Restricted Subalgebra Research Articles

On the subalgebra lattice of a restricted Lie algebra

In this paper we study the lattice of restricted subalgebras of a restricted Lie algebra. In particular, we consider those algebras in which this lattice is dually atomistic, lower or upper semimodular, or in which every restricted subalgebra is a quasi-ideal. The fact that there are one-dimensional subalgebras which are not restricted results in some of these conditions being weaker than for the corresponding conditions in the non-restricted case.

Restricted Lie algebras having a distributive lattice of restricted subalgebras

Let L be a restricted Lie algebra over a field of characteristic p>0. We investigate the structure of L when its lattice of restricted subalgebras satisfies some prescribed properties. In particular, we establish when is distributive. The special form that this result takes when is either the field with p elements or an algebraically closed field is also discussed. Furthermore, we establish when is Boolean or the two-elements lattice.

On modular representations of finite-dimensional Lie superalgebras

In this paper, we studied representations of finite-dimensional Lie superalgebras over an algebraically closed field \begin{document} $\mathbb{F}$ \end{document} of characteristic p > 2. It was shown that simple modules of a finite-dimensional Lie superalgebra over \begin{document} $\mathbb{F}$ \end{document} are finite-dimensional, and there exists an upper bound on the dimensions of simple modules. Moreover, a finite-dimensional Lie superalgebra can be embedded into a finite-dimensional restricted Lie superalgebra. We gave a criterion on simplicity of modules over a finite-dimensional restricted Lie superalgebra \begin{document} ${\mathfrak{g}}$ \end{document} , and defined a restricted Lie super subalgebra, then obtained a bijection between the isomorphism classes of simple modules of \begin{document} ${\mathfrak{g}}$ \end{document} and those of this restricted subalgebra. These results are generalization of the corresponding ones in Lie algebras of prime characteristic.

Criteria for the existence of a Jordan–Chevalley–Seligman decomposition

Let (L,[p]) be a finite dimensional restricted Lie algebra over a field K of positive characteristic p. A Jordan–Chevalley–Seligman decomposition of x∈L is a unique expression of x as a sum of commuting semisimple and nilpotent elements in L. It is well-known that each x∈L has such a decomposition when K is perfect. When K is non-perfect, the present paper gives several criteria for the existence of a Jordan–Chevalley–Seligman decomposition for a given x∈L as well as for determining when an element in the restricted subalgebra generated by x is semisimple or nilpotent.

Normal enveloping algebras

Let A be an algebra with involution ∗ over a field F. We recall that A is said to be normal if xx∗ = x∗x for every x ∈ A. Over the decades, normal algebras with involutions have been extensively investigated on their own (see e.g. [1], [3], [4], [5], [7], [8], [9], [10], [11]) and, moreover, they have several applications in linear algebra and functional analysis as well (see e.g. [2], [6], [11], [12], [13]). It is well-known that any normal algebra with involution satisfies the standard polynomial identity of degree 4 (see [7, §5]). Moreover, in [11] Maxwell determined the structure of a normal simple algebra of matrices with entries in a field with involution. He also proved that a division algebra D with involution is normal if and only if D is either a field or a generalized quaternion algebra over its center. Furthermore, a characterization of group algebras which are normal under the standard involution was established by Bovdi, Gudivok, and Semirot in [3]. Subsequently, such a result has been extended to twisted group algebras in [4] and to group algebras under a Novikov involution in [5]. On the other hand, it seems that the rather natural problems of characterizing ordinary and restricted enveloping algebras which are normal under their canonical involutions have not been settled yet. The present paper is just devoted to answering these questions. For an arbitrary Lie algebra L we denote by U(L) the universal enveloping algebra of L. Moreover, if L is restricted with a p-map [p] over a field F of characteristic p > 0, then we denote by u(L) the restricted enveloping algebra of L. We consider U(L) and u(L) with the principal involution ∗, namely, the unique F-antiautomorphism such that x∗ = −x for every x in L. Note that ∗ is just the antipode of the F-Hopf algebras U(L) or u(L). We use the symbols Z(L) and L′ for the center and the derived subalgebra of L, respectively. If S ⊆ L then we denote by 〈S〉F and 〈S〉 the F-vector space and the subalgebra generated by S. Also, if L is restricted then 〈S〉p denotes the restricted subalgebra generated by S and we put S = {x |x ∈ S}. In our first main result we completely settle the restricted case:

Restricted Lie algebras in which every restricted subalgebra is an ideal

We characterize restricted Lie algebras over perfect fields all of whose restricted subalgebras are ideals.

Enumeration of maximal subalgebras in free restricted lie algebras

Given a finitely generated restricted Lie algebra L over the finite field \( \mathbb{F}_q \), and n ≥ 0, denote by an(L) the number of restricted subalgebras H ⊆ L with \( \dim _{\mathbb{F} _q} \)L/H = n. Denote by an(L) the number of the subalgebras satisfying the maximality condition as well. Considering the free restricted Lie algebra L = Fd of rank d ≥ 2, we find the asymptotics of an(Fd) and show that it coincides with the asymptotics of an(Fd) which was found previously by the first author. Our approach is based on studying the actions of restricted algebras by derivations on the truncated polynomial rings. We establish that the maximal subalgebras correspond to the so-called primitive actions. This means that “almost all” restricted subalgebras H ⊂ Fd of finite codimension are maximal, which is analogous to the corresponding results for free groups and free associative algebras.

Weyl Type Non-Associative Algebras Using Additive Groups I

A Weyl type algebra is defined in the book [4]. A Weyl type non-associative algebra [Formula: see text] and its restricted subalgebra [Formula: see text] are defined in various papers (see [1, 3, 11, 12]). Several authors find all the derivations of an associative (a Lie, a non-associative) algebra (see [1, 2, 4, 6, 11, 12]). We define the non-associative simple algebra [Formula: see text] and the semi-Lie algebra [Formula: see text], where [Formula: see text]. We prove that the algebra is simple and find all its non-associative algebra derivations.

AN EXTENDED NON-ASSOCIATIVE ALGEBRA

A Weyl type algebra is defined in the paper (see [2],[4], [6], [7]). A Weyl type non-associative algebra <TEX>$\bar{WN_{m,n,s}}$</TEX> and its restricted subalgebra <TEX>$\bar{WN_{m,n,s_r}}$</TEX> are defined in the papers (see [1], [14], [16]). Several authors find all the derivations of an associative (Lie or non-associative) algebra (see [3], [1], [5], [7], [10], [16]). We find Der(<TEX>$\bar_{WN_{0,0,1_n}}$</TEX>) of the algebra <TEX>$\bar_{WN_{0,0,1_n}}$</TEX> and show that the algebras <TEX>$\bar_{WN_{0,0,1_n}}$</TEX> and <TEX>$\bar_{WN_{0,0,s_1}}$</TEX> are not isomorphic in this work. We show that the associator of <TEX>$\bar_{WN_{0,0,1_n}}$</TEX> is zero.

DERIVATIONS OF A WEYL TYPE NON-ASSOCIATIVE ALGEBRA ON A LAURENT EXTENSTION

A Weyl type algebra is defined in the book ([4]). A Weyl type non-associative algebra <TEX>$\={WP_{m,n,s}}$</TEX> and its restricted sub-algebra <TEX>$\={WP_{m,n,s_{\gamma}}}$</TEX> are defined in various papers ([1], [12], [3], [11]). Several authors 0nd all the derivations of an associative (Lie or non-associative) algebra in the papers ([1], [2], [12], [4], [6], [11]). We find all the non-associative algebra derivations of the non-associative algebra <TEX>$\={WP_{0,2,0_B}$</TEX>, where <TEX>$B=\{{\partial}_0,\;{\partial}_1,\;{\partial}_2,\;{\partial}_{12},\;{\partial}^2_1,\;{\partial}^2_2\}$</TEX>.

GROWTH OF SUBALGEBRAS FOR RESTRICTED LIE ALGEBRAS AND TRANSITIVE ACTIONS

We study a growth of subalgebras for restricted Lie algebras over a finite field 𝔽q. This kind of growth is an analog of the subgroup growth in the group theory. Let L be a finitely generated restricted Lie algebra. Then an(L) is the number of restricted subalgebras H ⊂ L such that dim 𝔽q L/H = n, n ≥ 0. We compute the numbers an(Fd) explicitly and find asymptotics, where Fd is the free restricted Lie algebra of rank d, d ≥ 1. As an important instrument, we use the notion of transitive L-action on coalgebras and algebras.

Cohomology of generalized restricted Lie algebras*1

In this note, the generalized restricted Lie algebra, which was introduced by Shu Bin in [J. Algebra 194 (1997) 157–177], is studied. By generalizing the concept of restricted subalgebras and the concept of restricted homomorphism, we show that the second generalized restricted cohomology H ϕ L 2 ( L , M ) is isomorphic to the equivalence classes of those generalized restricted extension of M by L . For any generalized restricted Lie algebra ( L , B L , ϕ L ) and any generalized restricted L -module M , we show that the sequence 0→H ϕ L 1 (L,M)→H 1 (L,M)→ hom F L,M L →H ϕ L 2 (L,M)→H 2 (L,M)→ hom F L,H 1 (L,M) is exact.

Restricted Lie Algebras and Their Envelopes

AbstractLet L be a restricted Lie algebra over a field of characteristic p. Denote by u(L) its restricted enveloping algebra and by ωu(L) the augmentation ideal of u(L). We give an explicit description for the dimension subalgebras of L, namely those ideals of L defined by Dn(L) - L∩ωu(L)n for each n ≥ 1. Using this expression we describe the nilpotence index of ωU(L). We also give a precise characterisation of those L for which ωu(L) is a residually nilpotent ideal. In this case we show that the minimal number of elements required to generate an arbitrary ideal of u(L) is finitely bounded if and only if L contains a 1-generated restricted subalgebra of finite codimension. Subsequently we examine certain analogous aspects of the Lie structure of u(L). In particular we characterise L for which u(L) is residually nilpotent when considered as a Lie algebra, and give a formula for the Lie nilpotence index of u(L). This formula is then used to describe the nilpotence class of the group of units of u(L).

Enveloping algebras satisfying a polynomial identity

In this paper we study ordinary and restricted enveloping algebras satisfying a polynomial identity. We first show how the Δ-methods of J. Bergen and D. S. Passman ( J. Algebra, in press) can be used to handle U( L) in all characteristics and u( L) if the latter ring is prime. In particular, we offer a simpler proof of the result of Yu. A. Bachturin ( J. Austral. Math. Soc. 18, 1974, 10–21) on ordinary enveloping algebras in characteristic p > 0. Our main theorem asserts that, in general, u( L) is p.i. if and only if L has a restricted subalgebra of finite codimension which is (essentially) commutative. These results are clearly the Lie algebra analogs of the p.i. group ring theorems of M. K. Smith ( J. Algebra 18, 1971, 477–499) and D. S. Passman ( Pacific J. Math. 36, 1971, 467–483).

Simple Lie algebras of toral rank one

Let L be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic p &gt; 7 p &gt; 7 . Let L have Cartan decomposition L = H + ∑ γ ∈ Γ L γ L = H + {\sum _{\gamma \in \Gamma }}{L_\gamma } . If Γ \Gamma generates a cyclic group then L is isomorphic to sl ( 2 , F ) {\text {sl}}(2,F) or to one of the simple Lie algebras of generalized Cartan type W ( 1 : n ) W(1:{\mathbf {n}}) or H ( 2 : n : Φ ) ( 2 ) H{(2:{\mathbf {n}}:\Phi )^{(2)}} .

Cartan subalgebras of simple Lie algebras

Let L be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic p &gt; 7 p &gt; 7 . Let H be a Cartan subalgebra of L, let L = H + Σ γ ∈ Γ L γ L = H + {\Sigma _{\gamma \in \Gamma }}{L_\gamma } be the Cartan decomposition of L with respect to H, and let H ¯ \bar H be the restricted subalgebra of Der L generated by ad H. Let T denote the maximal torus of H ¯ \bar H and I denote the nil radical of H ¯ \bar H . Then H ¯ = T + I \bar H = T + I . Consequently, each γ ∈ Γ \gamma \in \Gamma is a linear function on H.