $m,n \in \mathbb{N}$.
In this note we compute the homology of the Lie algebra $\mathfrak{gl}(\infty,R)$ where $R$ is an associative unital $k$-algebra which is used in higher dimensional soliton theory. When $k$ is a field of characteristic $0$, our result justifies an old result of Feigin and Tsygan appeared in 1983. The special case when $R=k$ is the complex number field $\mathbb{C}$ appeared first in soliton theory.
For the exceptional finite-dimensional modular Lie superalgebras $\mathfrak{g}(A)$ with indecomposable Cartan matrix $A$, and their simple subquotients, we computed non-isomorphic Lie superalgebras constituting the homologies of the odd elements with zero square. These homologies are~key ingredients in the Duflo--Serganova approach to the representation theory. There were two definitions of defect of Lie superalgebras in the literature with different ranges of application. We suggest a third definition and an easy-to-use way to find its value. In positive characteristic, we found out one more …
$C^*$-rigidity of the Heisenberg group
本文证明海森堡群可由其群 $C^*$-代数与其他连通单连通李群区分，关键步骤是利用余伴随轨道的拓扑性质在可解李群中刻画幂零李群。
We study the hairy graph homology of a cyclic operad; in particular we show how to assemble corresponding hairy graph cohomology classes to form cocycles for ordinary graph homology, as defined by Kontsevich. We identify the part of hairy graph homology coming from graphs with cyclic fundamental group as the dihedral homology of a related associative algebra with involution. For the operads Comm, Assoc and Lie we compute this algebra explicitly, enabling us to apply known results on dihedral homology to the computation of hairy graph homology. In addition we determine the image in hairy graph …
2-local derivation is a generalized derivation for a Lie algebra, which plays an important role to the study of local properties of the structure of the Lie algebra. In this paper, we prove that every 2-local derivation on the twisted Heisenberg-Virasoro algebra is a derivation.
Irreducible representations of the twisted Heisenberg-Virasoro algebra of Hom type
本文针对Hom-李代数推广背景，研究Hom型扭Heisenberg-Virasoro代数的不可约表示，提出可由其诱导李代数的不可约表示诱导得到的方法，并构造了若干类该代数的不可约表示。
Given a Lie algebra $\mathfrak{g}$ over a field of characteristic zero $k$, let $μ(\mathfrak{g})=\min\{\dim π: π\text{is a faithful representation of}\mathfrak{g}\}$. Let $\mathfrak{h}_{m}$ be the Heisenberg Lie algebra of dimension $2m+1$ over $k$ and let $k[t]$ be the polynomial algebra in one variable. Given $m\in\mathbb{N}$ and $p\in k[t]$, let $\mathfrak{h}_{m,p}=\mathfrak{h}_m\otimes k[t]/(p)$ be the current Lie algebra associated to $\mathfrak{h}_m$ and $k[t]/(p)$, where $(p)$ is the principal ideal in $k[t]$ generated by $p$. In this paper we prove that $ mu(\mathfrak{h}_{m,p}) = m °p …
Let K be a connected compact Lie group, and G be its complexification. The homology of the based loop group ΩK with integer coefficients is naturally a \ZZ-Hopf algebra. After possibly inverting 2 or 3, we identify H_*(ΩK,\ZZ) with the Hopf algebra of algebraic functions on B^\vee_e, where B^\vee is a Borel subgroup of the Langlands dual group scheme G^\vee of G and B^\vee_e is the centralizer in B^\vee of a regular nilpotent element e\in\Lie B^\vee. We also give a similar interpretation for the equivariant homology of ΩK under the maximal torus action.
The Adler Kostant Symes [A-K-S] scheme is used to describe mechanical systems for quadratic Hamiltonians of $\mathbb R^{2n}$ on coadjoint orbits of the Heisenberg Lie group. The coadjoint orbits are realized in a solvable Lie algebra $\mathfrak g$ that admits an ad-invariant metric. Its quadratic induces the Hamiltonian on the orbits, whose Hamiltonian system is equivalent to that one on $\mathbb R^{2n}$. This system is a Lax pair equation whose solution can be computed with help of the Adjoint representation. For a certain class of functions, the Poisson commutativity on the coadjoint orbits …
Let $G$ be a compact connected Lie group, and let $\mathrm{Hom}(\mathbb{Z}^m,G)$ be the space of pairwise commuting $m$-tuples in $G$. We study the problem of which primes $p$ $\mathrm{Hom}(\mathbb{Z}^m,G)_1$, the connected component of $\mathrm{Hom}(\mathbb{Z}^m,G)$ containing the element $(1,\ldots,1)$, has $p$-torsion in homology. We will prove that $\mathrm{Hom}(\mathbb{Z}^m,G)_1$ for $m\ge 2$ has $p$-torsion in homology if and only if $p$ divides the order of the Weyl group of $G$ for $G=SU(n)$ and some exceptional groups. We will also compute the top homology of $\mathrm{Hom}(\mathbb{Z}^…
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