A coplactic class in the symmetric group S_n consists of all permutations in S_n with a given Schensted Q-symbol, and may be described in terms of local relations introduced by Knuth. Any Lie element in the group algebra of S_n which is constant on coplactic classes is already constant on descent classes. As a consequence, the intersection of the Lie convolution algebra introduced by Patras and Reutenauer and the coplactic algebra introduced by Poirier and Reutenauer is the Solomon descent algebra.
We derive a general result about commuting actions on certain objects in braided rigid monoidal categories. This enables us to define an action of the Brauer algebra on the tensor space $V^{\otimes k}$ which commutes with the action of the orthosymplectic Lie superalgebra $\spo(V)$ and the orthosymplectic Lie color algebra $\spo(V,β)$. We use the Brauer algebra action to compute maximal vectors in $V^{\otimes k}$ and to decompose $V^{\otimes k}$ into a direct sum of submodules $T^λ$. We compute the characters of the modules $T^λ$, give a combinatorial description of these characters in terms o…
In this paper, we mainly study some properties of elementary n-Lie algebras, and prove some necessary and sufficient conditions for elementary n-Lie algebras, we also give the relations between elementary n-algebras and E-algebras.
We introduce bijections between generalized type $A_n$ noncrossing partitions (that is, associated to arbitrary standard Coxeter elements) and fully commutative elements of the same type. The latter index the diagram basis of the classical Temperley-Lieb algebra, while for each choice of standard Coxeter element the corresponding noncrossing partitions also index a basis, given by the images in the Temperley-Lieb algebra of the simple elements of the dual Garside structure (associated to this choice of standard Coxeter element) of the Artin braid group on $n+1$ strands. We then show that our b…
Let L be a restricted Lie algebra over a field of characteristic p > 2 and denote by u(L) its restricted enveloping algebra. We determine the conditions under which the set of symmetric elements of u(L) with respect to the principal involution is Lie solvable, Lie nilpotent, or bounded Lie Engel.
Let $L$ be a restricted Lie algebra over a field of characteristic $p>2$ and denote by $u(L)$ its restricted enveloping algebra. We establish when the symmetric or skew elements of $u(L)$ under the principal involution are Lie metabelian.
In this paper, the notion of a uniformly distributed systems of elements on the variety of metabelian Lie algebras is introduced. This notion is analogous to one of a measure preserving systems of elements on group varieties. As the main result of the paper it was shown that on the variety of metabelian Lie algebras a system of elements is primitive iff it is uniformly distributed.
We report on some computations with reachable elements in simple Lie algebras of exceptional type within the SLA package of GAP4. These computations confirm the classification of such elements by Elashvili and Grelaud. Secondly they answer a question from Panyushev. Thirdly they show in what way a recent result of Yakimova for the Lie algebras of classical type extends to the exceptional types.
An extremal element $x$ in a Lie algebra $\mathfrak{g}$ is an element for which the space $[x, [x, \mathfrak{g}]]$ is contained in the linear span of $x$. Long root elements in classical Lie algebras are examples of extremal elements. Lie algebras generated by extremal elements lead to geometries with as points the $1$-spaces generate by extremal elements an as lines the $2$-spaces whose non-zero elements are pairwise commuting extremal elements. In this paper we show that the finitary special linear Lie algebras can be characterized by their extremal geometry. Moreover, we also show that the …
A nonzero element $x$ in a Lie algebra $\mathfrak{g}$ with Lie product $[ , ]$ is called extremal if $[x,[x,y]]$ is a multiple of $x$ for all $y$. In this paper we characterize the (finitary) symplectic Lie algebras as simple Lie algebras generated by their extremal elements satisying the condition that any two noncommuting extremal elements $x,y$ generate an $\mathfrak{sl}_2$ and any third extremal element $z$ commutes with at least one extremal element in this $\mathfrak{sl}_2$.
By definition, a quadratic Lie superalgebra is a Lie superalgebra endowed with a non-degenerate supersymmetric bilinear form which satisfies the even and invariant properties. In this paper we calculate all of the second cohomology group of elementary quadratic Lie superalgebras which have been classified in \cite{DU14} by applying the super-Poisson bracket on the super exterior algebra. Besides, we give the classification of 8-dimensional solvable quadratic Lie superalgebras having 6-dimensional indecomposable even part. The method is based on the double extension and classification results o…
We call a finite, spanning set of a semi-simple real Lie algebra a distinguished set if it satisfies the following property: The Lie bracket of any two elements out of the set is, up to some constant, another element in the set; conversely, for any element in the set, there are two elements out of the set whose Lie bracket is, up to some constant, the given element. We show in the paper that every semi-simple real Lie algebra has a distinguished set.
Let $R$ be a commutative ring of characteristic zero and $G$ an arbitrary group. In the present paper we classify the groups $G$ for which the set of symmetric elements with respect to the classical involution of the group ring $RG$ is Lie metabelian.
An algebra with identities $a(bc)=b(ac),$ $(ab)c=(ac)b$ is called bicommutative. We construct list of identities satisfied by commutator and anti-commutator products in a free bicommutative algebra. We give criterions for elements of a free bicommutative algebra to be Lie or Jordan.
For a field $F$ of characteristic not 2 and a directed row-finite graph $Γ$ let $L(Γ)$ be the Leavitt path algebra with the standard involution $*.$ We study the Lie algebra $K=K(L(Γ),*)$ of $*-$skew-symmetric elements and find necessary and sufficient conditions for the Lie algebra $[K,K]$ to be simple.
We study some algebraic and combinatorial features of two algebras that arise as quotients of Temperley-Lieb algebras of type $\tilde{C}$, namely, the two-boundary Temperley-Lieb algebra and the symplectic blob algebra. We provide a monomial basis for both algebras. The elements of these bases are parameterized by certain subsets of fully commutative elements. We enumerate these elements according to their affine length.
Let L be a simple finite-dimensional Lie algebra of characteristic distinct from 2 and from 3. Suppose that L contains an extremal element that is not a sandwich, that is, an element x such that [x, [x, L]] is equal to the linear span of x in L. In this paper we prove that, with a single exception, L is generated by extremal elements. The result is known, at least for most characteristics, but the proofs in the literature are involved. The current proof closes a gap in a geometric proof that every simple Lie algebra containing no sandwiches (that is, ad-nilpotent elements of order 2) is in fac…
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.
We give an overview of some properties of Lie algebras generated by at most 5 extremal elements. In particular, for any finite graph Γ and any field K of characteristic not 2, we consider an algebraic variety X over K whose K-points parametrize Lie algebras generated by extremal elements. Here the generators correspond to the vertices of the graph, and we prescribe commutation relations corresponding to the nonedges of Γ. We show that, for all connected undirected finite graphs on at most 5 vertices, X is a finite-dimensional affine space. Furthermore, we show that for maximal-dimensional Lie …
The class of minimal non-elementary Lie algebras over a field F are studied. These are classified when F is algebraically closed and of characteristic different from 2,3. The solvable algebras in this class are also characterised over any perfect field.
A nonzero element x in a Lie algebra g over a field F with Lie product [ , ] is called a extremal element if [x, [x, g]] is contained in Fx. Long root elements in classical Lie algebras are examples of extremal elements. Arjeh Cohen et al. initiated the investigation of Lie algebras generated by extremal elements in order to provide a geometric characterization of the classical Lie algebras generated by their long root el- ements. He and Gabor Ivanyos studied the so-called extremal geometry with as points the 1-dimensional subspaces of g generated by extremal elements of g and as lines the 2-d…
A finite-dimensional Lie algebra $L$ over a field $F$ of characteristic zero is called elementary if each of its subalgebras has trivial Frattini ideal; it is an $A$-algebra if every nilpotent subalgebra is abelian. This paper is a continuation of the study of these algebras initiated by the authors in `Elementary Lie Algebras and Lie A-algebras', J. Algebra 312 (2007), 891--901. If we denote by $\mathcal{A}$, $\mathcal{G}$, $\mathcal{E}$, $\mathcal{L}$, $Φ$ the classes of $A$-algebras, almost algebraic algebras, $E$-algebras, elementary algebras and $ϕ$-free algebras respectively, then it is …
The operad of anti-pre-Lie algebras and its Lie elements
Nankai University（南开大学）；Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Institut de Recherche Mathématique Avancée, UMR7501, Université de Strasbourg et CNRS（斯特拉斯堡大学CNRS联合数学高级研究所）；Université de Strasbourg（斯特拉斯堡大学）
Results about the following classes of finite-dimensional Lie algebras over a field of characteristic zero are presented: anisotropic (i.e., Lie algebras for which each adjoint operator is semisimple), regular (i.e., Lie algebras in which each nonzero element is regular in the sense of Bourbaki), minimal nonabelian (i.e., nonabelian Lie algebras all whose proper subalgebras are abelian), and algebras of depth 2 (i.e., Lie algebras all whose proper subalgebras are abelian or minimal nonabelian).
Let $q$ be a scalar that is not a root of unity. We show that any polynomial in the Casimir element of the Fairlie-Odesskii algebra $U_q'(\mathfrak{so}_3)$ cannot be expressed in terms of only Lie algebra operations performed on the generators $I_1,I_2,I_3$ in the usual presentation of $U_q'(\mathfrak{so}_3)$. Hence, the vector space sum of the center of $U_q'(\mathfrak{so}_3)$ and the Lie subalgebra of $U_q'(\mathfrak{so}_3)$ generated by $I_1,I_2,I_3$ is direct.
We initiate the investigation of the projective variety $E(r,g)$ of elementary subalgebras of dimension $r$ of a ($p$-restricted) Lie algebra $g$ for some $r > 0$ and demonstrate that this variety encodes considerable information about the representations of $g$. For various choices of $g$ and $r$, we identify the geometric structure of $E(r,g)$. We show that special classes of (restricted) representations of $g$ lead to algebraic vector bundles on $E(r,g)$. For $g = Lie(G)$ the Lie algebra of an algebraic group $G$, rational representations of $G$ enable us to realize familiar algebraic vecto…
Seaweed Lie algebras are a natural generalisation of parabolic subalgebras of reductive Lie algebras. The well-known Richardson Theorem says that the adjoint action of a parabolic group has a dense open orbit in the nilpotent radical of its Lie algebra \cite{richardson}. We call elements in the open orbit Richardson elements. In \cite{JSY} together with Yu, we generalized Richardson's Theorem and showed that Richardson elements exist for seaweed Lie algebras of type $\mathbb{A}$. Using GAP, we checked that Richardson elements exist for all exceptional simple Lie algebras except $\mathbb{E}_8$,…
We study Lie algebras generated by extremal elements (i.e., elements spanning inner ideals of L) over a field of characteristic distinct from 2. We prove that any Lie algebra generated by a finite number of extremal elements is finite dimensional. The minimal number of extremal generators for the Lie algebras of type An, Bn (n>2), Cn (n>1), Dn (n>3), En (n=6,7,8), F4 and G2 are shown to be n+1, n+1, 2n, n, 5, 5, and 4 in the respective cases. These results are related to group theoretic ones for the corresponding Chevalley groups.
Let $M(\gl)$ be a Verma module for a basic classical simple Lie superalgebra $\fg \neq G(3)$ defined using the distinguished Borel subalgebra, and let $\gc$ be an isotropic positive root of $\fg.$ As a special case of our first main result we show that if $μ, \gl \in \fh^*$ with $\gl-μ= \gc$ we have $$\dim \Hom_{\sfg}(M(μ),M(\gl))\le 1.$$ This result applies to the construction of Šapovalov elements for isotropic roots. The proof rests on a comparison with the corresponding result for a certain simple Lie algebra $G$.
Let $G$ be a group and let $F$ be a field of characteristic different from 2. Denote by $(FG)^+$ the set of symmetric elements and by $\mathcal{U}^+(FG)$ the set of symmetric units, under an oriented classical involution of the group algebra $FG$. We give some lower and upper bounds on the Lie nilpotency index of $(FG)^+$ and the nilpotency class of $\mathcal{U}^+(FG)$.
Anti-cocommutative elements were introduced by Wang, Zhang, Zhuang (2013) in their paper Coassociative Lie Algebras. We use this notion to extend universal enveloping algebras of Lie algebras with regards to their Hopf structure, and see if these connected Hopf algebras are enveloping algebras. Furthermore, we apply these results to compare global dimension of connected Hopf algebras and the dimension of their corresponding Lie algebras of primitive elements. Title has been renamed to "GK-Dimension of some Connected Hopf Algebras".
Anti-cocommutativity was introduced by Wang, Zhuang, Zhang (2013) in their paper Coassociative Lie algebras. Since universal enveloping algebras of Lie algebras are connected Hopf algebras, we extend enveloping algebras using the notion of anti-cocommutative elements. This concept is the main focus of this thesis. We separate the results (Chapter 4) into four parts: 1) Lie algebras containing anti-cocommutative elements; 2) Extending enveloping algebras using anti-cocommutative elements; 3) A global dimension problem for connected Hopf algebras; 4) Properties of the antipode of some connected …
Let $X$ be one of the finite-dimensional simple graded Lie superalgebras of Cartan type $W, S, H, K, HO, KO, SHO$ or $SKO$ over an algebraically closed field of characteristic $p>3$. In this paper we prove that $X$ can be generated by one element except the ones of type $W,$ $HO$, $KO$ or $SKO$ in certain exceptional cases, in which $X$ can be generated by two elements. As a subsidiary result, we also prove that certain classical Lie superalgebras or their relatives can be generated by one or two elements.
We study so called regular Lie algebras, i.e. Lie algebras in which each nonzero element is regular. We make a connection with an open problem whether any element of reduced trace zero in a simple associative algebra is a commutator.
We describe the isomorphism classes of certain infinite-dimensional graded Lie algebras of maximal class, generated by an element of weight one and an element of weight two, over fields of odd characteristic.
We introduce the notion of the \textit{principal element} of a Frobenius Lie algebra $\f$. The principal element corresponds to a choice of $F\in \f^*$ such that $F[-,-]$ non-degenerate. In many natural instances, the principal element is shown to be semisimple, and when associated to $\sl_n$, its eigenvalues are integers and are independent of $F$. For certain ``small'' functionals $F$, a simple construction is given which readily yields the principal element. When applied to the first maximal parabolic subalgebra of $\sl_n$, the principal element coincides with semisimple element of the prin…
The long-root elements in Lie algebras of Chevalley type have been well studied and can be characterized as extremal elements, that is, elements $x$ such that the image of $(\ad x)^2$ lies in the subspace spanned by $x$. In this paper, assuming an algebraically closed base field of characteristic not 2, we find presentations of the Lie algebras of classical Chevalley type by means of minimal sets of extremal generators. The relations are described by simple graphs on the sets. For example, for $C_n$ the graph is a path of length $2n$, and for $A_n$ the graph is the triangle connected to a path…
Let $\mathfrak{sp}_{2n}(\mathbb {K})$ be the symplectic Lie algebra over an algebraically closed field of characteristic zero. We prove that for any nonzero nilpotent element $X \in \mathfrak{sp}_{2n}(\mathbb {K})$ there exists a nilpotent element $Y \in \mathfrak{sp}_{2n}(\mathbb {K})$ such that $X$ and $Y$ generate $\mathfrak{sp}_{2n}(\mathbb {K})$.
In this paper we investigate the Lie structure of the derived Lie superalgebra [K, K], with K the set of skew elements of a semiprime associative superalgebra A with superinvolution. We show that if U is a Lie ideal of [K, K], then either there exists an ideal J of A such that the Lie ideal [J \cap K,K] is nonzero and contained in U, or A is a subdirect sum of A', A'', where the image of U in A' is central, and A'' is a subdirect product of orders in simple superalgebras, each at most 16-dimensional over its center.
For any finite graph Gamma and any field K of characteristic unequal to 2 we construct an algebraic variety X over K whose K-points parameterise K-Lie algebras generated by extremal elements, corresponding to the vertices of the graph, with prescribed commutation relations, corresponding to the non-edges. After that, we study the case where Gamma is a connected, simply laced Dynkin diagram of finite or affine type. We prove that X is then an affine space, and that all points in an open dense subset of X parameterise Lie algebras isomorphic to a single fixed Lie algebra. If Gamma is of affine t…
Let F_o be a non-archimedean locally compact field of residual characteristic not 2. Let G be a classical group over F_o (with no quaternionic algebra involved) which is not of type A_n for n>1. Let b be an element of the Lie algebra g of G that we assume semisimple for simplicity. Let H be the centralizer of b in G and h its Lie algebra. Let I and I_b denote the (enlarged) Bruhat-Tits buildings of G and H respectively. We prove that there is a natural set of maps j_b : I_b --> I which enjoy the following properties: they are affine, H-equivariant, map any apartment of I_b into an apartment of…
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.
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…
