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[EQUATION] Together with Equation ( 31 ), we conclude that [EQUATION] We have therefore bound the right-hand-side of Equation ( 29 ) as desired. |
In addition, Lemma 4 If [MATH] is an improvement cycle, then for every improvement step [MATH] and topic [MATH] such that [MATH] there exist [MATH] such that [MATH] and |
[EQUATION] Proof of Lemma Let [MATH] such that [MATH] . From Lemma we know that for every improvement step [MATH] [MATH] ; thus, [MATH] which by Proposition leads to |
[EQUATION] By definition of improvement step [MATH] ; hence together with Equation ( 33 ) we get that [MATH] Notice that [MATH] is a finite improvement path, and that the condition of Proposition holds; hence, by invoking it for [MATH] we conclude the existence of an index [MATH] such that [MATH] and |
[EQUATION] Combining this fact with Equation ( 34 ), we get [EQUATION] We are now ready to prove Theorem Proof of Theorem Similarly to Theorem , to show that every better-response dynamics converges it suffices to show that every improvement path is finite. Moreover, every improvement path cannot contain more than a fi... |
[EQUATION] Since there are only [MATH] topics and that the inequality above contains [MATH] elements, there are at least two elements which are identical; thus we obtain a contradiction. We deduce that an improvement cycle can not exist. |
The above suggests that every better-response dynamics must converge. Appendix D Omitted Proofs from Section Proof of Proposition |
To prove that [MATH] is [MATH] -learnable, one must show that every game induced by [MATH] and the utility function [MATH] has the FIP property. |
Let [MATH] be an arbitrary game. We now reduce [MATH] to a game [MATH] with [MATH] as the mediator, where the utility of any author under any strategy profile in [MATH] equals to her utility under the same strategy profile in [MATH] . If this holds, then every improvement step in [MATH] is also an improvement step in [... |
[EQUATION] Since both [MATH] consists of the exposure-targeted utility function, we omit the super-script [MATH] and use the super-script [MATH] to specify the utility of author [MATH] under the strategy profile [MATH] in [MATH] , i.e., [MATH] , and equivalently for [MATH] for [MATH] |
By definition of exposure-targeted utility and [MATH] , for every valid [MATH] and [MATH] it holds that [EQUATION] Since [MATH] possesses [MATH] as the mediator, Theorem guarantees that [MATH] has the FIP property. Since we showed [MATH] and [MATH] are strategically equivalent, [MATH] also has the FIP property, and in ... |
Proof of Theorem It is sufficient to show that for every [MATH] that satisfies the theorem’s conditions, we can find a game instance with an improvement cycle. While all it takes to prove the theorem is to construct a single counter example (and this is what we do), using the technique below we can actually construct a... |
Let [MATH] be a scoring mediator with the corresponding function [MATH] , which we assume exhibits [MATH] . Due to the Intermediate Value Theorem, there exist [MATH] such that [MATH] and |
[EQUATION] For brevity, denote [MATH] and [MATH] , and observe that [MATH] Consider a game with [MATH] authors, [MATH] topics and a quality matrix [MATH] such that |
[EQUATION] The only missing ingredient is the distribution [MATH] over the topics. The selection of such [MATH] is crucial: we shall select [MATH] to allow improvement cycles. In service of that, we prove the following claim. |
Claim 1 There exists [MATH] such that [MATH] and the following properties hold 1. [MATH] 2. [MATH] , and 3. [MATH] The proof of Claim appears after this proof. Now, let [MATH] be an arbitrary constant satisfying the properties of Claim , and define [MATH] such that |
[EQUATION] It can be verified that [MATH] is a valid distribution over the set of topics. We claim that the game we constructed above possesses an improvement cycle. Consider the strategy profiles |
[EQUATION] In the rest of this proof we show that the cycle [MATH] is an improvement cycle. More precisely, we prove that for every [MATH] [MATH] [MATH] |
[MATH] [MATH] : the deviating author is [MATH] . Observe that [MATH] and [MATH] . It holds that [EQUATION] thus, [MATH] is an improvement step. |
The above analysis implies that [MATH] in an improvement cycle. As a result, [MATH] is not [MATH] -learnable. Proof of Claim Since [MATH] , it follows that |
[EQUATION] Since the left-hand-side is strictly less than [MATH] , we denote by [MATH] a positive real number such that [EQUATION] |
In addition, notice that [MATH] ; thus, [MATH] . We denote by [MATH] a positive real number such that [EQUATION] Similarly, since [MATH] is constant and [MATH] , there exists [MATH] such that [MATH] , which implies that |
[EQUATION] The proof is completed by setting [MATH] Proof of Theorem It is sufficient to show that for every [MATH] that satisfies the theorem’s conditions, we can find a game instance with an improvement cycle. While all it takes to prove the theorem is to construct a single counter example (and this is what we do), u... |
Let [MATH] be a scoring mediator with the corresponding function [MATH] , which we assume exhibits [MATH] for some [MATH] . Due to the Intermediate Value Theorem, there exist [MATH] such that [MATH] and |
Claim 2 There exists [MATH] such that [MATH] and the following properties hold 1. [MATH] 2. [MATH] , and 3. [MATH] The proof of Claim appears after this proof. Now, let [MATH] be an arbitrary constant satisfying the properties of Claim , and define [MATH] such that |
[MATH] [MATH] : the deviating author is [MATH] . Observe that [MATH] and [MATH] .Since [MATH] we get that [EQUATION] implying that [MATH] ; thus, it holds that |
[EQUATION] thus, [MATH] is an improvement step. The above analysis implies that [MATH] in an improvement cycle. As a result, [MATH] is not [MATH] -learnable. |
Proof of Claim Since [MATH] , it follows that [EQUATION] Since the left-hand-side is strictly less than [MATH] , we denote by [MATH] a positive real number such that |
[EQUATION] In addition, notice that [MATH] ; thus, [MATH] . We denote by [MATH] a positive real number such that [EQUATION] Since [MATH] and [MATH] , for every [MATH] it holds that |
[EQUATION] The proof is completed by setting [MATH] Appendix E Non-Learnability under Action-Targeted Utility In this section we prove non-learnability of another family of scoring mediators under [MATH] We consider scoring mediators where the corresponding function [MATH] is bounded by (non-affine) linear functions. E... |
Theorem 7 Let [MATH] be a scoring mediator. If [MATH] is continuous function and there exist [MATH] such that [MATH] , then [MATH] is not [MATH] -learnable. |
Let [MATH] be a scoring mediator with the corresponding continuous function [MATH] , and let [MATH] and [MATH] such that [MATH] and |
[EQUATION] For brevity, denote [MATH] . Since [MATH] we know that [MATH] . Notice that for every, [MATH] , it must hold that [MATH] as [MATH] where [MATH] . Let [MATH] be an arbitrary quality such that [MATH] Due to the Intermediate Value Theorem, there exist [MATH] such that [MATH] and |
[EQUATION] Consider a game with [MATH] authors, [MATH] topics and a quality matrix [MATH] such that [EQUATION] The only missing ingredient is the distribution [MATH] over the topics. The selection of such [MATH] is crucial: we shall select [MATH] to allow improvement cycles. In service of that, we prove the following c... |
Claim 3 There exists [MATH] such that [MATH] and the following properties hold 1. [MATH] 2. [MATH] , and 3. [MATH] The proof of Claim appears after this proof. Now, let [MATH] be an arbitrary constant satisfying the properties of Claim , and define [MATH] such that |
[MATH] [MATH] : the deviating author is [MATH] . Observe that [MATH] and [MATH] .Since for every [MATH] [MATH] we get that [EQUATION] |
implying that [MATH] ; thus, it holds that [EQUATION] thus, [MATH] is an improvement step. The above analysis implies that [MATH] in an improvement cycle. As a result, [MATH] is not [MATH] -learnable. |
Proof of Claim As [MATH] , we can find [MATH] such that, [EQUATION] hence, we get that [EQUATION] Since [MATH] we can find [MATH] such that |
[EQUATION] Therefore, [EQUATION] In addition, we can find [MATH] such that [EQUATION] which implies that [EQUATION] The proof is completed by setting [MATH] |
# Source: arxiv 1806.05505 # Title: Non-degenerate invariant (super)symmetric bilinear forms on simple Lie (super)algebras # Sections: all # Downloaded: 2026-03-03T02:34:16.625642+00:00 |
Non-degenerate invariant (super)symmetric bilinear forms on simple Lie (super)algebras Abstract. We review the list of non-degenerate invariant (super)symmetric bilinear forms (briefly: NIS) on the following simple (relatives of) Lie (super)algebras: (a) with symmetrizable Cartan matrix of any growth, (b) with non-symm... |
Over algebraically closed fields of positive characteristic, we establish when the deform (i.e., the result of deformation) of the known finite-dimensional simple Lie (super)algebra has a NIS. Amazingly, in most of the cases considered, if the Lie (super)algebra has a NIS, its deform has a NIS with the same Gram matrix... |
Closely related with simple Lie (super)algebras with NIS is the notion of doubly extended Lie (super)algebras of which affine Kac–Moody (super)algebras are the most known examples. |
Key words and phrases: Killing form, positive characteristic, Lie superalgebra 2010 Mathematics Subject Classification: Primary 17B50; Secondary 17B20 |
S.B. and A.K. were partly supported by the grant AD 065 NYUAD. A.K. was partly supported by WCMCS post-doctoral fellowship. A part of this research was done while A.K. was visiting NYUAD; the financial support and warm atmosphere of this institute are gratefully acknowledged. We are thankful to J. Bernstein, P. Grozman... |
To Alexandre Kirillov-père, our teacher 1. Introduction This is a sequel to our talk at the conference “Representation Theory at the Crossroads of Modern Mathematics” in honor of Alexandre Kirillov-père, Reims, May 29 – June 2, 2017. At the talk, we reported two results: that of BLLSq (classification of the simple Lie ... |
We needed derivations and central extensions of simple Lie (super)algebras in the approach to the classification of simple [MATH] -graded Lie (super)algebras of depth 1 we used over [MATH] , see LSh ; we hope to adjust the same approach to classification of simple vectorial Lie (super)algebras for [MATH] and 2 (for [MA... |
Here, we consider two types of [MATH] -graded simple (or close to simple) Lie (super)algebras for which classification is obtained or at least conjectured: finite-dimensional and of polynomial growth; we mention several other types as well. We recall the known results (which of the algebras of these two types have a NI... |
1.1. General comments. Over fields [MATH] of characteristic [MATH] , simple Lie (super)algebras have overwhelmingly many deformations; for [MATH] small (3 or, even worse, 2), the number of deformations becomes appalling. We are interested, however, in deforms , i.e., the results of deformations, rather than in deformat... |
On the last page of Zus , there are given conditions for the deform of a Lie algebra [MATH] with the new bracket [MATH] , where [MATH] , to have a NIS [MATH] , provided [MATH] is a NIS on [MATH] . Namely, for any [MATH] we should have |
[EQUATION] In our computations (proofs of Claims with the aid of the SuperLie package Gr ) we checked when these conditions, and their (not obvious) superizations are fulfilled. We observed the following very interesting fact we can not explain. |
1.1.1. Fact. In all cases for [MATH] we know, except the serial deforms [MATH] and [MATH] , see table ( 72 ), if a given simple Lie (super)algebra has a NIS, then its deform also has a NIS. |
The results of computer experiments (the two series of exceptions from Fact, see Conjecture 5.4.1 ) trouble us, but these exceptions are also a fact (and “fact are stubborn things”). |
In PU , Proposition 5.10 (Proposition 4.4.1 in the arXiv version)] , it is proved, using simple arguments, that “a deformation of a finite-dimensional complex Lie algebra with a NIS, is equivalent to a deformation with the form unchanged…” The statement in PU is not astonishing: over a quadratically closed fields of ch... |
1.2. The types of Lie (super)algebras we consider. (A) Over [MATH] If [MATH] is symmetrizable and invertible, then the Lie (super)algebra [MATH] has a NIS, see ( ), provided either [MATH] , or [MATH] and [MATH] for any root [MATH] |
If [MATH] is non-symmetrizable corresponding to the Lie superalgebra [MATH] of polynomial growth, there are two types of such Lie superalgebras; on [MATH] , there is an odd NIS for [MATH] of one type, and no NIS for [MATH] of the other type, see § 7. |
Over algebraically closed fields [MATH] of characteristic [MATH] , finite-dimensional Lie algebras and Lie (super)algebras [MATH] with indecomposable Cartan matrix [MATH] have been classified, see BGL |
We investigate the existence of NIS on all true deforms of Lie algebras of the form [MATH] , and on their simple relatives. We do not know how many non-isomorphic Lie algebras are there among the deforms of a given [MATH] . To answer this question, one should apply (for example) the technique of Kuznetsov and Chebochko... |
(B) To Dzhumadildaev’s announcement Dz1 on classification of NISes on simple [MATH] -graded vectorial Lie algebras in characteristic [MATH] , reproduced (with explicit proof added) in the book by Strade and Farnsteiner SF we add |
(Ba) the examples of simple vectorial Lie (super)algebras for [MATH] and [MATH] found after Dz1 SF were published, and (Bb) deforms of simple finite-dimensional Lie (super)algebras [MATH] with indecomposable Cartan matrix [MATH] or simple subquotients (“relatives”) thereof; (for the cases these deforms have no Cartan m... |
(C) We extend these investigations of cases (A) and (B) to simple vectorial Lie super algebras, first of all over [MATH] , where the classification is obtained (see Sh5 Sh14 LSh and K10 CaKa and references therein) and also over [MATH] for [MATH] as far as a conjecture and partial classification results (see BGLLS BGLL... |
(D) We also consider the simple Lie superalgebras with an odd supersymmetric invariant bilinear form, and their non-simple relatives, both finite-dimensional and of polynomial growth. |
We do not consider odd parameters of deformations 1.3. NIS: selected applications. In (quantum) inverse scattering method for solving partial differential equations: for a general review, see FT , §1.1, §4.1] BSz ; for a discussion in the context of BV-quantization, see KK |
It is known (due to P. Etingof, I. Losev) that the (super)algebra of observables (deformed Fock space) of the Calogero models based on the root system is simple for almost all values of the coupling constants. Konstein et al. found exact values of coupling constants for which the bilinear forms induced by (super)traces... |
2. Double extension: an interesting notion Well-known examples of indecomposable double extensions to be defined shortly are affine Kac–Moody algebras [MATH] with indecomposable Cartan matrix [MATH] over [MATH] , see Kb [MATH] |
and [MATH] for any [MATH] in characteristic [MATH] ; Lie superalgebras [MATH] [MATH] , and [MATH] for any [MATH] . “Pendant plus de quarante ans, nous avons parlé de prose — et ce n’est pas connu!” |
2.1. Definition of double extension. In this Section, we explain how the construction of double extension works for Lie algebras over fields of any characteristic, and Lie superalgebras over fields of characteristic [MATH] . For the (rather non-trivial) case [MATH] , see BeBou |
Observe that for Lie algebras over fields [MATH] of characteristic [MATH] and Lie superalgebras for any [MATH] , an analog of well-known NIS, called the Killing form , should be sought in projective representations. In other words, the said analog should be related to a non-trivial central extension, see Kapp GP . It o... |
Given a Lie (super)algebra [MATH] , its double extension [MATH] simultaneously involves three ingredients: 1) a central extension [MATH] of [MATH] with the center spanned by [MATH] , so [MATH] |
2) a derivation [MATH] of [MATH] such that [MATH] , a semidirect sum, 3) an [MATH] -invariant NIS [MATH] on [MATH] . Observe that, in super setting, [MATH] can be odd. |
Then, under certain conditions, there is a NIS [MATH] on [MATH] , which extends [MATH] , and the interesting cases are the ones where the Lie algebra [MATH] is not a direct sum of ideals [MATH] and [MATH] ; we call such direct sums decomposable double extensions; it is called reducible in BeBou |
2.1.1. Lemma (On a central extension). (Lemma 3.6, page 73 in BB Let [MATH] be a Lie (super)algebra over a field [MATH] , let [MATH] be an [MATH] -invariant NIS on [MATH] , and [MATH] a derivation such that [MATH] is [MATH] -invariant, i.e., |
[EQUATION] Then the bilinear form [MATH] is a [MATH] -cocycle of the Lie (super)algebra [MATH] Thus, in the assumptions of Lemma 2.1.1 , we can construct a central extension [MATH] of [MATH] given by cocycle [MATH] so that [MATH] |
Let us find out what the conditions for [MATH] -invariance of the cocycle [MATH] are, i.e., when the operator [MATH] , such that [MATH] and [MATH] , is a derivation of the Lie (super)algebra [MATH] . For this, we have to verify that |
[EQUATION] We have [EQUATION] Since the form [MATH] is nondegenerate on [MATH] , it follows that equality ( ) holds for any [MATH] if and only if the operator [MATH] is either even, or odd and such that |
[MATH] . (Recall again that we do not consider Lie superalgebras for [MATH] .) Thus, we have a Lie (super)algebra [MATH] and its derivation [MATH] . Generally, given an arbitrary Lie superalgebra [MATH] and its even derivation [MATH] , we can always construct a semidirect sum [MATH] . If, however, [MATH] is odd, to con... |
On the Lie (super)algebra [MATH] , define a (super)symmetric form [MATH] by setting [EQUATION] 2.1.2. Lemma (On non-degenerate invariant symmetric forms). |
(Theorem 1, page 68 in BB ; for Lie algebras: Exercise 2.10 in Kb The form [MATH] defined by ( ) is a NIS on [MATH] Thus constructed Lie (super)algebra [MATH] with NIS [MATH] on it is called the double extension or, for emphasis, D-extension , of [MATH] |
Remark If the derivation [MATH] is inner, i.e., there exists an [MATH] such that [MATH] , then the operator [MATH] vanishes identically. Replacing [MATH] by [MATH] , we see that the cocycle [MATH] is also the zero one, and hence the Lie (super)algebra [MATH] is a direct sum of its ideal [MATH] and a 2-dimensional commu... |
2.2. Lie (super)algebra [MATH] that can be a double extension of a Lie (super)algebra [MATH] Let [MATH] be a Lie algebra over any field [MATH] or a Lie superalgebra over a field [MATH] of characteristic [MATH] ; let [MATH] be a non-degenerate invariant (super)symmetric bilinear form on [MATH] , and [MATH] a central ele... |
The invariance of the form [MATH] implies that [MATH] for any [MATH] , i.e., the space [MATH] contains the commutant [MATH] of [MATH] , and hence is an ideal. Since the form [MATH] is nondegenerate, the codimension of this ideal is equal to 1. |
If [MATH] , then the Lie (super)algebra [MATH] is just a direct sum [MATH] . This case is not interesting, i.e., decomposable Moreover, even if [MATH] , but [MATH] , then any subspace [MATH] , complementing [MATH] and containing [MATH] , is an ideal and the Lie (super)algebra [MATH] is a direct sum of ideals: [MATH] . ... |
2.2.1. Theorem. Let [MATH] be a Lie algebra over any field [MATH] or a Lie superalgebra over a field [MATH] of characteristic [MATH] ; let [MATH] be a non-degenerate invariant (super)symmetric bilinear form on [MATH] , and [MATH] the center of [MATH] . If [MATH] , then [MATH] is a double extension of a Lie (super)algeb... |
Proof. Let [MATH] be a central element of [MATH] lying in [MATH] . We have seen already that the space [MATH] contains [MATH] , and hence is an ideal of [MATH] . Since [MATH] , we have [MATH] . Non-degeneracy of [MATH] implies that, first, [MATH] , and, second, there exists an element [MATH] such that [MATH] . Since [M... |
[EQUATION] i.e., [MATH] is not central. Let [MATH] . The element [MATH] belongs to the kernel of the restriction [MATH] . Therefore [MATH] descends onto the quotient [MATH] , and remains [MATH] -invariant. Denote this restriction by [MATH] . The action of [MATH] also descend onto [MATH] , defining a derivation [MATH] o... |
[EQUATION] Besides, the [MATH] -invariance of [MATH] implies that [MATH] is [MATH] -invariant. Denote [MATH] . Then [MATH] (as linear spaces) and the natural projection [MATH] is an isomorphism of linear spaces, i.e., we may consider [MATH] as “embedding of [MATH] (as a space) into [MATH] ”. |
As a result, we see that [MATH] is cooked from [MATH] by means of a central extension and a derivation while the form [MATH] is obtained from the form [MATH] by precisely the rules of constructing double extensions. So it only remains to compute the cocycle [MATH] defining the central extension. |
Let [MATH] . Then [EQUATION] Now, we use the invariance condition for the triple [MATH] [EQUATION] But this is precisely the statement that [MATH] is a double extension of [MATH] |
2.3. On history. In 1984, the notion of double extension of Lie algebras (shorter and more suggestively called D-extension in BeBou ) was distinguished, see MR . Medina and Revoy inductively constructed a Lie algebra [MATH] with a non-degenerate [MATH] -invariant symmetric bilinear form [MATH] from an algebra [MATH] of... |
At almost the same time there was written a paper FS , in which the doubly extended Lie algebras [MATH] and [MATH] were considered up to isometry , i.e., an isomorphism [MATH] such that |
[EQUATION] It is reasonable to consider classes of double extensions up to an isometry , not individual double extensions. This proved useful, e.g., in BeBou where several new classes of double extensions explained some results in BGLL |
In BB , one of the first papers on double extensions, the notion was extended to Lie superalgebras for [MATH] . For a most recent of various generalizations of Theorem 2.2.1 , see ABB |
The paper BeBou gives a review of known examples of double extensions, but its main, new, and most interesting results are general constructions and examples of double extensions of Lie superalgebras over fields of characteristic [MATH] |
3. Lie (super)algebras with indecomposable symmetrizable Cartan matrices Any Lie (super)algebra [MATH] with symmetrizable Cartan matrix [MATH] with entries in the ground field [MATH] has an invariant (super)symmetric bilinear form; this form is non-degenerate if [MATH] is invertible. For a precise definition of Cartan ... |
(A) finite-dimensional and of polynomial growth (one stringy and “affine Kac–Moody” (super)algebras, see LSS HS BGL ) and (B) one class of exponential growth: “almost affine”, a.k.a. “hyperbolic”, Lie algebras and superalgebras, see CCLL |
We denote the positive elements of the Chevalley basis (for its definition, see CCLL ), by [MATH] , the corresponding negative ones are [MATH] ; we set [MATH] for the generators |
[MATH] only. 3.1. NIS on [MATH] with [MATH] symmetrizable. Let [MATH] , where [MATH] and [MATH] , be an [MATH] Cartan matrix, [MATH] . For the proof of existence and uniqueness (up to a scalar factor) of NIS in the non-super case over [MATH] , see Kb , Th.2.2, p.17] , the superization and generalization for algebraical... |
Define the (super)symmetric invariant bilinear form [MATH] inductively, starting with Chevalley generators [MATH] of degree [MATH] as follows, where indices in [MATH] are degrees relative the principal [MATH] -grading, whereas in [MATH] they denote the weights: |
[EQUATION] The form [MATH] is NIS on [MATH] for any symmetrizable [MATH] , even if [MATH] . (For example, if [MATH] is non-invertible of corank 1, the form induces NIS on [MATH] , where [MATH] and [MATH] is the center of [MATH] . This, however, is hardly interesting: the point is given a NIS on [MATH] , define a NIS on... |
In the next subsections we consider finite-dimensional modular Lie (super)algebras [MATH] with indecomposable Cartan matrix [MATH] classified in BGL . All such Lie algebras (resp. superalgebras) are rigid for [MATH] (resp. for [MATH] , with the exception of [MATH] ), see BGLd . Below we consider all non-rigid modular L... |
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