We introduce the notion of a generalized representation of a Jordan algebra with unit. The greneralized representation has the following properties:
(1) Usual representations and Jacobson representations correspond to special cases of generalized representations.
(2) Every simple Jordan algebra has infinitely many nonequivalent generalized representations.
(3) There is a one-to-one correspondence between irreducible generalized representations of a Jordan algebra A and irreducible representations of a graded Lie algebra L(A)=U_{-1}\oplus U_0\oplus U_1 corresponding to A (the Lie algebra L(A) coincides with the TKK construction when A has a unit).
The latter correspondence allows to use the theory of representations of Lie algebras to study generalized representations of Jordan algebras. In particular, one can classify irreducible generalized representations of semisimple Jordan algebras and also obtain classical results about usual representations and Jacobson representations in a simple way.
We classify two classes of B_2-graded Lie algebras which have a second compatible grading by an abelian group A: (a) graded-simple Lie algebras for A torsion-free and (b) division-A-graded Lie algebras. Our results describe the centreless cores of a class of affine reflection Lie algebras, hence apply in particular to the centreless cores of extended affine Lie algebras, the so-called Lie tori, for which we recover results of Allison-Gao and Faulkner. Our classification (b) extends a recent result of Benkart-Yoshii. Both classifications are consequences of a new description of Jordan algebras …
Let $k$ be a field of characteristic $0$. For a superspace $V=V_\bar{0}\oplus V_\bar{1}$ over $k$, we call the vector $(\dim_k V_\bar{0} ,\dim_k V_\bar{1})$ the (${\mathbb Z}_2$-)graded dimension of $V$. Let $J(D_1|D_2)$ be the free Jordan superalgebra generated by $D_1$ even generators and $D_2$ odd generators. In this paper, we study the graded dimensions of the $n$-components of $J(D_1|D_2)$ and find the connection between them and the homology of Tits-Allison-Gao Lie superalgebra of $J(D_1|D_2)$ following the method given by I.Kashuba and O.Mathieu in [KM], where they deal with the free Jo…
A $G$-grading on an algebra is called multiplicity free if each homogeneous component of the grading is 1-dimensional, where $G$ is an abelian group. We introduce skew root systems of Lie type and skew root systems of Jordan type respectively, and use them to construct multiplicity free gradings on semisimple Lie algebras and on semisimple Jordan algebras respectively. Under certain conditions the corresponding Lie (resp. Jordan) algebras are simple. Three families of skew root systems of Lie type (resp. of Jordan type) are constructed and the corresponding Lie (resp. Jordan) algebras are iden…
Let g be a simple Lie algebra of rank n over C. We show that the n-dimensional abelian ideals of a Borel subalgebra of g are limits of Jordan Lie subalgebras. Combining this with a classical result by Kostant, we show that the g-module spanned by all n-dimensional abelian Lie subalgebras of g is actually spanned by the Jordan Lie subalgebras.
For any finite-dimensional Lie algebra we introduce the notion of Jordan-Kronecker invariants, study their properties and discuss examples. These invariants naturally appear in the framework of the bi-Hamiltonian approach to integrable systems on Lie algebras and are closely related to Mischenko-Fomenko's argument shift method.
We prove that if $\mathfrak{s}$ is a solvable Lie algebra of matrices over a field of characteristic 0, and $A\in\mathfrak{s}$, then the semisimple and nilpotent summands of the Jordan-Chevalley decomposition of $A$ belong to $\mathfrak{s}$ if and only if there exist $S,N\in\mathfrak{s}$, $S$ is semisimple, $N$ is nilpotent (not necessarily $[S,N]=0$) such that $A=S+N$.
For an arbitrary representation $ρ$ of a complex finite-dimensional Lie algebra, we construct a collection of numbers that we call the Jordan-Kronecker invariants of $ρ$. Among other interesting properties, these numbers provide lower bounds for degrees of polynomial invariants of $ρ$. Furthermore, we prove that these lower bounds are exact if and only if the invariants are independent outside of a set of large codimension. Finally, we show that under certain additional assumptions our bounds are exact if and only if the algebra of invariants is freely generated.
Let $\g$ be a finite dimensional Lie algebra over a field $k$ of characteristic zero. An element $x$ of $\g$ is said to have an \emph{abstract Jordan-Chevalley decomposition} if there exist unique $s,n\in\g$ such that $x=s+n$, $[s,n]=0$ and given any finite dimensional representation $π:\g\to\gl(V)$ the Jordan-Chevalley decomposition of $π(x)$ in $\gl(V)$ is $π(x)=π(s)+π(n)$. In this paper we prove that $x\in\g$ has an abstract Jordan-Chevalley decomposition if and only if $x\in [\g,\g]$, in which case its semisimple and nilpotent parts are also in $[\g,\g]$ and are explicitly determined. We d…
We give new applications of graded Lie algebras to: identities of standard polynomials, deformation theory of quadratic Lie algebras, cyclic cohomology of quadratic Lie algebras, $2k$-Lie algebras, generalized Poisson brackets and so on.
The theories of $π$-points and modules of constant Jordan type have been a topic of much recent interest in the field of finite group scheme representation theory. These theories allow for a finite group scheme module $M$ to be restricted down and considered as a module over a space of small subgroups whose representation theory is completely understood, but still provide powerful global information about the original representation of $M$. This paper provides an extension of these ideas and techniques to study finite dimensional supermodules over a classical Lie superalgebra $\mathfrak{g} = \…
In this master's thesis, we recall the definitions and basic results for Lie superalgebras. We specify the definition for Klein graded Lie algebras and, motivated by well known results for Lie superalgebras, we prove similar results for Klein graded Lie algebras. More precisely, we state the theorems of Poincare-Birkhoff-Witt and Ado's, as well as Schur's lemma. Moreover, we present two examples of Klein graded algebra.
We provide a uniform construction of $L^2$-models for all small unitary representations in degenerate principal series of semisimple Lie groups which are induced from maximal parabolic subgroups with abelian nilradical. This generalizes previous constructions to the case of a maximal parabolic subgroup which is not necessarily conjugate to its opposite, and hence the previously used Jordan algebra methods have to be generalized to Jordan pairs. The crucial ingredients for the construction of the $L^2$-models are the Lie algebra action and the spherical vector. Working in the so-called Fourier …
We give a block decomposition of the equivariant derived category arising from a cyclically graded Lie algebra. This generalizes certain aspects of the generalized Springer correspondence to the graded setting.
Cuspidal character sheaves on graded exceptional Lie algebras: stable gradings
School of Mathematics and Statistics, University of Melbourne（墨尔本大学数学与统计学院）；Department of Mathematics and Statistics, University of Helsinki（赫尔辛基大学数学与统计系）
We call a linear operator on a vector space over a field Jordanable if it has a Jordan canonical form. An almost Abelian Lie algebra has only one non-vanishing Lie bracket, which is given by a linear operator. If the latter is Jordanable then we call the almost Abelian Lie algebra Jordanable. The existence of canonical form allows to describe important structural properties of these Lie algebras in explicit terms. Lie subalgebras, ideals, automorphisms and derivations, as well as quadratic Casimir elements are given in explicitly in a suitable basis.
We consider a fixed block for the equivariant perverse sheaves with nilpotent support in the $1$-graded ccomponent of a semisimple cyclically graded Lie algebra. We give a combinatorial parametrization of the simple objects in that block.
The aim of this paper is to study relations between regular reductive PVs with one-dimensional scalar multiplication and the structure of graded Lie algebras. We will show that the regularity of such PVs is described by an $\mathfrak{sl}_2$-triplet of a graded Lie algebra.
In this paper we introduce generalized Spencer cohomology for finite depth Z-graded Lie (super)algebras. We develop a method of finding filtered deformations of such Z-graded Lie (super)algebras based on this cohomology. As an application we determine all simple filtered deformations of certain Z-graded Lie superalgebras classified in [K3], thus completing the last step in the classification of simple infinite-dimensional linearly compact Lie superalgebras.
Späth showed that the Alperin-McKay conjecture in the representation theory of finite groups holds if the so-called inductive Alperin-McKay condition holds for all finite simple groups. In a previous article, we showed that the Bonnafé-Rouquier equivalence for blocks of finite groups of Lie type can be lifted to include automorphisms of groups of Lie type. We use our results to reduce the verification of the inductive condition for groups of Lie type to quasi-isolated blocks.
An important problem in the representation theory of affine and toroidal Lie algebras is to classify all possible irreducible integrable modules with finite dimensional weight spaces. Recently the irreducible integrable modules having finite dimensional weight spaces with non-trivial central action have been classified for a more general class of Lie algebras, namely the graded Lie tori. In this paper, we classify all the irreducible integrable modules with finite dimensional weight spaces for this graded Lie tori where the central elements act trivially. Thus we ultimately obtain all the simp…
We define a pair of symplectic Dirac operators $(D^+,D^-)$ in an algebraic setting motivated by the analogy with the algebraic orthogonal Dirac operators in representation theory. We work in the settings of $\mathbb Z/2$-graded quadratic Lie algebras $\mathfrak g=\mathfrak k+\mathfrak p$ and of graded affine Hecke algebras $\mathbb H$.
In a previous paper by the authors, we obtain the first example of a finitely freely generated simple $\mathbb Z$-graded Lie conformal algebra of linear growth that cannot be embedded into any general Lie conformal algebra. In this paper, we obtain, as a byproduct, another class of such Lie conformal algebras by classifying $\mathbb Z$-graded simple Lie conformal algebras ${\cal G}=\oplus_{i=-1}^\infty{\cal G}_i$ satisfying the following, (1) ${\cal G}_0\cong{\rm Vir}$, the Virasoro conformal algebra; (2) Each ${\cal G}_i$ for $i\ge-1$ is a ${\rm Vir}$-module of rank one. These algebras includ…
Any graded restricted simple Lie algebra of Cartan type contains a subalgebra isomorphic to the Witt algebra over a field of prime characteristic. As some analogue of study on branching rules for restricted non-classical Lie algebras, it is shown that each graded restricted simple Lie algebra of Cartan type can be decomposed into a direct sum of restricted baby Verma modules and simple modules as an adjoint module over the Witt algebra. In particular, the composition factors are precisely determined.
We generalize Baranov and Shlaka's results about bar-minimal Jordan-Lie and regular inner ideals of finite dimensional associative algebras. Let A be a finite dimensional 1-perfect associative algebras A over an algebraically closed field \mathbb{F} of characteristic p\ge0 and let A' be a subalgebra of A. We prove that for any bar-minimal Jordan-Lie inner ideal B' a A', there is a bar-minimal Jordan-Lie inner ideal of A that contains B' and if B' is regular, then is a regular inner ideal of A that contains B'. We also prove that for any strict orthogonal pair (e',f') in A', there is a strict o…
A Moore--Penrose inverse of an arbitrary complex matrix A is defined as a unique matrix A' such that AA'A=A, A'AA'=A', and AA', A'A are Hermite matrices. We show that this definition has a natural generalization in the context of shortly graded simple Lie algebras corresponding to parabolic subgroups with aura (abelian unipotent radical) in simple complex Lie groups, or equivalently in the context of simple complex Jordan pairs. We give further generalizations and applications.
We study certain "σ-commuting varieties" associated with a pair of commuting involutions of a semisimple Lie algebra $\g$. The usual commuting variety of $\g$ and commuting varieties related to one involution are particular cases of our construction. We develop a general theory of σ-commuting varieties and point out some cases, when they have especially good properties. We show that, for a special choice of commuting involutions, the σ-commuting variety is isomorphic to the commuting variety of a simple Jordan algebra. As a by-product of our theory, we show that if $J$ is the Jordan algebra of…
For any finitely generated abelian group $Q$, we reduce the problem of classification of $Q$-graded simple Lie algebras over an algebraically closed field of "good" characteristic to the problem of classification of gradings on simple Lie algebras. In particular, we obtain the full classification of finite-dimensional $Q$-graded simple Lie algebras over any algebraically closed field of characteristic $0$ based on the recent classification of gradings on finite dimensional simple Lie algebras. We also reduce classification of simple graded modules over any $Q$-graded Lie algebra (not necessari…
The Bonnafé-Rouquier equivalence can be seen as a modular analogue of Lusztig's Jordan decomposition for groups of Lie type. In this paper, we show that this equivalence can be lifted to include automorphisms of the finite group of Lie type. Moreover, we prove the existence of a local version of this equivalence which satisfies similar properties.
We develop general results on centroids of Lie algebras and apply them to determine the centroid of extended affine Lie algebras, loop-like and Kac-Moody Lie algebras, and Lie algebras graded by finite root systems.
The filiform and the quasi-filiform Lie algebras form a special class of nilpotent Lie algebras. The aim of this paper is to compute the index and provide regular vectors of this two class of nilpotent Lie algebras. we consider the graded filiform Lie algebras $L_{n}$, $Q_{n}$, the $n$-dimensional filiform Lie algebras for $n$ $<8$, also the graded quasi-filiform Lie algebras and finally Lie algebras whose nilradical is $Q_{2n}$.
We determine the Lie superalgebras over fields of characteristic zero that are graded by the root system A(n,n) of the special linear Lie superalgebra psl(n+1,n+1).
With this paper we start a programme aiming at connecting two vast scientific areas: Jordan algebras and representation theory. Within representation theory, we focus on non-compact, real forms of semisimple Lie algebras and groups as well as on the modern theory of their induced representations, in which a central role is played by the parabolic subalgebras and subgroups. The aim of the present paper and its sequels is to present a Jordan algebraic interpretations of maximal parabolic subalgebras. In this first paper, we confine ourselves to maximal parabolic subalgebras of the non-compact re…
We use fermionic representations to obtain a class of BC$_{{}_{\text N}}$-graded Lie algebras coordinatized by quantum tori with nontrivial central extensions.
We compute the cohomology with trivial coefficients of two graded infinite-dimensional Lie algebras of maximal class, give explicit formulas for their representative cocycles. Also we discuss the relations with combinatorics and representation theory.
We show that the Temperley-Lieb algebra of type $A$ and the blob algebra (also known as the Temperley-Lieb algebra of type $ B$) at roots of unity are $ \mathbb Z$-graded algebras.We moreover show that they are graded cellular algebras, thus making their cell modules, or standard modules, graded modules for the algebras.
Let g_A (respectively, g_A(μ)) be the graded multi-loop Lie algebra (respectively graded twisted multi-loop Lie algebra)" associated with the simple finite dimensional Lie algebra g over the complex field C. In this paper, we prove that irreducible integrable g_A(μ)-modules with finite dimensional weight spaces are either highest weight modules or their duals and classify the isomorphism classes of irreducible integrable g_A-modules and g_A(μ)-modules with finite dimensional weight spaces.
In this note we propose a method to classify homogeneous nilpotent elements in a real $Z_m$-graded semisimple Lie algebra $g$. Using this we describe the set of orbits of homogeneous elements in a real $Z_2$-graded semisimple Lie algebra. A classification of 4-vectors (resp. 4-forms) on $R^8$ can be given using this method.
Modules with few Jordan blocks for rank $1$ groups of Lie type and related groups
Università di Padova（帕多瓦大学）；University of Louisiana at Lafayette（路易斯安那拉斐特大学）；University of Birmingham（伯明翰大学）；Loughborough University（拉夫堡大学）；University of Manchester（曼彻斯特大学）
This paper is primarily concerned with generalized reduced Verma modules over $\mathbb{Z}$-graded modular Lie superalgebras. Some properties of the generalized reduced Verma modules and the coinduced modules are obtained. Moreover, the invariant forms on the generalized reduced Verma modules are considered. In particular, we prove that the generalized reduced Verma module is isomorphic to the mixed product for modules of $\mathbb{Z}$-graded modular Lie superalgebras of Cartan type.
We establish a Gelfand-Naimark-Segal construction which yields a correspondence between cyclic unitary representations and positive definite superfunctions of a general class of $\mathbb Z_2^n$-graded Lie supergroups.
In this note we present a symbolic pseudo-differential calculus on graded nilpotent Lie groups and, as an application, a version of the sharp Garding inequality. As a corollary, we obtain lower bounds for positive Rockland operators with variable coefficients as well as their Schwartz-hypoellipticity.
In this paper we investigate Besov spaces on graded Lie groups. We prove a Nikolskii type inequality (or the Reverse Hölder inequality) on graded Lie groups and as consequence we obtain embeddings of Besov spaces. We prove a version of the Littlewood-Paley theorem on graded Lie groups. The results are applied to obtain embedding properties of Besov spaces and multiplier theorems for both spectral and Fourier multipliers in Besov spaces on graded Lie groups. In particular, we give a number of sufficient conditions for the boundedness of Fourier multipliers in Besov spaces.
In this paper we study the Cauchy problem for diffusion equations associated to a class of strongly hypoelliptic pseudo-differential operators on graded Lie groups. To do so, we develop a global complex functional calculus on graded Lie groups in order to analyse the corresponding energy estimates. One of the main aspects of this complex functional calculus is that for the $(ρ,δ)$-Euclidean Hörmander classes we recover the standard functional calculus developed by Seeley [38]. In consequence the Gårding inequality that we prove for arbitrary graded Lie groups absorbs the historical 1953's ineq…
