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6.5. Loop superalgebras. For any (simple, or a relative of a simple) finite-dimensional Lie (super)algebra [MATH] over [MATH] , we consider spaces [MATH] of “loops”, i.e., [MATH] -valued functions on the circle [MATH] expandable into Laurent polynomials (or their completions, Laurent series); here [MATH] , where [MATH]... |
Recall that if [MATH] is an order [MATH] automorphism of [MATH] , and [MATH] , where [MATH] , are eigenspaces of [MATH] with eigenvalue [MATH] , where [MATH] , then |
[EQUATION] Non-isomorphic loop superalgebras [MATH] with values in simple finite-dimensional Lie superalgebras [MATH] over [MATH] , and twisted loops [MATH] corresponding to order [MATH] automorphisms [MATH] of [MATH] are classified in Se |
6.5.1. NIS on loop superalgebras. The NIS “ [MATH] ” for any [MATH] on the target space [MATH] from the left column in ( 101 ) is [MATH] in the purely even case; [MATH] or [MATH] for an appropriate representation [MATH] for all superalgebras with an even NIS, except for [MATH] in which case “ [MATH] ” is [MATH] |
Each of the twisted loop superalgebras [MATH] listed in ( 101 ), and simple loop algebras listed in , pp. 54, 55] , has a NIS induced by “ [MATH] ” on [MATH] |
[EQUATION] where [MATH] is the coefficient of [MATH] in the Laurent polynomial [MATH] 6.5.2. Central extensions of loop superalgebras. |
The central extensions of the loop superalgebra with a NIS on the target space [MATH] is given by the formula [EQUATION] where [MATH] ) is the coefficient of [MATH] in the 1-form [MATH] Observe that |
[EQUATION] Loop superalgebras with values in [MATH] for [MATH] [MATH] and [MATH] for [MATH] , and [MATH] for [MATH] None of these target algebras has a NIS; the only non-trivial automorphism exists only for [MATH] , see ( 100 ), but twisted loop superalgebra corresponding to this automorphism is isomorphic to the non-t... |
Loop superalgebras with values in [MATH] for [MATH] [MATH] for [MATH] [MATH] for [MATH] , and [MATH] Each of these target algebras [MATH] has one non-trivial central extension, whereas [MATH] has three of them. Therefore, [MATH] has infinitely many central extensions, given by loops with values in each of the centers o... |
6.5.3. Derivations of loop superalgebras. For [MATH] simple, the outer derivations the loop superalgebra [MATH] constitute the semidirect sum [MATH] , where [MATH] and [MATH] is the space of loops with values in [MATH] . The popular affine Kac–Moody algebras correspond (in the non-super case) to just one derivation, [M... |
6.6. Lie superalgebras of polynomial growth with nonsymmetrizable Cartan matrices. Hoyt and Serganova HS proved Serganova’s conjecture ( LSS ) that there are only two types of indecomposable nonsymmetrizable Cartan matrices corresponding to Lie superalgebras of polynomial growth. These Lie superalgebras are the followi... |
1) The series of twisted loops with values in [MATH] corresponding to the 2nd order automorphism [MATH] for any [MATH] . Let [MATH] be the change of parity operator; set |
[EQUATION] A double extension [MATH] of [MATH] determined by an even central extension and an odd outer derivation has Cartan matrices with the same Dynkin graphs as that of the loop algebra [MATH] , but — in the simplest case — one of the nodes of the Dynkin graph for [MATH] being “gray”, the other Cartan matrices bei... |
2) The exceptional Lie superalgebra [MATH] with Cartan matrices ( 95 ) is NOT the double extension of the stringy Lie superalgebra (which is simple for [MATH] |
[EQUATION] in the sense of definition in Subsection 2.1 [MATH] has properties 1) and 2) but not 3); there is no NIS on either [MATH] or [MATH] |
6.6.1. Root system of [MATH] For Chevalley generators of the Lie superalgebra we denote [MATH] take [EQUATION] By means of the isotropic reflections, see CCLL , we establish that [MATH] has the following Cartan matrices: |
[EQUATION] In order to compare these matrices, reduce them to the following normal forms having renumbered rows/columns and rescaled (by definition [MATH] , in particular, [MATH] , so the fractions are well defined), respectively: |
[EQUATION] Evidently, the maps [MATH] and [MATH] establish isomorphisms of the Lie superalgebras corresponding to these Cartan matrices, so one may assume that [MATH] |
In manual computations, it is convenient to work not in the Chevalley basis of [MATH] , but in the basis consisting of the following [MATH] -homogeneous elements |
[EQUATION] There is only one outer derivation of [MATH] for [MATH] (for a proof, see Po ) given (modulo the space of inner derivations) by the even operator |
[EQUATION] There is only one non-trivial central extension (for a proof, see KvL ) given by the even cocycle [MATH] with the following non-zero values: |
[EQUATION] where [EQUATION] Theorem (No NIS on [MATH] On [MATH] with any of Cartan matrices ( 95 ), there is no NIS compatible with the principal grading [MATH] for all [MATH] Conjecturally, there is no NIS whatsoever |
Proof. Induction, beginning with degree 0, see ( ), leads to a contradiction. ∎ 6.7. The exceptional simple vectorial Lie superalgebras for [MATH] |
6.7.1. [MATH] for [MATH] For [MATH] and any [MATH] , we have the following description: [MATH] and [MATH] with the natural [MATH] -action on [MATH] and the bracket of odd elements is given by |
[EQUATION] where we identify [EQUATION] For [MATH] , when [MATH] , we have the following odd NIS on [MATH] [EQUATION] Having in mind Fact (Subsection 1.1.1 ) and the fact that [MATH] is a deform of [MATH] , it is no wonder that there is an odd NIS on [MATH] |
6.7.2. Lie superalgebras indigenous to [MATH] In BoL1 BoGL , there are constructed two super analogs of the Melikyan algebra (one, as a Cartan prolong of a non-positive part of the exceptional Lie superalgebra [MATH] , the other one analogous of the original Melikyan’s construction, with [MATH] for [MATH] instead of [M... |
[EQUATION] that could have a NIS are [MATH] [MATH] [MATH] [MATH] [MATH] , and [MATH] . The last one has a NIS thanks to ( ), whereas SuperLie proves that each of the other algebras has a NIS. These NISes are corollaries of the necessary , but not sufficient, conditions of Lemma 5.1 ; the ambient algebras do not have NI... |
For simplicity, we give the answer only for [MATH] [MATH] Observe that [MATH] is a subalgebra of [MATH] , the latter preserving to the distribution singled out by the form (see BoGL |
[EQUATION] On [MATH] , there is no NIS, see Subsection 5.6 . The odd NIS on [MATH] is given by pairing [MATH] with the component [MATH] of highest degree [MATH] , and extending to other pairs of elements by invariance. For example, for [MATH] we have [MATH] and |
[EQUATION] [MATH] For any [MATH] and Cartan matrix of [MATH] , the Cartan prolongs of the non-positive part in any its [MATH] -grading return |
[MATH] , except for [MATH] where for the Cartan matrix [MATH] in ( 98 and its [MATH] -grading [MATH] , the Cartan prolongs yield the simple subalgebras of series [MATH] , and the exceptional algebra [MATH] whose double extension [MATH] has 5 inequivalent Cartan matrices, see BGL |
The Lie superalgebra [MATH] over [MATH] has the following four non-equivalent Cartan matrices, see CCLL BGL ; we use Chevalley generators corresponding to matrix 1): |
[EQUATION] From an explicit description of [MATH] and its first fundamental representation, e.g., see FH , we deduce an explicit form of the non-positive elements of [MATH] which we give in terms of the generating functions with respect to the contact bracket corresponding to the contact form |
[EQUATION] where the [MATH] [MATH] and [MATH] are odd; our notation match FH , p. 354, while our [MATH] and [MATH] correspond to [MATH] and [MATH] of FH , p. 340, respectively. |
We also set [MATH] [MATH] [MATH] [MATH] To describe the [MATH] -module [MATH] , only the highest weight vector suffices: [EQUATION] |
As expected, for [MATH] and [MATH] , the CTS prolong is isomorphic to [MATH] For [MATH] , the Lie algebra [MATH] is not simple, but has a simple Lie subalgebra isomorphic to [MATH] |
[MATH] , and [MATH] Let [MATH] , where the center of [MATH] is [MATH] The [MATH] -module [MATH] splits into two irreducible components: a [MATH] -dimensional, and [MATH] -dimensional with lowest weight vectors, respectively: |
[EQUATION] Since [MATH] generates the positive part of the CTS prolong, [MATH] , the [MATH] -module [MATH] is irreducible, and [MATH] , the standard criterion for simplicity ensures that the generalized Cartan prolong is simple; we denote it |
[MATH] The positive components of [MATH] are all of dimension 8 (the direct sums [MATH] of irreducible [MATH] -modules of dimension 1 and 7, up to parity), except the one of the highest degree ( [MATH] ), which is of dimension 1, and the second highest degree, which is of dimension 7. Let [MATH] and |
[MATH] be the lowest weight vectors (with respect to [MATH] ) of [MATH] and [MATH] respectively for [MATH] [EQUATION] A subalgebra of [MATH] : let [MATH] be [MATH] , as |
[MATH] -module. Let [MATH] denote the partial prolong. This is a simple Lie superalgebra of dimension [MATH] with [EQUATION] and [MATH] |
6.8. Abomination expressions. In what follows elements of different parity are separated by a bar: (even [MATH] odd). [MATH] The odd NIS on [MATH] is given by paring of [MATH] with [MATH] , and extending to other pairs of elements by invariance: |
[EQUATION] [MATH] We realize [MATH] as a Lie subsuperalgebra of [MATH] in indeterminates (even [MATH] odd) [MATH] The odd NIS on [MATH] is given by paring of [MATH] with |
[MATH] , and extending to other pairs of elements by invariance, where [MATH] and [MATH] ; for a basis in [MATH] we take the even element [MATH] |
[EQUATION] [MATH] We realize [MATH] as a Lie subsuperalgebra of [MATH] in indeterminates (even [MATH] odd) [MATH] . The odd NIS on |
[MATH] is given by paring of [MATH] with [MATH] and extending to other pairs of elements by invariance, where [MATH] and [MATH] [EQUATION] |
[MATH] We realize [MATH] as a Lie subsuperalgebra of [MATH] in indeterminates (even [MATH] odd) [MATH] The odd NIS on [MATH] is given by paring of [MATH] with [MATH] , and extending to other pairs of elements by invariance, where [MATH] and [MATH] |
[EQUATION] 7. Summary: NIS on simple Lie (super)algebras Some NISes listed below under even (resp. odd) Subsection are actually odd (resp. even) depending on a parameter, as indicated. |
7.1. NIS even. 1) Lie (super)algebras [MATH] over [MATH] with any symmetrizable Cartan matrix [MATH] have a NIS; these algebras are simple if and only if [MATH] is invertible. (Aside: These algebras are [MATH] -graded, and as such they fall into three classes: finite-dimensional, of polynomial growth and of exponential... |
2) [MATH] over [MATH] . There is an even NIS on [MATH] . Its double extension is [MATH] 2p) Over [MATH] of characteristic [MATH] , there is a NIS on [MATH] for [MATH] , and we see that [MATH] . The double extension of [MATH] is [MATH] except for [MATH] and [MATH] , where there are 3 non-isometric double extensions, see... |
3) (Twisted) loop superalgebras over [MATH] . These are Lie superalgebras [MATH] corresponding to lines 1 through 10 of table ( 101 ). (Aside: The twisted loop superalgebras corresponding to lines 5, 6, 8, 9 of table ( 101 ) and loop superalgebras [MATH] do not have any Cartan matrix.) |
4) Stringy Lie superalgebra [MATH] over [MATH] has a NIS. No central extensions, no outer derivations. 5) The following simple vectorial Lie superalgebras in positive characteristic have NIS: |
[MATH] if [MATH] , see ( 84 ), where [MATH] , and [MATH] the Melikyan algebra (for [MATH] ); several of Skryabin algebras for [MATH] , namely, [MATH] [MATH] [MATH] , see GL |
the Lie algebra [MATH] has a NIS if and only if either [MATH] and [MATH] or [MATH] ; if [MATH] , then [MATH] has a NIS if and only if [MATH] , see Subsection 5.2 |
the simple Lie algebra [MATH] has a NIS, see Subsection 5.5.1 for deforms of [MATH] for [MATH] , see Claim 3.2.1 for deforms of Lie algebras [MATH] and Lie superalgebra [MATH] for [MATH] , and [MATH] , where [MATH] , for [MATH] , see Claim 3.3 |
for deforms of Lie algebras [MATH] and [MATH] , where [MATH] , for [MATH] , see Claim 3.4 7.2. NIS odd. 1) [MATH] over [MATH] . These are [MATH] for [MATH] and |
[MATH] for [MATH] . The corresponding double extensions are [MATH] and [MATH] 1p) Over [MATH] of characteristic [MATH] and [MATH] odd, where there are 2 non-isometric double extensions of [MATH] , see BeBou |
2) (Twisted) loop superalgebras over [MATH] . These are [MATH] and [MATH] and Lie superalgebras corresponding to lines 10 and 11 of table ( 101 ). Observe that a (there are several, their isometry classes are unknown) double extension corresponding to line 10 does have Cartan matrices, albeit non-symmetrizable ones. |
3) Stringy Lie superalgebra [MATH] over [MATH] has a NIS. No central extensions, no outer derivations. Observe that [MATH] has a Cartan matrix, albeit non-symmetrizable one, and [MATH] has a non-trivial central extension and an outer derivation, but no NIS; no NIS on [MATH] , either. |
4) For [MATH] , the queerification [MATH] (see BLLSq ) of any simple restricted Lie algebra [MATH] with a NIS has an odd NIS, see Subsection 6.2 |
5) There is a NIS on [MATH] and [MATH] , where [MATH] , see Subsection 6.4 6) On [MATH] [MATH] [MATH] [MATH] [MATH] , see Subsection 6.7.2 |
7) on [MATH] for [MATH] , see Subsection 6.7.1 7.3. Open problems. 1) Extend to any [MATH] the results and formulas obtained in Subsections 6.7.2 and 6.8 for [MATH] |
2) Conjecturally, all non-trivial central extensions of (twisted) loop superalgebras with values in finite-dimensional simple Lie superalgebras are those listed in Subserction 6.5.2 . Which of these extensions can be extended to double extensions and which of these double extensions belong to one isometry class? |
3) Prove conjectures 5.4.1 6.5.2 and 6.6.2. Theorem 4) Investigate if there is a NIS on deforms of [MATH] , see Subsection 4.15 5) Related to this paper are several conjectures, listed in the order of feasibility. |
5a) Classification of simple [MATH] -graded Lie superalgebras of polynomial growth over [MATH] : they are (a) the finite-dimensional ones, (b) vectorial (with polynomial coefficients), (c) stringy (with Laurent coefficients) and (d) (twisted) loop superalgebras. Since the time this conjecture was formulated in LSS it w... |
5b) Classification of simple finite-dimensional Lie algebras over an algebraically closed field [MATH] of characteristic [MATH] conjecturally these are the examples obtained by means of the Kostrikin-Shafarevich procedure, examples listed in GL , and deforms of these two types of examples. |
5c) Classification of simple finite-dimensional Lie superalgebras over an algebraically closed field [MATH] of characteristic [MATH] . For a formulation of the conjecture, see BGLLS |
6) Consider deforms with odd parameters; for the cases where they are classified, see BGLd 7) Consider general Lie algebras of matrices of complex size [MATH] , where [MATH] is the Lie algebra associated with the associative algebra [MATH] and [MATH] is the 2nd order Casimir, and the ortho/symplectic subalgebras of [MA... |
8) Simple Lie (super)algebras for [MATH] . For a general theory of double extensions — quite different from that for [MATH] — and interesting examples, see BeBou . Additionally, for selected examples, see Subsections 3.1 3.3 3.4 ; for a wide series of examples, see Subsection 6.2 |
9) Lie algebras of pseudodifferential operators, see Dz4 8. Tables In Tables ( 100 ) and ( 101 ), let [MATH] for any [MATH] and some even invertible [MATH] |
# Source: arxiv 1806.05529 # Title: On the nilpotency class of finite groups with a Frobenius group of automorphisms # Sections: all # Downloaded: 2026-03-02T09:09:34.761192+00:00 |
On the nilpotency class of finite groups with a Frobenius group of automorphisms (Date: June 11, 2018 ) Abstract. Suppose that a metacyclic Frobenius group [MATH] , with kernel [MATH] and complement [MATH] , acts by automorphisms on a finite group [MATH] , in such a way that [MATH] is trivial and [MATH] is nilpotent. I... |
Key words and phrases: Frobenius group; automorphism; Lie algebra; finite group; nilpotency class 1. Introduction Let [MATH] be a group acting on a group [MATH] . We denote by [MATH] the centralizer of [MATH] in [MATH] , namely the fixed-point subgroup of [MATH] . Experience shows that in many cases the properties of [... |
We recall that a finite Frobenius group [MATH] , with kernel [MATH] and complement [MATH] , can be characterized as a semidirect product of a normal subgroup [MATH] by [MATH] such that [MATH] for every non trivial element [MATH] of [MATH] . The structure of Frobenius groups is well known. In particular, by Thompson’s t... |
Suppose that a Frobenius group [MATH] acts on a finite group [MATH] in such a way that [MATH] is trivial. By Belyaev and Hartley’s theorem BH96 , the group [MATH] is soluble. Moreover, by Khukhro–Makarenko–Shumyatsky’s theorem KMS14 , Theorem 2.7] if [MATH] is nilpotent, then [MATH] is nilpotent. In the same article, t... |
For bounding the nilpotency class of the group [MATH] , Lie ring methods are used. Indeed, the proof is based on the analogous results for Lie rings and algebras which are also important in their own right. The advantage of this method lies in the fact that it is usually easier to deal with Lie rings as they are more l... |
Until now, the question whether the bound for the nilpotency class of [MATH] could be made independent of the order of the Frobenius complement [MATH] remained unsolved. Indeed, there were no explicit examples showing the opposite. In this paper, we shall give a negative answer to this question by proving the following |
Theorem 1 There exists a family [MATH] of finite nilpotent groups, of unbounded nilpotency class, whose members [MATH] satisfy the conditions: |
(1) [MATH] admits a metacyclic Frobenius group of automorphisms; (2) the centralizer of the kernel in [MATH] is trivial; (3) the centralizer of the complement in [MATH] is abelian. |
This result is based on an analogous one for Lie algebras. In this case, the transition to finite groups is obtained from the Lazard correspondence. |
The main part of this paper is devoted to proving the result for Lie algebras. This will consist of the explicit construction of a family [MATH] of Lie algebras, satisfying analogous properties. As a preliminary step, for every prime number [MATH] we will construct a Lie algebra [MATH] which admits a metacyclic Frobeni... |
Let us denote by [MATH] and [MATH] the smallest invariant ideals generated respectively by [MATH] and by the derived subalgebra of [MATH] . We will study the former in Section and the latter in Section . Since they are both homogeneous, we will study their intersections with the homogeneous components, bounding the dim... |
In Section we will build the family [MATH] , whose elements are the quotient algebras [MATH] . We will prove the existence, for any natural number [MATH] , of a prime [MATH] such that the corresponding algebra [MATH] in [MATH] has nilpotency class at least [MATH] . For this part, the previous bounds in Propositions and... |
2. The family of Lie algebras [MATH] A crucial step towards the proof of Theorem consists of the following Proposition 2 There exists a family [MATH] of nilpotent Lie algebras, of unbounded nilpotency class, such that any member [MATH] satisfies the conditions: |
It will be proved in Section . In the same section we will also see that, in order to produce the analogous result for groups, we will need to consider the case where the characteristic of the underlying field changes with [MATH] . This section instead is devoted to the definition and the study of another family of Lie... |
Let [MATH] be a field of characteristic coprime with [MATH] , containing a primitive [MATH] th root of unity. For generality, we will not specify its characteristic at this stage, because the whole construction works independently of it. However, we stress that, although this is not made explicit by notation, the chara... |
Let [MATH] be an ordered set and let [MATH] denote the free metabelian Lie algebra over [MATH] with free generating set [MATH] . Following Bak87 , we know that [MATH] has a basis consisting of [MATH] together with the left-normed Lie brackets of the form |
[EQUATION] In particular, for every fixed number [MATH] , the above set represents a basis for the degree [MATH] homogeneous component of [MATH] , which is the [MATH] -vector space spanned by all Lie brackets of length [MATH] involving the generators. |
Renaming the generators in such a way that [MATH] and [MATH] , we get that the set in ( ) consists of the following Lie brackets: |
[MATH] , for all [MATH] , and [MATH] [MATH] [MATH] , for all [MATH] and for all [MATH] [MATH] [MATH] , for all [MATH] where [MATH] and [MATH] |
Let [MATH] denote the subalgebra [MATH] . Similarly, denote by [MATH] the subalgebra [MATH] , and by [MATH] the ideal [MATH] Consider the quotient algebra [MATH] . Since [MATH] is a homogeneous ideal, this Lie algebra inherits from [MATH] its [MATH] -grading. The generators form a basis for its degree [MATH] homogeneou... |
[MATH] , for all [MATH] [MATH] , for all [MATH] [MATH] , for all [MATH] where [MATH] . Indeed, it is not hard to prove that [MATH] is [MATH] |
Finally, define the Lie algebra [MATH] to be [MATH] . Similarly, since the ideal [MATH] is homogeneous, the algebra [MATH] is [MATH] -graded. A basis for it consists of [MATH] together with the Lie brackets |
[EQUATION] for every [MATH] . By construction, it follows that the Lie algebra [MATH] is a semidirect sum of the abelian ideal [MATH] and the abelian subalgebra [MATH] |
In the next proposition, we determine the dimension of the homogeneous components of [MATH] Proposition 3 For every prime [MATH] and natural number [MATH] , let [MATH] denote the dimension of [MATH] , the homogeneous component of [MATH] of degree [MATH] . We have |
[EQUATION] Proof. Since the set [MATH] is a basis for the subspace [MATH] , the first equality trivially holds. Now assume [MATH] . By Formula ( ), all possible Lie brackets in [MATH] with a fixed initial entry [MATH] are determined by the [MATH] -combinations with repetition from [MATH] elements. The expression for [M... |
The Lie algebra [MATH] admits a Frobenius group of automorphisms of the form [MATH] , also denoted simply by [MATH] . Let [MATH] denote a generator of the Frobenius kernel [MATH] . For each [MATH] , we define [MATH] and [MATH] , where [MATH] is a complex primitive [MATH] th root of unity. Let [MATH] be a generator of t... |
Let [MATH] and [MATH] respectively denote the centralizers of the Frobenius kernel and the Frobenius complement. They are both [MATH] -graded subspaces, since the homogeneous components [MATH] are [MATH] -invariant. A basis for [MATH] is obtained by those Lie brackets |
[MATH] in [MATH] satisfying the condition [MATH] The next proposition presents a basis for the subspace [MATH] Proposition 4 The basis [MATH] of [MATH] is permuted by [MATH] . The number of its orbits is equal to the dimension of the subspace [MATH] , namely [MATH] . Indeed, the set |
[MATH] is a basis for such subspace. Proof. First we need to prove that the generator [MATH] of [MATH] acts as a permutation of [MATH] . This follows directly from the formula for [MATH] . Indeed, for every Lie bracket [MATH] in this set, its image under [MATH] is [MATH] , again contained in [MATH] , possibly after a r... |
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