Eugen Paal
Generalized Lie-Cartan theorem for linear birepresentations of an analytic Moufang loop is considered. The commutation relations of the generators of the birepresentation were found. In particular, the Lie algebra of the multiplication group of the birepresentation is explicitly given.
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The continuous Moufang loops are characterized as the algebraic systems where the associativity law is perturbed minimally. The minimal perturbation of associativity is characterized by the first- order partial differential equations, which in a natural way generalize the Lie and Maurer-Cartan equations from the theory of Lie groups.
The differential equations for a continuous birepresentation of a local analytic Moufang loop are established. The commutation relations for the infinitesimal operators of the representation are found. These commutation relations can be seen as a (minimal) generalization of the Maurer-Cartan equations and do not depend on the particular birepresentation.
Integrability of generalized Lie equations of continuous Moufang transformations is inquired.
Integrability of generalized Lie equations of a local analytic Moufang loop is inquired.
It is shown how integrability of the generalized Lie equations of continous Moufang transformatiosn is related to the reductivity conditions and Sagle-Yamaguti identity.
It is shown how integrability of the generalized Lie equations of a local analytic Moufang loop is related to the reductivity conditions and Sagle-Yamaguti identity.
Triple closure of the infinitesimal translations of an analytic Moufang loop is inquired. This property is equivalent to reductivity and relates Mal'tsev algebras to the Lie triple systems.
Reductivity in the Ma'tsev algebras is inquired. This property relates the Mal'tsev algebras to the general Lie triple systems.
The purpose of this paper is to answer the question whether it is possible to realize simultaneously the relations $N_{α,β}=-N_{-α,-β}$, $N_{α,β}=N_{β,-α-β}=N_{-α-β,α}$ and $N_{α,β}N_{-α,-β}=-\frac{1}{2}q(p+1)\langleα,H_α\rangle$ by the structure constants of the Lie algebra ${\mathfrak g}_2$. We show that if the structure constants obey the first relation, the three last ones are violated, and vice versa. Contrary to the second case, the first one uses the Cartan matrix elements to derive the structure constants in the form of $\langleβ,H_α\rangle$. The commutation relations corresponding to …
In this paper, we study pre-Lie analogues of Poisson-Nijenhuis structures and introduce ON-structures on bimodules over pre-Lie algebras. We show that an ON-structure gives rise to a hierarchy of pairwise compatible O-operators. We study solutions of the strong Maurer-Cartan equation on the twilled pre-Lie algebra associated to an O-operator, which gives rise to a pair of ON-structures which are naturally in duality. We show that KVN-structures and HN-structures on a pre-Lie algebra g are corresponding to ON-structures on the bimodule $(\mathfrak g^*;\mathrm{ad}^*,-R^*)$, and $KVΩ$-structures …
A Vogan diagram is a Dynkin diagram with a Cartan involution of twisted affine superlagebras based on maximally compact Cartan subalgebras. This article construct the Vogan diagrams of twisted affine superalgebras. It is a part of completion of classification of vogan diagrams to superalgebras cases.
We derive explicit formulas for the inverses of the Cartan matrices of the simple Lie algebras and the basic classical Lie superalgebras, as well as for their infinite generalizations.
This paper describes finite dimensionall irreducible representations of both twisted and untwisted Cartan map Lie superalgebras.
For the two Cartan type S subalgebras of the Witt algebra $\W_n$, called Lie algebras of divergence-zero vector fields, we determine all module structures on the universal enveloping algebra of their Cartan subalgebra $\h_n$. We also give all submodules of these modules.
Finite dimensional modular Lie superalgebras over algebraically closed fields with indecomposable Cartan matrices are classified under some technical, most probably inessential, hypotheses. If the Cartan matrix is invertible, the corresponding Lie superalgebra is simple otherwise the quotient of the derived Lie superalgebra modulo center is simple (if its rank is greater than 1). Eleven new exceptional simple modular Lie superalgebras are discovered. Several features of classic notions, or notions themselves, are clarified or introduced, e.g., Cartan matrix, several versions of restrictedness …
We give an overview of our effort to introduce (dual) semicanonical bases in the setting of symmetrizable Cartan matrices.
In this paper, we study non-abelian extensions of 3-Lie algebras through Maurer-Cartan elements. We show that there is a one-to-one correspondence between isomorphism classes of non-abelian extensions of 3-Lie algebras and equivalence classes of Maurer-Cartan elements in a DGLA. The structure of the Leibniz algebra on the space of fundamental objects is also analyzed.
We establish the conjugacy of Cartan subalgebras for generic Lie tori "of type A". This is the only conjugacy problem of Lie tori related to Extended Affine Lie Algebras that remained open.
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.
In this review article, first we give the concrete formulas of representations and cohomologies of associative algebras, Lie algebras, pre-Lie algebras, Leibniz algebras and 3-Lie algebras and some of their strong homotopy analogues. Then we recall the graded Lie algebras and graded associative algebras that characterize these algebraic structures as Maurer-Cartan elements. The corresponding Maurer-Cartan element equips the graded Lie or associative algebra with a differential. Then the deformations of the given algebraic structures are characterized as the Maurer-Cartan elements of the result…
Recently Alberto Elduque listed all simple and graded modulo 2 finite dimensional Lie algebras and superalgebras whose odd component is the spinor representation of the orthogonal Lie algebra equal to the even component, and discovered one exceptional such Lie superalgebra in characteristic 5. For this Lie superalgebra all inequivalent Cartan matrices (in other words, inequivalent systems of simple roots) are listed together with defining relations between analogs of its Chevalley generators.
In this short note, we give a description of semisimple orbits in the restricted Cartan type Lie algebras $W, S, H$.
We completely describe presentations of Lie superalgebras with Cartan matrix if they are simple Z-graded of polynomial growth. Such matrices can be neither integer nor symmetrizable. There are non-Serre relations encountered. In certain cases there are infinitely many relations. Our results are applicable to the Lie algebras with the same Cartan matrices as the Lie superalgebras considered.
Let $L$ be a restricted Cartan type Lie algebra over an algebraically closed field $k$ of characteristic $p>3$, and let $G$ denote the automorphism group of $L$. We prove that there are no nontrivial invariants of $L^*$ under the coadjoint action of $G$, i.e., $k[L^*]^G=k$. This property characterises the Cartan type algebras among the restricted simple Lie algebras.
Let $\Lie{g}$ be a simple complex Lie algebra and $\Lie{h}$ a Cartan subalgebra. In this article we explain how to obtain the principal basis of $\Lie{h}$ starting form a set of generators $\{p_1,...,p_r\}$,$r=\rank(\Lie{g})$, of the invariants polynomials $\Sgg$. For each invariant polynomial $p$, we define a $G$-equivariant map $Dp$ form $\Lie{g}$ to $\Lie{g}$. We show that the Gram-Schmidt orthogonalization of the elements $\{Dp_1(ρ^\vee), ... Dp_r(ρ^\vee) \}$ gives the principal basis of $\Lie{h}$. Similarly the orthogonalization of the elements $\{Dp_1(ρ), >... Dp_r(ρ) \}$ produces the pr…
That finite-dimensional simple Lie algebras over the complex numbers can be classified by means of purely combinatorial and geometric objects such as Coxeter-Dynkin diagrams and indecomposable irreducible root systems, is arguably one of the most elegant results in mathematics. The definition of the root system is done by fixing a Cartan subalgebra of the given Lie algebra. The remarkable fact is that (up to isomorphism) this construction is independent of the choice of the Cartan subalgebra. The modern way of establishing this fact is by showing that all Cartan subalgebras are conjugate. For …
Root-reductive Lie algebras are direct limits of finite-dimensional reductive Lie algebras under injections which preserve the root spaces. It is known that a root-reductive Lie algebra is a split extension of an abelian Lie algebra by a direct sum of copies of finite-dimensional simple Lie algebras as well as copies of the three simple infinite-dimensional root-reductive Lie algebras sl_infty, so_infty, and sp_infty. As part of a structure theory program for root-reductive Lie algebras, Cartan subalgebras of the Lie algebra gl_infty were introduced and studied in a paper of Neeb and Penkov. I…
Let $\mathbb{k}$ be an algebraically closed field of characteristic $ p>0. $ In this short note, we illustrate a class of Lie superalgebras over $ \mathbb{k} $ such that the category of restricted supermodules is of one block. As an application, if $ p>3 $ and $ \mathfrak{g} $ is a graded restricted Cartan type Lie superalgebra of type W, S and H, then the category of restricted $ \mathfrak{g} $ supermodules is of one block.
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…
The roots of exceptional modular Lie superalgebras with Cartan matrix
本研究针对带不可分解Cartan矩阵的例外李超代数，给出对应最多两两正交迷向奇单根的不等价Cartan矩阵的根列表与Chevalley基，工具为SuperLie代码与Python。
Using tensor categories, we present new constructions of several of the exceptional simple Lie superalgebras with integer Cartan matrix in characteristic $p = 3$ and $p = 5$ from the complete classification of modular Lie superalgebras with indecomposable Cartan matrix and their simple subquotients over algebraically closed fields by Bouarroudj, Grozman, and Leites in 2009. Specifically, let $\mathbfα_p$ denote the kernel of the Frobenius endomorphism on the additive group scheme $\mathbb{G}_a$ over an algebraically closed field of characteristic $p$. The Verlinde category $\mathrm{Ver}_p$ is …
The analogs of Chevalley generators are offered for simple (and close to them) Z-graded complex Lie algebras and Lie superalgebras of polynomial growth without Cartan matrix. We show how to derive the defining relations between these generators and explicitly write them for a "most natural" ("distinguished" in terms of Penkov and Serganova) system of simple roots. The results are given mainly for Lie superalgebras whose component of degree zero is a Lie algebra (other cases being left to the reader). Observe presentations of exceptional Lie superalgebras and Lie superalgebras of hamiltonian ve…
Actions of monoidal categories and representations of Cartan type Lie algebras
本文通过交叉同态建立幺半范畴作用，统一构造Lie-Rinehart代数及Leibniz对的弱表示与容许表示，并引入交叉同态上同调理论以研究其形变，同时为广义Witt代数等提供新表示。
Let $A_{m,n}$ be the tensor product of the polynomial algebra in $m$ even variables and the exterior algebra in $n$ odd variables over the complex field $\C$, and the Witt superalgebra $W_{m,n}$ be the Lie superalgebra of superderivations of $A_{m,n}$. In this paper, we classify the non-trivial simple bounded weight $W_{m,n}$ modules with respect to the standard Cartan algebra of $W_{m,n}$. Any such module is a simple quotient of a tensor module $F(P,L(V_1\otimes V_2))$ for a simple weight module $P$ over the Weyl superalgebra $\mathcal K_{m,n}$, a finite-dimensional simple $\gl_m$-module $V_1…
Let $G$ be a simply connected Lie group with Lie algebra $\mathfrak{g}$. We show that the following categories are naturally equivalent. The category $\mathsf{Mod}(C(G))$, of sufficiently smooth modules over the DG-algebra of singular chains on $G$. The category $\mathsf{Rep}(Tg)$ of representations of the DG-Lie algebra $T\mathfrak{g}$, which is universal for the Cartan relations. This equivalence extends the correspondence between representations of $G$ and representations of $\mathfrak{g}$. In a companion paper we show that in the compact case, the equivalence can be extended to an $\mathsf…
We establish the conjugacy of Cartan subalgebras for extended affine Lie algebras whose centreless core is "of type A", i.e., matrices over a quantum torus Q whose trace lies in the commutator space of Q. This settles the last outstanding part of the conjugacy problem for Extended Affine Lie Algebras that remained open.
Simple prolongs of the non-positive parts of graded Lie algebras with Cartan matrix in characteristic 2
针对特征2情形，研究李代数延拓方法的相关步骤，发现新例外单李代数系列及其隐藏超对称性，揭示部分单李代数与李超代数的去超化关联，指出分类复杂度超预期。
As a sequel to [Duan-Shu-Yao], we introduce here a category $\mathscr{LC}$ arising from the BGG category $\mathcal{O}$ defined in [Duan-Shu-Yao] for Lie algebras of polynomial vector fields. The objects of $\mathscr{LC}$ are so-called Lie-Cartan modules which admit both Lie-module structure and compatible $R$-module structure ($R$ denotes the corresponding polynomial ring). This terminology is natural, coming from affine connections in differential geometry through which the structure sheaves in topology and the vector fields in geometry are integrated for differential manifolds. In this paper…
Inverses of Cartan matrices of Lie algebras and Lie superalgebras
本文计算了多类李（超）代数不可分解Cartan矩阵的逆，发现三个新现象，同时纠正了前人对相关结果的误解并补充了更早期的文献依据。
In the present work the properties of Cartan subalgebras and their connection with regular elements in finite dimensional Lie algebras are extended to the case of Leibniz algebras. It is shown that Cartan subalgebras and regular elements of a Leibniz algebra correspond to Cartan subalgebras and regular elements of a Lie algebra by a natural homomorphism. Conjugacy of Cartan subalgebras of Leibniz algebras is shown.
We will construct standard pentads which are analogues of Cartan subalgebras, and moreover, we will study graded Lie algebras corresponding to these standard pentads. We call such pentads pentads of Cartan type and describe them by two positive integers and three matrices. The structures of their corresponding Lie algebras are related with contragredient Lie algebras.
Let $L$ be one of the finite dimensional Lie algebras $W_n({\bf m}),$ $S_n({\bf m}),$ $ H_n({\bf m})$ of Cartan type over an algebraically closed field of prime characteristic $p>0.$ For an elements $F$ of the symmetrical algebra $S(L)$ we found necessary and sufficient condition in order to the element $ad(\partial_1)^{p^{m_1}-1} ad(\partial_2)^{p^{m_2}-1}... ad(\partial_n)^{p^{m_n}-1}(F)$ belongs to the symmetrical invariants algebra $S(L)^L.$ Also, for $p=3,5$ the algebra of symmetrical invariants $S(H_2)^{H_2}$ is calculated in explicit way.
The generalized Cartan type $\mathbf{S}$ Lie algebras in char 0 with the Lie bialgebra structures involved are quantized, where the Drinfel'd twist we used is proved to be a variation of the Jordanian twist. As the passage from char 0 to char p, their quantization integral forms are given. By the modular reduction and base changes, we obtain certain quantizations of the restricted universal enveloping algebra $\mathbf u(\mathbf{S}(n;\underline{1}))$ (for the Cartan type simple modular restricted Lie algebra $\mathbf{S}(n;\underline{1})$ of $\mathbf{S}$ type). They are new Hopf algebras of trun…
This paper is a continuation of earlier work on generators of simple Lie algebras in arbitrary characteristic (see arXiv:0708.1711). We show that, in contrast to classical Lie algebras, simple graded Lie algebras of Cartan type S,H or K never enjoy the 'one-and-a-half generation' property. The methods rely on a study of centralisers in Cartan type Lie algebras.
Representations of affine Nappi-Witten Lie algebras over polynomial algebras
本文分类了Nappi-Witten李代数H4及其仿射代数上的秩一Cartan自由模，给出了不可约充要条件，并应用于仿射-Virasoro Nappi-Witten李代数。
In this paper we initiate the study of the maximal subalgebras of exceptional simple classical Lie algebras \g over algebraically closed fields k of positive characteristic p, such that the prime characteristic is good for \g. In this paper we deal with what is surely the most unnatural case; that is, where the maximal subalgebra in question is a simple subalgebra of non-classical type. We show that only the first Witt algebra can occur as a subalgebra of \g and give explicit details on when it may be maximal in \g.
Let $ W(n) $ be Jacobson-Witt algebra over algebraic closed field $ \mathbb{K} $ with positive characteristic $ p>2. $ It is difficult to classify all Borel subalgebras of $ W(n) $ or non-classical restricted simple Lie algebras. The present paper and \cite{S7} study two kinds of subalgebras which are easily to understand and highly related to Borel subalgebras. In \cite{S7}, the last author investigates a class of special Borel subalgebras of $W(n)$ which is called homogeneous Borel subalgebras. The present paper focuses on subalgebras of $ W(n) $ which are related to Borel subalgebras such t…
Extensions of Restricted Simple Modules for Restricted Lie Algebras of Cartan Type
本文确定了特征 p≥5 的 Cartan 型受限单李代数受限单模的一阶扩张群，回答了 Benkart-Feldvoss 问题，并通过有限受限上同调复形计算扩张，给出了例外模的闭式公式，且确定了特征七下 S(3;1)^(1) 的箭图。
Let $G$ be a simply connected Lie group with Lie algebra $\mathfrak{g}$ and denote by $\mathrm{C}_{\bullet}(G)$ the DG Hopf algebra of smooth singular chains on $G$. In a companion paper it was shown that the category of sufficiently smooth modules over $\mathrm{C}_{\bullet}(G)$ is equivalent to the category of representations of $\mathbb{T} \mathfrak{g}$, the DG Lie algebra which is universal for the Cartan relations. In this paper we show that, if $G$ is compact, this equivalence of categories can be extended to an $\mathsf{A}_{\infty}$-quasi-equivalence of the corresponding DG categories. A…
Cartanification of contragredient Lie superalgebras
本文定义反变李超代数的 Cartan 化 $B^W$，推广张量层级代数构造，并证明其与同广义 Cartan 矩阵给出的 $W$ 同构的条件。
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 …
We give an analog of a Chevalley-Serre presentation for the Lie superalgebras W(n) and S(n) of Cartan type. These are part of a wider class of Lie superalgebras, the so-called tensor hierarchy algebras, denoted W(g) and S(g), where g denotes the Kac-Moody algebra A_r, D_r or E_r. Then W(A_{n-1}) and S(A_{n-1}) are the Lie superalgebras W(n) and S(n). The algebras W(g) and S(g) are constructed from the Dynkin diagram of the Borcherds-Kac-Moody superalgebras B(g) obtained by adding a single grey node (representing an odd null root) to the Dynkin diagram of g. We redefine the algebras W(A_r) and …
Over algebraically closed fields of characteristic p>2, prolongations of the simple finite dimensional Lie algebras and Lie superalgebras with Cartan matrix are studied for certain simplest gradings of these algebras. Several new simple Lie superalgebras are discovered, serial and exceptional, including superBrown and superMelikyan superalgebras. Simple Lie superalgebras with Cartan matrix of rank 2 are classified.
In this article we analyze the quotients of the maximal compact subalgebras of the split real Kac-Moody algebras of the En series resulting from the generalized spin representations introduced in part 1. It turns out that these quotients satisfy a Cartan-Bott periodicity. Our findings are also meaningful in the finite-dimensional cases of A2 + A1, A4, D5, E6, E7, E8, where it turns out that the generalized spin representation is injective. Consequently the observed Cartan-Bott periodicity provides a structural explanation for the seemingly sporadic isomorphism types of the maximal compact Lie …
Parabolic BGG categories and their block decomposition for Lie superalgebras of Cartan type
本文研究复数域上Cartan型阶化李超代数的抛物BGG范畴，证明仅有极小和极大两类抛物子代数，并将问题简化为极小抛物范畴，进而证明射影覆盖存在性以建立块理论，最终完成块的分类及不可分解倾斜模与射影模的特征标公式计算。
The finite dimensional simple modular Lie algebras with Cartan matrix cannot be deformed if the characteristic p of the ground field is equal to 0 or greater than 3. If p=3, the orthogonal Lie algebra o(5)is one of the two simple modular Lie algebras with Cartan matrix that have deformations (the Brown algebras br(2; a) are among these 10-dimensional deforms and hence are not counted separately); the 29-dimensional Brown algebra br(3) is the only other simple Lie algebra with Cartan matrix that has deformations. Kostrikin and Kuznetsov described the orbits (isomorphism classes) under the actio…
Characters for extended affine Lie algebras; a combinatorial approach
本文用组合方法研究扩展仿射李代数的特征标与嘉当自同构，证明几乎所有此类代数中有限阶嘉当自同构都是对角的，且对应组合映射为特征标。
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