School of Education, Culture and Communication
Division of Applied Mathematics
MASTER THESIS IN MATHEMATICS / APPLIED MATHEMATICS
Hom-Lie algebras and deformations
by
Germ ´an Garc´ıa Butenegro
Masterarbete i matematik / tillampad matematik
M ¨ALARDALEN UNIVERSITY SE-721 23 V ¨ASTERAS, SWEDEN
School of Education, Culture and Communication
Division of Applied Mathematics
Master thesis in mathematics / applied mathematics
Date:
2019-05-31
Project name:
Hom-Lie algebras and deformations
Author:
Germ´an Garc´ıa Butenegro
Supervisor(s): Sergei D. Silvestrov Reviewer: Richard Bonner Examiner: Christopher Engstr¨om Comprising: 30 ECTS credits
Abstract
The purpose of this thesis is to review and reconsider the theory of Hom-Lie algebras. We search on more general hypotheses for morphisms σ and τ in order to build as wide a family of deformations as possible. Central extensions of Hom-Lie and quasi-Hom-Lie alge-bras will be discussed as well, with emphasis on deformed Witt algealge-bras and their Virasoro-type central extensions (if any).
The work in this document is based on and extends the results obtained on [1] by Silve-strov et al. in 2006. We attempt to propose a method to establish conditions on possible Virasoro-type extensions of non-linearly deformed Witt algebras.
Acknowledgements
To my parents, for their continued support. To Carmen and Carlos, for discovering and helping me discover my passion for mathematics. To Helena, for her love and support. To Chema and Pilar, for driving me into this crazy adventure that algebra is. To Sergei, for his supervision and advice.
Contents
1 Introduction 4
2 Theoretical background 8
2.1 Algebras and commutators . . . 9
2.2 Derivatives and deformations . . . 10
3 Derivations and Hom-Lie algebras 11 3.1 Hom-algebra structures . . . 11
3.2 (σ , τ)−derivations . . . 12
3.3 Bracket on σ -derivations . . . 18
3.4 Hom-Lie algebras . . . 19
3.5 Morphisms on (σ , τ)−derivations . . . 22
3.6 Extensions of Hom-Lie algebras . . . 22
3.7 Induced twisted structures onA . . . 24
4 Extensions of the Witt algebra 30 4.1 q−deformed Witt algebras . . . 31
4.2 (σ , τ)−derivations on Witt algebras . . . 34
4.2.1 Two (q, −q)−deformations . . . 34
4.2.2 Considerations on σ and τ . . . 36
4.2.3 Another level of q−deformation . . . 37
4.2.4 A twice non-linearly deformed Witt algebra . . . 38
4.2.5 A quasi (p, q)-deformation . . . 39
5 The Virasoro algebra 45 5.1 q−Deformation of the Virasoro algebra . . . 45
5.1.1 Uniqueness . . . 45
5.1.2 Existence . . . 48
5.2 Another non-linear deformation . . . 49
5.2.1 Quotients and finite sums . . . 54
Chapter 1
Introduction
Algebras are one of the fundamental objects in mathematics.
From quantum planes to representations of symmetry groups to determining a certain type of physical constants such as the speed of light, there is one or more underlying algebras with different properties to step on.
Along this document, focus will be put in three particular algebras, say Lie algebras, Witt algebras and Virasoro algebras. Lie algebras appear naturally under the notion of commuta-tor of a vector field and are the most intuitive example of a non-commutative algebra. The commutator indicates how operators act in terms of products in the algebra.
This is one of the key properties in differentiation theory: derivation operators are defined via the Leibniz rule
D( f g) = D( f )g + f D(g), f , g ∈A .
This rule not only defines derivations but also characterizes them. It describes the action of the operator on products in terms of the values it takes on each individual element ofA . During the last 50 years, deformations and twistings have been the driving force behind Mathematics and Physics: in the case of Lie algebras, q−deformations and quantum groups associated to them are widely known, and new deformations and properties to them appear by the minute.
In this direction, one can take an intuitive approach to deformed structures in the form of twisted Leibniz rules. The concept behind Leibniz rule admits different deformations and discretizations, which give place to twisted derivations. One of them is the Jackson
q-derivative( [2], p. 316):
Dq( f )(t) = f(qt) − f (t)
qt− t and tDq( f )(t) =
f(qt) − f (t) q− 1
defined over the Lie algebra of linear operators D on C[t,t−1] which is known as the Witt algebra. This operator is not a derivative, but it holds a q−twisted version of the Leibniz rule.
Dq( f g) = Dq( f )g + σq( f )D(g)
where σqis the automorphism defined by σq( f )(t) = f (qt). This deformation ( [2], Theorem 27) provides a natural q−deformation of the Witt algebra, which will be covered along these pages. The commutation relations obtained don’t look quite natural, as we have made a pre-cise choice for our coefficient algebra and the automorphism σ , these beingA = C[t,t−1] and σqas above in our construction for arbitrary σ −derivations.
Along these pages, we will develop multiple examples of deformations of the Witt alge-bra, parametrized by trios of integers which define particular pairs of morphisms (σ , τ) in A . This condition will be stated to be
c(σ ( f g) − σ ( f )σ (g)) = c(τ( f g) − τ( f )τ(g))
for some unit c ∈A . This approach has two different fundamental strengths: On the one hand, it provides more general results1 compared to the fundamentals of the theory on (σ , τ)−derivations introduced in [1] by Silvestrov et al. in 2006. On the other hand, the de-formations obtained by different choices of the coefficient algebra, morphisms σ and τ and the correspondent (σ , τ)−derivation are exactly the natural algebraic structures for differen-tial calculus and geometry under the corresponding generalized derivation and difference-type operators.
The approach we follow consists on choosingA = C[t,t−1] as the enveloping algebra and switching different (σ , τ)−derivations which can help us construct new structures based on the corresponding product rule. This scheme will be developed along sections 4 and 5, and it is a continuation of the methods used by the authors in [2]. The authors provide structural re-sults based on σ −derivations that can be backtracked into the non-deformed structures. The approach here revolves around establishing new constructions for quasi-Lie algebras based on (σ , τ)−derivations - which can be backtracked into both σ −deformed and non-deformed algebras.
Example 1 (The Virasoro algebra, [2], p. 316). The non-deformed Witt algebra admits a unique (up to multiplication by a scalar) one-dimensional central extension - the Virasoro Lie
1The condition on σ and τ is weaker than endomorphism, which opens the door to more flexibility in the
algebra. Its universal enveloping algebra, also refered to as the Virasoro algebra when no confusion is possible, is the algebra generated by the infinite set{dn: n ∈ Z} ∪ {c} satisfying the following relations:
[dn, dl] = dndl− dldn= (n − l)dn+l+ δn+l,0 1
12(n + 1)(n)(n − 1)c, [c, dl] =cdl− dlc = 0, n, l ∈ Z.
This document intends to provide a method for constructing central extension of non-linearly deformed Witt algebras built on (σ , τ)−derivations. In section 2, we will explore the fundamental results allowing quasi-Lie algebra constructions on (σ , τ)−derivations from [1]. It will be proven that there is a more general condition on σ and τ that provides the same constructions.
In order to do this, we will show that the deformations we build verify both skew-symmetry and a twisted version of the usu Jacobi identity for Lie algebras. This Jacobi-like identity has six terms, three twisted from the inside and three from the outside.
The form of these terms will help us define a class of Hom-Lie algebras with a skew-symmetric multiplication and a Jacobi-like identity, which includes the regular Jacobi iden-tity (and thus, Lie algebras) as the untwisted case. This class can be endorsed into a su-perclass of quasi-Lie algebras ( [1], Definition 1.10, p. 660) that admit different symmetry properties.
Along these pages, the work will focus on those quasi-Lie algebras where the bracket is skew-symmetric: these form a class that will be refered to as quasi-Hom-Lie algebras. This concept is introduced in Hellstrom et al. (2000).
If all six terms can be coupled (i.e. the twisting can be put inside of each bracket instead of outside) into three, we get a deformed Jacobi identity of the form
n,m,lh(id + ς )(dn
), hdm, dliσiσ = 0
which motivates the term Hom-Lie algebras. If the twisting σ is trivial, we get twice the Jacobi identity and thus a regular Lie algebra.
The purpose of this thesis revolves around revisiting and redefining the fundamentals of Hom-Lie algebras and (σ , τ)−derivations in order to provide new more general examples of deformed Witt and Virasoro algebras. In chapter 2 we will introduce the fundamentals on non-associative algebras as well as Lie algebras. Exhaustive definitions and results are de-tailed in Rodriguez (2007). A look will be taken at Leibniz’s product rule and its importance.
Chapter 3 gathers the more theoretical sections. An in-depth reading into the original ar-ticle by Silvestrov et al. (2006, [2]), shows endomorphisms as an intuitive approach to the problem, due to their product-related properties. This fact is reinforced along [2] from the very beginning.
Endomorphism property of σ and τ can be looked upon as
c(σ ( f g) − σ ( f )σ (g)) = 0 = c(τ( f g) − τ( f )τ(g))
for some element c in the center ofA . Following the proof to Theorem 4 ( [2], p. 320) suggests that the desired property - the one being used implicitly - is
c(σ ( f g) − σ ( f )σ (g)) = c(τ( f g) − τ( f )τ(g))
instead. Section 3.2 will use this property extensively in order to tie up a wider class of results regarding twisted derivations.
Brackets introduced in section 3.3 are fundamental for the construction of generalized alge-braic structures ( [1], section 2). The intuitive approach uses skew-symmetry of the brackets providing Hom or quasi-Lie algebras. These are introduced in 3.4 and 3.6.
Application of twisted derivations on non-associative algebras involves determining a sub-module of the space of derivations over the algebra. Interior brackets will be used along 3.7 to determine conditions on morphisms σ and τ giving place to deformed algebraic structures. Results in this direction are established in [1], section 2.2. They accept the new conditions over σ and τ, thus allowing their use to build Hom-Lie algebras. The value of δ , whether it is a constant or a twisted derivation itself, is used to dictate if we can override a Hom-Lie algebra over a certain derivation. The former is covered in [1], while the latter is explored in [6], providing a more general class of quasi-Lie algebras to build upon.
Said results are instrumental, and will be used widely through Chapters 4 and 5, where we take this theory and apply it into building examples of deformed Witt and Virasoro algebras. Sections 4 and 5 start from the canonical q−deformations seen in [2] and apply the dif-ferent techniques and results explored towards finding more general Witt and Virasoro-like algebras.
Chapter 2
Theoretical background
Notation
Theoretical sections:
– A : Non-associative algebra un-less specified otherwise.
– L, ˆL : Lie or Hom-Lie algebra(s). – [·, ·] : Commutator in Lie algebra. – h·, ·i : Defined bracket.
– h·, ·iσ ,τ: Bracket defined in terms of σ and τ.
– h·, ·iL( ˆL) : Bracket defined on L( ˆL).
– : Cyclic sum.
– Dσ ,τ(A ) : Set of (σ,τ)−derivations onA .
– c : An element in the center ofA . – D, ∆ : (σ , τ)−derivations.
– δ : An element inA . – f , g, h : Elements inA . – r : Any gcd((τ − σ )(A )).
– ς , ˆς : The twist associated to a Hom-Lie algebra.
– U FD : Unique Factorization Do-main.
Witt and Virasoro algebras: – A : Witt algebra.
– δ , δa: Elements ofDσ ,τ(A ). – δx,y: Kr¨onecker delta function. – Ln, ˆLn : The n−th generator of L or ˆL. – {n}q : The n−th q−number: qn− 1 q− 1 Products:
– Juxtaposition: denotes the ring product in the algebra.
– The product dot (·): denotes the module product in the space of (σ , τ)−derivations.
– The circle product ◦ : denotes composition of functions in the algebra.
2.1
Algebras and commutators
We follow the definitions from [5], chapter 1.
Definition 2 ( [5], Definition 1.1, p. 7). An algebra over a field F is a vector space A over F with a product operation over it
A × A → A (x, y) 7→ xy
noted by juxtaposition, and verifying the following properties:
1. Distributivity: x(y + z) = xy + xz, (x + y)z = xz + yz for all x, y, z ∈A . 2. Product by scalars: λ (xy) = (λ x)y = x(λ y) for λ ∈ F, x, y ∈ A .
If there is no possible confusion, we will identify the algebra with the associated vector spaceA .
Definition 3 ( [5], Definition 1.2, p. 8). An algebraA is associative if associativity is also respected, i.e. x(yz) = (xy)z for all x, y, z ∈A .
Definition 4 ( [5], Definition 1.3, p. 8). An algebraA is said to be commutative if xy = yx for all x, y ∈A .
Related to this property comes the notion of commutator. Commutators are a fundamen-tal part in understanding how operators behave in different algebras - intuitively, a commu-tator responds to how related xy and yx are as elements ofA .
Definition 5. A commutator [·, ·] :A × A → A is the operator defined by [x,y] = xy − yx. The values of this commutator give place to many different algebraic structures: the first division is made when [x, y] = 0 ∀x, y ∈A which provides commutative algebras.
Another important example of the action of commutators is given by Hermann Weyl (1930s) under the name of Lie algebra.
Definition 6 (Lie algebra). A nonassociative algebraA with an operator [·,·] : A ×A → A verifying the following properties:
1. For all x, y ∈A ,[x,y] = −[y,x]
2. For all x, y, z ∈A ,[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0 is referred to as a Lie algebra.
The first property is direct consequence on the definition of the commutator. Other alge-bras, like Heisenberg algebras (see [6]), admit some form of skewed commutators, where the commutation property is [x, y] = q[y, x] for some q ∈A . Other more general structures will be shown to have even more general commutation rules.
The circular sum that constitutes the second property is the Jacobi identity for Lie algebras: from now on, for the sake of clarity, we will write it as
x,y,z[x, [y, z]] = 0.
2.2
Derivatives and deformations
Leibniz’s product rule is the fundamental property of derivation operators in any algebra. Let f, g ∈A .
Theorem 7 (Leibniz’s product rule). A derivation operator D :A → A satisfies D( f g) = D( f )g + f D(g).
Due to other structural properties, we can find ourselves in a situation where Leibniz’s product rule is not convenient. These spaces are usually deformed by the action of one or two morphisms, namely σ and τ - they can be used to define a new class of twisted derivation operators obeying a generalized version of Leibniz’s product rule.
Definition 8. A (σ , τ)−derivation is an operator D :A → A satisfying a generalize Leib-niz’s product rule
D( f g) = D( f )τ(g) + σ ( f )D(g).
Example 9. One relevant example of these twisted derivation operators, underlying the foun-dations of q−analysis, is the Jackson q−derivative, defined by
Dq( f )(t) = f(qt) − f (t)
qt− t and MtDq( f )(t) =
f(qt) − f (t) q− 1 which act on C[t,t−1] or several other function spaces. It satisfies
D( f g) = D( f )g + σq( f )D(g), for σq( f )(t) = f (qt),
and becomes the usual Witt algebra when considering q= 1. This is a natural q−deformation of the Witt algebra.
Chapter 3
Derivations and Hom-Lie algebras
Along the next pages we will explore the fundamental results on Hom-Lie algebras based on different versions of the Leibniz rule defined by (σ , τ)−derivations.
Conditions will be given on the form of σ and τ in order to build different (σ , τ)−derivations. We will also discuss the action of morphisms on said derivations and the structure of the D−subspace they generate.
Definitions are detailed in [1], section 2.
Following up, we attend bracket operators based on σ and (σ , τ)−derivations on commuta-tive Lie algebras and Witt algebras and establish the conditions for the construction of proper Hom-Lie algebras attending to UFD properties in the underlying coefficient algebra.
In the last subsection central extensions of Hom-Lie algebras will be discussed. An abelian Hom-Lie algebra can be used to define a proper morphism that preserves Hom-Lie algebra properties. The last section of this document is dedicated to the Virasoro algebra of deriva-tions over the Witt algebra, which appears naturally as a central extension and is closely connected to q−calculus and other physical entities.
This type of extensions are established by Silvestrov et al. in [2]. We will follow their steps in order to generalize the method and propose a wider scope of deformed Virasoro-like algebras.
3.1
Hom-algebra structures
The following is an intuitive approach to the nature of Hom-algebra structures.
struc-ture is twisted by a linear space homomorphism.
Following this definition, plenty of well-known algebras admit some degree of deforma-tion: this gives place to Hom-Leibniz, Hom-Poisson and of course Hom-Lie algebras.
Definition 11 ( [9], Definition 1.1, p. 3). A Hom-associative algebra over a vector spaceA is a triple(A , µ,α), where µ : A × A → A is a bilinear map and α : A → A is a linear map satisfying the following condition:
µ (α (x), µ (y, z)) = µ (µ (x, y), α (z)) .
The tensor product of Hom-associative algebras (A1, µ1, α1) and (A2, µ2, α2) is defined in the intuitive way:
(A1, µ1, α1) ⊗ (A2, µ2, α2) = (A1⊗A2, µ1⊗ µ2, α1⊗ α2).
Example 12 ( [9], Definition 1.5, p. 4). A triple (A , µ,α), with a bilinear map µ : A ×A → A and a linear map α : A → A such that
[[x, y], α(z)] = [[x, z], α(y)] + [α(x), [y, z]] is known asHom-Leibniz algebra.
Definition 13. A linear map φ :A1→A2is amorphism of Hom-associative algebras if µ2◦ (φ ⊗ φ ) = φ ◦ µ1and φ ◦ α1= α2◦ φ .
3.2
(σ , τ)−derivations
Definition 14 ( [2], p. 318). A (σ , τ)−derivation D onA is a F−linear map satisfying D(ab) = D(a)τ(b) + σ (a)D(b),
where a, b ∈A . The set of all (σ,τ)−derivations on A is denoted by D(σ ,τ)(A ). Note that, ifA is commutative, then every (σ,τ)−derivation is also a (τ,σ)−derivation and thereforeD(σ ,τ)(A ) = D(τ,σ )(A ).
Definition 15 ( [2], p. 318). A σ −derivation onA is a (σ,id)−derivation, i.e., a F−linear map D satisfying
D(ab) = D(a)b + σ (a)D(b)
These two definitions represent two examples of variations on the Leibniz rule for deriva-tions. Note that we have not included any hypothesis on σ and τ besides the natural them being (strong) morphisms ofA . The upcoming results recover the original results obtained by Silvestrov et al. in [2](section 2.1). The proofs to said results are based on σ , τ being endomorphisms, which is quite a strong hypothesis to keep - Up next, we uncover a more general result, based on how far σ and τ are from the homorphism properties.
Homomorphisms are indeed included in the class of morphisms we have considered1. The original results on these morphisms can be found in [1], p.659.
Lemma 16. Let c be in the center ofA , σ,τ : A → A two morphisms verifying c(σ ( f g) − σ ( f )σ (g)) = c (τ( f g) − τ( f )τ(g)) .
The map D:A → A defined by D( f ) = c(σ( f ) − τ( f )) is a (σ,τ)−derivation. This lemma is parallel to Proposition 1.6 in [1].
Proof. Let D( f g) = c(σ ( f g) − τ( f g)). By definition,
D( f g) = D( f )τ(g) + σ ( f )D(g) = c(σ ( f ) − τ( f ))τ(g) + σ ( f )c(σ (g) − τ(g)) = c(σ ( f )τ(g) − τ( f )τ(g) + σ ( f )σ (g) − σ ( f )τ(g)) = c(σ ( f )σ (g) − τ( f )τ(g))
= c(σ ( f g) − τ( f g)).
Lemma 17. Let σ , τ :A → A such that c(σ( f g) − σ( f )σ(g)) = c(τ( f g) − τ( f )τ(g)) and D be a(σ , τ)−derivation. The following two are equivalent:
1. αD is a (α ◦ σ , α ◦ τ)−derivation.
2. c(α(D( f )τ(g)) − α(D( f ))α(τ(g))) = −c(α(σ ( f )D(g)) − α(σ ( f ))α(D(g))). Proof.
c((α ◦ D)( f g)) = c((α ◦ D)( f ) · (α ◦ τ)(g) + (α ◦ σ )( f ) · (α ◦ D)(g)) c(α(D( f g))) = c(α(D( f ) · τ(g)) + α(σ ( f ) · D(g)))
By definition, the two terms in the left are equal:
c((α ◦ D)( f )(α ◦ τ)(g)) + c((α ◦ σ )( f )(α ◦ D)(g)) = c(α(D( f )τ(g))) + c(α(σ ( f )D(g)))
Reordering terms, we get to:
c(α(D( f )τ(g)) − α(D( f ))α(τ(g))) = (−c)(α(σ ( f )D(g)) − α(σ ( f ))α(D(g))), as sought. The proof is one-directional, but the calculations are reversible. Reciprocally, this condition implies that α ◦ D is indeed a (α ◦ σ , α ◦ τ)−derivation.
Proposition 18 ( [1], Lemma 1.4, p.659). If σ , τ ∈ End(A ) and c is in the center of A , then D:A −→ A , f 7→ D( f ) = c(σ( f ) − τ( f ))
is a(σ , τ)−derivation.
Proof. Since σ , τ ∈ End(A ), c(σ( f g) − σ( f )σ(g)) = 0 = c(τ( f g) − τ( f )τ(g)) and con-clusion follows.
Proposition 19. Let D be a (σ , τ)−derivation, c in the center of A and α : A → A such that
c(α(D( f )τ(g)) − α(D( f ))α(τ(g))) + c(α(σ ( f )D(g)) − α(σ ( f ))α(D(g))) = 0. Then, the following holds:
1. α ◦ D is a (α ◦ σ , α ◦ τ)−derivation. 2. D◦ α is a (σ ◦ α, τ ◦ α)−derivation.
3. If σ is invertible, then σ−1◦ D is a (σ−1◦ τ)−derivation.
4. If τ is invertible, then τ−1◦ D is a (τ−1◦ σ , id)−derivation. IfA is commutative, it is also a(τ−1◦ σ )−derivation.
Proof. I hereby include the demonstration for property 1. Property 2 is proven in the same way, while 3 and 4 come immediately from 1 and 2.
(α ◦ D)( f g) = α(D( f g)) by definition. By hypothesis,
c(α(D( f )τ(g)) − α(D( f ))α(τ(g))) + c(α(σ ( f )D(g)) − α(σ ( f ))α(D(g))) = 0 c(α(D( f )τ(g)) − α(D( f ))α(τ(g))) = −c(α(σ ( f )D(g)) − α(σ ( f ))α(D(g)))
We reorder terms and get to
c(α(D( f )τ(g)) + α(σ ( f )D(g))) = c(α(D( f ))α(τ(g)) + α(σ ( f ))α(D(g))),
where the left term is the expansion for α(D( f g)) and the right term shows that it is, indeed, a (α ◦ σ , α ◦ τ)−derivation.
Well-known corollaries of this result are Proposition 1.5 and Corollary 1.6 from [1]. They can be proven immediately and consecutively, which motivates including them as a single proposition instead of two separate results.
Example 20. We let α :A → A be a morphism in such a way that
c((α ◦ σ )( f g) − (α ◦ σ )( f ) · (α ◦ σ )(g)) = c((α ◦ τ)( f g) − (α ◦ τ)( f ) · (α ◦ τ)(g)) Then D( f ) := c((α ◦ σ )( f ) − (α ◦ τ)( f )) is indeed a (α ◦ σ , α ◦ τ)−derivation.
In the following chapters, we will assume each Unique Factorization Domain (UFD) we use as a commutative associative algebra over C with unity 1 and no zero-divisors. This includes both regular and Laurent polynomial algebras over C.
Note that, when σ (x)a = aσ (x) for all x, a ∈A 2, Dσ ,τ(A ) carries a natural A −module structure defined by the following product:
(a, D) 7−→ a · D x7−→ aD(x)
We shall write a|b ,for a, b ∈A if there is c ∈ A such that ac = b.
Definition 21 ( [2], p.319). If S ⊆A , a greatest common divisor of S, gcd(S), is an element g∈A such that
g|a for a ∈ S,
b|a for a ∈ S ⇒ b|gcd(S)
As a direct consequence of that, S ⊆ T ⊆A implies that gcd(T)|gcd(S)3
Lemma 22. LetA be a commutative algebra, σ,τ two algebra morphisms on A such that c(σ ( f g) − σ ( f )σ (g)) = c(τ( f g) − τ( f )τ(g))
and D a(σ , τ)−derivation onA . Then
D(x)(τ(y) − σ (y)) = 0
for all x∈ ker(τ − σ ), y ∈A . Moreover, if A has no zero-divisors and σ 6= τ then ker(τ − σ ) ⊆ ker(D).
2The same applies to τ, andA being commutative is a particular case.
3As long as both exist. However, in UFDs it not only exists, but is also unique - thus, we shall speak about
This lemma serves as support for the following result, which will be our bread and butter later on. It provides a concrete constructive method to build Hom-Lie algebras from a UFD and a pair of suitable morphisms defined on it. It is based on [1], p. 319, but incorporates new, more flexible conditions on the morphisms involved.
Theorem 23 ( [2], Theorem 4, p. 319). Let σ , τ be different algebra morphisms on a UFD A verifying
c(σ ( f g) − σ ( f )σ (g)) = c(τ( f g) − τ( f )τ(g))
for some c in the center ofA . Then D(σ ,τ)(A ) is free of rank 1 as an A −module, and is generated by the following operator:
∆ := (τ − σ )
g : x 7−→
(τ − σ )(x)
r ,
where r= gcd((τ − σ )(A )).
Proof. Firstly, we check that cD = cτ − σ
r is a (σ , τ)−derivation: cD( f g) = c(τ − σ )( f g) r = c(τ( f g) − σ ( f g)) r = c(τ( f )τ(g) − σ ( f )σ (g)) r = = c((τ( f ) − σ ( f ))τ(g) − σ ( f )(τ(g) − σ (g))) r = c (τ − σ )( f ) r σ (g)+ + cτ( f )(τ − σ )( f ) r = (cD( f )τ(g) + σ ( f )cD(g)).
If cD is a (σ , τ)−derivation over a commutative, associative and algebra A with unit then D is as well as long as c is a unit.
Secondly, assume x · cτ − σ
r = 0. If σ 6= τ, there is y ∈A such that σ(y) 6= τ(y) → x· c(τ − σ )(y)
r = 0 ⇒ x = 0 ⇒A ·cD is a free A -module - And since cD is the only gener-ator, it has rank 1.
Lastly, one must show that Dσ ,τ(A ) ⊆ A · cD. Indeed, let ∆ be a (σ,τ)−derivation in A , and let a∆∈A such that ∆( f ) = a∆c(τ − σ )( f )
r , f ∈A . Two conditions must be satisfied: 1. (τ − σ )( f ) divides D( f )r.
2. D( f )r c(τ − σ )( f ) =
D(g)r
For the first part, let f , g ∈A with (τ −σ)( f ) 6= 0 6= (τ −σ)(g). We know the following:
0 = ∆( f g − g f ) = ∆( f )τ(g) + σ ( f )∆(g) − ∆(g)τ( f ) − σ (g)∆( f ) =
= ∆( f )(τ(g) − σ (g)) − ∆(g)(τ( f ) − σ ( f )) ⇒ ∆( f )(τ(g) − σ (g)) = ∆(g)(τ( f ) − σ ( f )) Now we define the following function:
h:A × A → A
(z, w) 7−→ gcd(τ(z) − σ (z), τ(w) − σ (w))
By election of f , g, h( f , g) 6= 0, so we can divide by h( f , g) in the equation above:
∆( f )τ (g) − σ (g)
h( f , g) = ∆(g)
τ ( f ) − σ ( f ) h( f , g) ⇒
Now, it is true that gcd τ (g) − σ (g) h( f , g) , τ ( f ) − σ ( f ) h( f , g) = 1, so τ ( f ) − σ ( f ) h( f , g) divides ∆( f ) ⇒ ⇒ (τ − σ )( f )|D( f ) · h( f , g).
Finally, let S =A \ker(τ − σ). Then (τ − σ)( f )|D( f ) · h( f ,S). But
gcd(h( f , S)) = gcd({gcd((τ − σ )( f ), (τ − σ )(s)|s ∈ S}) = gcd((τ − σ )(S) ∪ {(τ − σ )( f ))}) = = gcd((τ − σ )(A ) ∪ {(τ − σ)( f )}) = r ⇒ (τ − σ)( f )|D( f )r.
In order to prove the second point we recall
0 = ∆( f g − g f ) = ∆( f )(τ(g) − σ (g)) − ∆(g)(τ( f ) − σ ( f )). From which it is immediate that
∆( f )c(τ(g) − σ (g)) r = ∆(g) c(τ( f ) − σ ( f )) r ⇒ D( f )r c(τ( f ) − σ ( f )) = D(g)r c(τ(g) − σ (g)), as sought. Note thatA being a UFD allows to drop the c as long as it is a unit, and therefore the result applies for D as well.
This result allows a wider spread of possibilities regarding the morphisms σ and τ we can choose in order to generate new algebraic structures.
3.3
Bracket on σ -derivations
LetA be a commutative associative algebra over F with unity 1. For clarity, we will denote the module multiplication by · and the algebra multiplication by juxtaposition.
Definition 24 ( [2], p.322). The annihilator Ann(D) of an element D ∈Dσ(A ) is the set of all a∈A verifying a · D = 0.
This is clearly an ideal inA .
Theorem 25 ( [2], Theorem 5, p. 323). Let σ :A → A , D ∈ Dσ(A ) and element δ ∈ A , such that the following holds:
σ (Ann(D)) ⊆ Ann(D), D(σ (a)) =δ σ (D(a)), a ∈A
LetA · D = {a · D : a ∈ A } be the cyclic A −submodule of Dσ(A ) generated by D. The map below:
h·, ·iσ :A · D × A · D −→ A · D
defined by the following setting (◦ denotes composition of functions): ha · D, b · Diσ = (σ (a) · D) ◦ (b · D) − (σ (b) · D) ◦ (a · D), a, b ∈A . is a well-defined F−algebra product on A · D. It verifies the following properties:
ha · D, b · Diσ = (σ (a)D(b) − σ (b)D(a)) · D, ha · D, b · Diσ = −hb · D, a · Diσ.
Additionally to this skew-symmetric property, the bracket h·, ·iσ holds to a Jacobi-like identity defined as
hσ (a) · D, hb · D, c · Diσiσ+ δ ha · D, hb · D, c · Diσiσ+ hσ (b) · D, hc · D, a · Diσiσ+ δ hb · D, hc · D, a · Diσiσ+ hσ (c) · D, ha · D, b · Diσiσ+ δ hc · D, ha · D, b · Diσiσ = 0. From now on, we will employ the usual cyclic sum notation
a,b,chσ (a) · D, hb · D, c · Diσ
to refer to these Jacobi-like identities.
IfA is not commutative, one additional condition on D is required: [a, b] D(c) = 0 for all a, b, c ∈A .
In the following sections, this will be refered to as a quasi-Lie algebra.
Proposition 26 ( [2], Proposition 9, p. 328). LetA be a commutative F−algebra without zero-divisors, σ :A −→ A an algebra endomorphism. If 0 6= D ∈ Dσ(A ) and 0 6= D0∈ Dσ(A ) generate the same cyclic A −submodule M of Dσ(A ), there is a unit u ∈ A such that
hx, yiσ ,D= u · hx, yiσ ,D0. Furthermore, if u∈ F then
(M ,h·,·iσ ,D) ∼= M ,h·,·iσ ,D0
This proposition provides the tools needed for a proper definition of a generalized Witt algebra.
Definition 27 ( [2], Definition 10, p.328). LetA be a commutative and associative algebra, σ :A −→ A an algebra endomorphism and D a σ− derivation on A . Then a (A ,σ,D)− Witt algebra or generalized Witt algebra is the non-associative4 algebra (A · D,h·,·iσ ,D) together with the product defined by
ha · D, b · Diσ ,D= (σ (a)D(b) − σ (b)D(a)) · D.
Example 28 ( [2], Example 11, p.328). LetA = Ct,t−1, σ = idA the identity operator onA , D = dtd and δ = 1.
In this conditions it is clear thatA · D is equal to the whole Dσ(A ). The skew symmetry property comes trivially, and the Jacobi-like identity becomes twice the usual Jacobi identity.
3.4
Hom-Lie algebras
Definition 29 ( [2], p. 330). A hom-Lie algebra (L, ς ) is a non-associative algebra L together with an algebra homomorphism ς : L −→ L such that
hx, yiL= −hy, xiL,
x,y,zh(id + ς )(x), hy, ziLiL= 0 for all x, y, z ∈ L. Here h·, ·iL denotes the product in L.
Example 30. If we take ς = id we get the exact definition of regular Lie algebras.
Example 31. A Hom-Leibniz algebra (A ,[·,·],α) in which the bracket is skew-symmetric is by definition a Hom-Lie algebra. The twisting morphism is α in this case.
This example motivates the name Hom-Lie algebras, as they can be seen as a deforma-tion of Lie algebras via a proper homomorphism. They are introduced by Silvestrov et al. in [2] as the natural extension of Lie algebras: the Hom-Jacobi identity introduces a twist by the morphism ς in the first term of the bracket - taking ς := id returns twice the original Jacobi identity.
Hom-Lie algebras lie within the superclass of quasi-Lie algebras, which helps to link dif-ferent extensions and revisions of Lie algebras. These include (Hom)color Lie algebras and other known structures. Naturally Lie algebras fall into this superclass, but also quasi-Hom-Lie algebras, as defined by Larsson and Silvestrov in [8](p. 4306), where the Hom-Jacobi identity is twisted yet again by a morphism σ based on a σ −derivation.
Definition 32 ( [1], Def. 1.10, p. 660). A quasi-Lie algebra is a tuple (A ,[·,·],α,β,ω,θ) where
1. A is an algebra.
2. [·, ·] :A ⊗ A → A is a bilinear map, known as product or bracket. 3. α, β :A → A are linear maps.
4. ω : Dω →LR(V ), θ : Dθ →LR(V ) are maps defined on Dω, Dθ ⊆A ⊗ A . 5. ω-symmetry: The product verified a version of the skew-symmetry condition:
[x, y] = ω(x, y)[y, x] for all (x, y) ∈ Dω.
Definition 33. A quasi-Lie algebra where ω := −idA is known asquasi-Hom-Lie algebra. Example 34 ( [2], Example 16, p.330). Let a be any vector space. If we consider
hx, yia= 0
for any x, y ∈ a, then (a, ςa) is trivially a Hom-Lie algebra - the above conditions hold imme-diately. These are known asabelian or commutative Hom-Lie algebras.
Definition 35. A morphism of Hom-Lie algebras is a linear map ϕ : (L1, ς1) −→ (L2, ς2) such that
2. ϕ ◦ ς1= ς2◦ ϕ.
If only the first condition holds, ϕ is known as a weak morphism.
Theorem 36 ( [1], Theorem 1.11, p.660). Let (L, ς ) be a Hom-Lie algebra, ρ : L → L a weak morphism, i.e. ρ ◦ [·, ·] = [·, ·] ◦ (ρ ⊗ ρ). Then (L, [·, ·]ρ = ρ ◦ [·, ·], ςρ = ρ ◦ ς ) is a Hom-Lie algebra.
Definition 37. A homomorphism of Hom-Lie algebras ϕ : (L1, ς1) −→ (L2, ς2) is an algebra homomorphism from L1to L2such that ϕ ◦ ς1= ς2◦ ϕ.
Proposition 38 ( [1], Proposition 1.12, p. 661). Let ψ be an isomorphism between two Hom-Lie algebras(L, [·, ·], ς ) and (L, [·, ·]ρ, ςρ). Then ρ = ψ ◦ ς ◦ ψ−1◦ ς−1.
This result has important consequences, as it establishes a fundamental structural dif-ference between Lie and Hom-Lie algebras - the nature of the twist operation ρ makes this possible.
Corollary 39 ( [1], Corollary 1.13, p. 661). Let (L, id) be a Lie algebra, and (L, [·, ·]ρ, ρ) a Hom-Lie algebra isomorphic to it. Then the twist is trivial, i.e. ρ = id.
An important task when given a Hom-Lie algebra is the possibility to transfer it via a proper morphism into a more general algebraic structure. The following proposition opens that possibility, with the added flavour that a proper restriction of the chosen morphism provides another Hom-Lie algebra.
Proposition 40 ( [2], Proposition 18, p.331). Let (L, ς ) be a Hom-Lie algebra, N any non-associative algebra, and ϕ : L −→ N an algebra homomorphism. The following conditions are equivalent:
1. There is a linear subspace U⊆ N containing ϕ(L) and a linear map k : U −→ N such that ϕ ◦ ς = k ◦ ϕ.
2. kerϕ ⊆ ker(ϕ ◦ ς ).
Moreover, under any of these conditions the following holds: • k is uniquely determined on ϕ(L) by ϕ and ς .
• k|ϕ (L)is a homomorphism.
• (ϕ(L), k|ϕ (L)) is a Hom-Lie algebra.
• ϕ is a homomorphism of Hom-Lie algebras.
This proposition motivates the concept of extensions of Hom-Lie algebras, which we will explore later on.
3.5
Morphisms on (σ , τ)−derivations
The nature of morphisms of Hom-Lie algebras (see Definition 34) suggest that they have a commutativity-measuring effect between (σ , τ)−derivations and Hom-Lie algebra mor-phisms - which is perfectly solid provided that the former is but a particular case of the latter. This is by all means non-trivial, as it ensures that the composition of functions is well-defined in both directions.
Proposition 41. Let φ be a Hom-Lie algebra morphism, D a (σ , τ)−derivation, α another Hom-Lie algebra morphism in such a way that φ (D( f g)) = α(D(φ ( f )φ (g))). Then the following holds:
1. D◦ φ is a (σ ◦ φ , τ ◦ φ )−derivation. 2. α ◦ D is a (α ◦ σ , α ◦ τ)−derivation.
3. α ◦ D ◦ φ is a (α ◦ σ ◦ φ , α ◦ τ ◦ φ )−derivation. 4. φ ◦ D = α ◦ D ◦ φ .
Proof. The first three are immediate by Proposition 14. We use 2 and 3 in succession: φ (D( f g)) = (α ◦ D)(φ ( f )φ (g)))
φ (D( f g)) = (α ◦ D)(φ ( f ))(α ◦ τ )(φ (g)) + (α ◦ σ )(φ ( f ))(α ◦ D)(φ (g)) φ (D( f g)) = (α ◦ D ◦ φ )( f g)
The morphism α is a sort of default of commutation: In a commutative associative alge-bra, α = id - in non-associative algebras, there are some extra considerations:
• If φ is invertible, then α ◦ D = φ ◦ D ◦ φ−1, and thus α ◦ D is the conjugate of D in the algebra.
• If both φ and D are invertible, then α = φ ◦ D ◦ φ−1◦ D−1, which can be computed directly.
3.6
Extensions of Hom-Lie algebras
In this section we will focus on the fundamental results behind Hom-Lie algebra extensions. In section 5 this theory will be put into practice in the case of the q−deformed Witt al-gebra: it can be extended into the q−deformed Virasoro algebra of derivations on Laurent
polynomials. Then, we will develop a method to establish extensions of more general alge-bras.
First of all, let U and V be vector spaces over F, and Alt2(U,V ) denote the space of alternating mappings U ×U −→ V.
Definition 42 ( [2], Definition 20, p.333). An extension of a Hom-Lie algebra (L, ς ) by an abelian Hom-Lie algebra(a, ςa) is a commutative diagram with exact rows
l pr
0 −→ a −→ ˆL −→ L −→ 0 ςa↓ ς ↓ˆ ς ↓
0 −→ a −→ ˆL −→ L −→ 0, where( ˆL, ˆς ) is a Hom-Lie algebra.
The extension is saidcentral if l(a) ⊆ Z( ˆL) = {x ∈ ˆL : 0 = hx, ˆLiˆL}. This definition raises multiple questions. The first one comes naturally: Under which conditions does the above diagram provide a central extension?
For the sake of clarity, we will omit diagrams completely. Instead, we will develop this theory in terms of linear sections.
Definition 43 ( [2], p. 333). A section s : L → ˆL is a linear morphism between the two algebras such that
pr◦ s = idL.
Theorem 44 ( [2], Theorem 21, p. 335). Let (L, ς ) and (a, ςa) be Hom-Lie algebras, a abelian. If there exists a central extension( ˆL, ˆς ) by (a, ςa) then for every section s : L −→ ˆL there is some gs∈ Alt2(L, a) and a linear map f
s: ˆL −→ a verifying fs◦ l = ςa,
gs(ς (x), ς (y)) = fs(hs(x), s(y)iˆL) and
x,y,zgs((id + ς )(x), hy, ziL) = 0
The proof to this result is pretty straight up calculation, the reader can find it in [2], p. 333-336.
Definition 45 ( [2], Definition 22, p. 336). A central Hom-Lie algebra extension ( ˆL, ˆς ) of (L, ς ) by (a, ςa) is known as trivial if there is a linear section s : L −→ ˆL such that gs(x, y) = 0 for all x, y ∈ L.
Theorem 46 ( [2], Theorem 24, p. 336). Let (L, ς ), (a, ςa) be Hom-Lie algebras and a abelian. Then for every g∈ Alt2(L, a) and every linear map f : L ⊕ a −→ a verifying
f(0, a) = ςa(a) for a ∈ a, g(ς (x), ς (y)) = f (hx, yiL, g(x, y))
x,y,zg((id+ς )(x), hy, ziL) = 0
for x, y, z ∈ L there is a Hom-Lie algebra ( ˆL, ˆς ) which is a central extension of (L, ς ) by (a, ςa).
3.7
Induced twisted structures on
A .
Let σ :A → A be a morphism of A . The interior bracket defined in 3.3 naturally provides a Hom-Lie algebra based on σ .
Theorem 47. The set Dσ(A ) of σ−derivations on A is a free A −module of rank one, generated by any mapping of the form
id− σ gcd((id − σ )(A )) This set can be equipped with the bracket defined by
h f · D, g · Diσ = (σ ( f ) · D) ◦ (g · D) − (σ (g) · D) ◦ ( f · D). The bracket satisfies the following properties:
h f · D, g · Diσ = (σ ( f )D(g) − σ (g)D( f )) · D, h f · D, g · Diσ = −hg · D, f · Diσ,
As well as the cyclic sum, for f, g, h ∈A :
h(σ ( f ) + f ) · D, hg · D, h · Diσiσ+ h(σ (g) + g) · D, hh · D, f · Diσiσ+ h(σ (h) + h) · D, h f · D, g · Diσiσ = 0.
This bracket provides a Hom-Lie algebra structure, together with the following homo-morphism:
ς :Dσ(A ) −→ Dσ(A ) f· D 7−→ σ ( f ) · D
Proof. The proof to this result can be consulted in [2], Theorem 4, p. 320.
The intuitively analogous result is indeed valid for (σ , τ)−derivations. This will be con-sidered again later on, in order to remain consistent with the inner logic of this thesis. Now, introduce a new morphism τ :A → A . Firstly, assume that it is invertible. Then τ−1◦ D is a (τ−1◦ σ )−derivation ( [1], Corollary 1.6). This allows to rewrite the bracket (we omit τ−1◦ σ ) from the notation to aliviate the reading:
τ (h f · τ−1◦ D, g · τ−1◦ Di) =
= (σ ( f ) · D) ◦ (b · τ−1◦ D) − (σ (g) · D) ◦ ( f · τ−1◦ D) =: hτ( f ) · D, τ(g) · Di
Considering this as our new bracket, together with τ going inside the bracket, allows the following theorem to introduce a quasi-Lie algebra structure onA .
Theorem 48 ( [1], Theorem 2.1, p.662). LetA be a commutative associative algebra with unity, σ , τ :A → A , τ invertible and D a (σ,τ)−derivation. If the inclusion
(σ ◦ τ−1)(Ann(D)) ⊂ Ann(D) holds, the maph·, ·iσ ,τ :A · D × A · D → A · D defined by:
h f ·D, g·Diσ ,τ:= ((σ ◦τ−1)( f )·D)◦(τ−1(g)·τ−1◦D)−((σ ◦τ−1)(g)·D)◦(τ−1( f )·τ−1◦D) for f, g ∈A is a well-defined algebra bracket on the R-linear space A · D. It satisfies the following identities:
h f · D, g · Diσ ,τ = ((σ ◦ τ−1)( f )(D ◦ τ−1)(g) − (σ ◦ τ−1)(g)(D ◦ τ−1)( f )) · D, hg · D, f · Diσ ,τ = −h f · D, g · Diσ ,τ.
Moreover, if there is δ ∈A such that
D◦ τ−1◦ σ ◦ τ−1= δ (σ ◦ τ−1◦ D ◦ τ−1), then f,g,h hσ (τ −1( f )) · D, hg · D, h · Di σ ,τiσ ,τ+ δ h f · D, hg · D, h · Diσ ,τiσ ,τ = 0.
This equation will be refered to as quasi-Jacobi identity. It defines a quasi-Lie algebra structure onA · D, with α = σ ◦ τ−1, β = δ , θ = idA and ω = −idA.
Proof. The proof for this theorem is straight-up calculation: first we prove that the bracket is well-defined, and afterwards the quasi-Jacobi identity. Details can be consulted in [1], p.662-663.
Corollary 49 ( [1], Corollary 2.3, p.663). Under the same hypothesis, if (σ ◦ τ−1)(δ ) = δ and D(δ ) = 0(i.e. when δ = λ idA is a multiple of the unit ofA ), the previous quasi-Jacobi identity collapses into a Hom-Jacobi identity and A · D is a Hom-Lie algebra with twist α = σ ◦ τ−1+ δ idA
Removing invertibility on τ forces an interior bracket in the form h f · D, g · Diσ ,τ := (σ ( f )D(g) − σ (g)D( f )) · D.
This bracket satisfies a quasi-Jacobi identity, as stated by Silvestrov et al. in [1] (section 2.2, p. 664-665). Full details on the form of δ and its effect on the Jacobi-like identity can be found in [6], p. 4306-4307. The upcoming result gathers Theorem 2.6 from [1] and Theorem 1 from [6].
Theorem 50 ( [1], Theorem 2.6, p.665). LetA be a commutative associative algebra with unity, D a(σ , τ)−derivation ofA with σ,τ : A → A being two algebra morphisms satis-fying, for every f ∈A :
σ (τ ( f )) = τ (σ ( f )) D(σ ( f )) = δ σ (D( f ))
D(τ( f )) = δ τ(D( f )) for some δ ∈Dσ ,τ(A ). The bracket h·,·iσ ,τ defined by
h f · D, g · Diσ ,τ = (σ ( f )D(g) − σ (g)D( f )) · D
is a well-defined product in A . It satisfies skew-symmetry and a twisted Jacobi-like identity:
n,m,lhσ (dn
), hdm, dliσ ,τiσ ,τ+ δ hdn, hdm, dliσ ,τiσ ,τ = 0.
This provides a quasi-Lie algebra structure on the space(A · D,h·,·iσ ,τ, σ + τ). If addi-tionally δ ∈A , the Jacobi-like identity collapses into a three-term Hom-Jacobi identity and thus we have a Hom-Lie algebra.
Proof. Outline: We extend the interior bracket and apply the commutation properties di-rectly: most terms in the sum cancel, and the remaining ones become a quasi-Jacobi identity. If δ ∈A , the sum collapses into a three-term Hom-Jacobi identity in terms of σ + τ. The proof is sketched in [2], section 2.2, p. 664-665. For the final analysis on the form of δ , the reader can consult [6], p. 4306-4307.
Unique Factorization property inA immediately provides the commutation relation be-tween D, σ and τ established in Theorem 48 ( [1], Theorem 2.2, p. 662).
Proposition 51 ( [1], Proposition 2.4, p.663). IfA is a UFD, σ,τ : A → A are two different algebra morphims and τ is invertible, then equation
D◦ τ−1◦ σ ◦ τ−1= δ (σ ◦ τ−1◦ D ◦ τ−1)
holds with δ =(σ ◦ τ −1)(r)
r , where r= gcd((τ − σ )(A )).
Importance of this result lies in it providing a direct way for calculating δ . The form of δ is extremely concrete, which suggest a generalization is possible for different morphisms τ - this is indeed the case if τ and σ commute, as stated in the following result.
Proposition 52 ( [1], Proposition 2.8, p.665). IfA is a UFD and σ,τ : A → A are two morphisms ofA such that στ = τσ, then
D(σ ( f )) = δ σ (D( f )) D(τ( f )) = δ τ(D( f )) implies δ = σ (r) r = τ (r) r , where r= gcd((τ − σ )(A )). Proof. Let y ∈A and
x:= τ − σ r (y) =
τ (y) − σ (y) r Since τ and σ commute,
σ (rx) = σ
rτ (y) − σ (y) r
= σ (τ(y) − σ (y)) = σ (τ(y)) − σ2(y) = (τ − σ )(σ (y)) τ (rx) = τ
rτ (y) − σ (y) r
= τ(τ(y) − σ (y)) = τ2(y) − τ(σ (y)) = (τ − σ )(τ(y)) Since r|τ(r) − σ (r), if it divides one of them it divides both. Assume that to be the case, replace the expression on x and divide by r:
σ (r) r σ τ − σ r (y) = τ − σ r (σ (y)) ⇐⇒ σ (r) r σ (D(y)) = D(σ (y)) τ (r) r τ τ − σ r (y) = τ − σ r (τ(y)) ⇐⇒ τ (r) r τ (D(y)) = D(τ (y)).
If τ commutes with σ , we can use the previous proposition to define a Hom-Lie algebra structure onDσ ,τ(A ) = A · D when A is a UFD.
This result will be used constantly along section 4. The Witt algebra of Laurent polyno-mials over C is a UFD and also commutative, which sets the perfect conditions for arbitrary usage of this method. A pair of properly chosen morphisms σ and τ will provide an adequate quasi-Lie structure.
Example 53. Morphisms on Laurent polynomials are either traslations t → t − j, scaling functions t→ txor a combination of both.
An arbitrary pair of morphisms is unlikely to provide the desired structural properties. Given a morphism α, it is known we can build (α ◦ σ , α ◦ τ)−derivations attending to a certain composition rule with σ and τ: these derivations open the possibility that apparently unmatching pairs of morphisms provide Hom or quasi-Lie algebra constructions via compo-sition with a proper α.
Up next, we set our sights in the Witt algebra: morphisms in this algebra are either scal-ing morphisms t → t − j, shifting morphisms t → tx or a combination of both. We consider both possibilities for α over pairs of scaling morphisms in order to establish proper condi-tions for this composition to work.
Consider two scaling morphisms σ : t → qta, τ : t → ptbfor some a, b ∈ Z and q, p ∈ C. We take two possible scaling t→ tx and traslation t→ t − j morphisms, searching whether or not we can use them to build new (α ◦ σ , α ◦ τ)−derivations. Would this be the case, it will provide a quick and standard method to find new twisted derivations.
We start considering the fundamentalquasi-product rule on σ and τ :
c(σ ( f g) − σ ( f )σ (g)) = c(τ( f g) − τ( f )τ(g))
In order to provide a(α ◦ σ , α ◦ τ)−derivation, we demand the same property from α ◦ σ and α ◦ τ :
Let f = tn, g = tl for some n, l ∈ Z: c((α ◦ σ )( f g) − (α ◦ σ )( f )(α ◦ σ )(g)) = c((α ◦ τ)( f g) − (α ◦ τ)( f )(α ◦ τ)(g)) c qn+lα (t(n+l)a) − qn+lα (tna)α(tla) = cpn+lα (t(n+l)b) − pn+lα (tnb)α(tlb) c qn+lα (t(n+l)a) − pn+lα (t(n+l)b) = cqn+lα (tna)α(tla) − pn+lα (tnb)α(tlb) .
Now we observe that every operator of the form α : t → ytx is a homomorphism, and then both sides of the equation become zero.
We let α : t → t − j for some j ∈ C\{0}.
cqn+lt(n+l)a− j − (tna− j)(tla− j)= cpn+lt(n+l)b− j − (tnb− j)(tlb− j) cqn+lt(n+l)a− j − t(n+l)a+ jtna+ jtla− j2= cpn+lt(n+l)p− j − t(n+l)b+ jtnb+ jtlb− j2 cqn+lj tna+ tla− 1 − j= cpn+lj tnb+ tlb− 1 − j We get to p q n+l =t na+ tla− 1 − j tnb+ tlb− 1 − j
These two are polynomials, and thus both have the same degree: either na= nb, la = lb⇒ a = b or na = lb, la = nb for all values of n and l - but the last option implies l = n, which is a contradiction. We conclude a= b, but there is more.
a= b ⇒ tna+ tla− 1 − j = tnb+ tlb− 1 − j ⇒ p q
n+l = 1.
One can conclude p= ±q, but the equality holds for all entire values of n and l, which implies p= q.
This allows to conclude that arbitrary composition of morphisms does not always lead to a derivation with the desired properties. Scaling functions always provide said derivations, but traslation only does if both traslation5morphisms are the same.
Chapter 4
Extensions of the Witt algebra
In the following section, we provide a theoretical construction for q−deformations of the Witt algebra. The fundamental central extension of the Witt algebra is the Virasoro algebra, whose properties will be shown and developed in terms of q−numbers - They allow a proper and clean representation of the Hom-Jacobi identity of Virasoro algebras.
The Witt algebra, infinite-dimensional Lie algebra of complex polynomial vector fields on the unit circle, is a relevant case of Lie algebra, as some deformations that will be seen connect naturally to the algebraic structures behind differential and integral calculi and ge-ometry - the difference lying on the corresponding derivation and difference operators.
Definition 54 (Witt algebra). The Witt algebra is the Lie algebra of derivations in the algebra of complex Laurent polynomials in one variable, i.e. the following Lie algebra:
• Enveloping algebra: DA, the algebra of linear operators D onA . • Lie algebra product: h f , gi = f g − g f .
• Derivation rule: D( f g) = D( f )g + f D(g).
The Witt algebra is isomorphic to an infinite-dimensional associative algebra with gen-erators {dn: n ∈ Z} satisfying the relations
hdn, dli = dndl− dldn= (n − l)dn+l, for n, l ∈ Z,
These relations are the starting point for a generalization of the Witt algebra - twisting the Leibniz rule via proper morphisms makes it posible for more general twisted derivations to be considered.
4.1
q−deformed Witt algebras
Consider the complex algebraA of Laurent polynomials in one variable t A = Ct,t−1 .
Let q ∈ C\{0, 1} and σ be the endomorphism on A determined by σ (t) = qt.1
At this moment, we have all the tools we need to fully explain how Jackson q−derivative is the perfect example of how these structures are built, as well as provide some insight on how this result can be stretched into (σ , τ)−derivations.
Example 55 ( [2], Example 3.1, p. 338). The set Dσ(A ) of σ−derivations on A is a freeA −module of rank one (see first section for details), and the q − derivative mapping D:A −→ A given by D( f (t)) = tσ ( f (t)) − f (t) σ (t) − t = f(qt) − f (t) q− 1 provides a generator.
Indeed,A being a UFD means a generator of Dσ(A ) has the form id− σ
gcd((id − σ )(A )) by theorem 18.
Now, any gcd((id − σ )(A )) divides any element of (id − σ)(A ), so it particularly di-vides (id − σ )(t) = −(q − 1)t - which is a unit when q 6= 1. This implies it has the form ctk, c ∈ C\{0}, k ∈ Z, and then q − 1 is a gcd((id − σ )(A )). Putting the last two equations together,
D( f (t)) = f(qt) − f (t) q− 1 = −
id− σ
q− 1 ( f (t)), so D= −id−σq−1 is a generator forDσ(A ).2
D is a polynomial over C in σ , so D and σ commute. We denote h·, ·iσ the following product onDσ(A ):
h f · D, g · Diσ = (σ ( f ) · D) ◦ (g · D) − (g · D) − (σ (g) · D) ◦ ( f · D).
1Immediately, σ ( f (t)) = f (qt), for f (t) ∈A . 2Particularly, t−1D= D
It trivially satisfies the following properties:
h f · D, g · Diσ = (σ ( f )D(g) − σ (g)D( f )) · D, h f · D, g · Diσ = −hg · D, f · Diσ,
as well as the cyclic sum, for f, g, h ∈A :
f,g,hh(σ ( f ) + f ) · D, hg · D, h · Diσ iσ = 0
The last two identities show thatDσ(A ) is indeed a Hom-Lie algebra, together with the following homomorphism:
ς :Dσ(A ) −→ Dσ(A ) f· D 7−→ σ ( f ) · D
As a C−linear space, Dσ(A ) has a basis {dn: n ∈ Z}. Those generators have the form dn= −tn· D. Note that σ (−tn) = −qntn⇒ ς (dn) = qndn. Furthermore, D(−tn) = −q ntn+ tn q− 1 = −{n}qt n.3
Our bracket, defined above, can be described in terms of generators as hdn, dliσ = qndndl− qldldn.
Now, let f(t) = −tn, g(t) = −tl. We examine how they behave under our bracket and the relations given for Hom-Lie algebras:
hdn, dliσ = ((−qntn)(−{l}qtl) − (−qltl)(−{n}qtn)) · D = qlq n− 1 q− 1 − q nql− 1 q− 1 (−tn+l) · D = q n− ql q− 1 dn+l= ({n}q− {l}q)dn+l. 3Here {n} q:= qn− 1 q− 1 is the n−th q−number.
Example 56. If q = 1 we can define D := t · ∂ , ∂ =dtd, d generatesDid(A ). In this case, the relation
hdn, dliσ = ({n}q− {l}q)dn+l becomes the standard commutation relation in the Witt algebra
hdn, dliσ = (n − l)dn+l. It is immediate that
hdn, dliσ = −hdl, dniσ,
We substitute f (t) = −tn, g(t) = −tl, h(t) = −tm into the Jacobi-like identity obtaining the following q−deformation:
(qn+ 1)hdn, hdl, dmiσiσ+ (q
l+ 1)hd
l, hdm, dniσiσ+ (q
m+ 1)hd
m, hdn, dliσiσ = 0. This construction can be gathered in the following theorem, ennounced and proven by Silvestrov et al. in [2] (p. 340-341).
Theorem 57. LetA = C[t,t−1]. The C−linear space Dσ(A ) =
M
n∈Z C · dn
with D= −id−σq−1 , dn= −tnD and σ (t) = qt can be equipped with the bracket multiplica-tion
h·, ·iσ :Dσ(A ) × Dσ(A ) −→ Dσ(A ) defined in terms of its generators as
hdn, dmiσ = qndndm− qmdmdn with the following commutation relations:
hdn, dmiσ = ({n}q− {m}q)dn+m. This bracket satisfies skew-symmetry
hdn, dmiσ = −hdm, dniσ and a σ −deformed Jacobi identity
This construction carries its operator representation, which provides good definition -Hence, we behold a well-defined associative algebra. As one natural outcome of this ap-proach, this parametric family of algebras is a deformation of the Witt algebra - i.e. a gener-alization, where we can recover the untwisted Witt algebra by taking q = 1.
Following this structure, a whole family of Witt algebras can be created.
Definition 58. Let A be a commutative and associative algebra, σ : A → A an algebra endomorphism and D a σ −derivation onA . A generalized Witt algebra or (A ,σ,D)−Witt algebra is the non-associative algebra (A · D,h·,·iσ ,D) with the product defined in the fol-lowing way:
ha · D, b · Diσ ,D= (σ (a)D(b) − σ (b)D(a)) · D.
4.2
(σ , τ)−derivations on Witt algebras
Up next, some examples of (σ , τ)−derivations will be given for common operators over the Witt algebra and worked in a similar fashion to one-dimensional q−deformations, which are based in σ −derivations, in order to provide multi-dimensional deformations of Witt algebras based on (σ , τ)−derivations.
One can look at σ −derivations as (σ , id)−derivations. In this direction, q−deformations appear when choosing σ : t → qt and τ = id, as stated in 4.1 when choosing the generator D:= id− σ
q− 1 .
In the upcoming section we explore some (σ , τ)−derivations on Witt algebras based on well-known examples of extensions and q−deformations, using the results visited up to this point extensively and the fundamental properties of the Witt algebra.
The work structure along this section is parallel to what has been shown in 4.1, and inherits from [2]: firstly, we determine a value for δ . Said value will help decide whether Dσ ,τ(A ) can be endorsed with a Hom or quasi-Lie algebra structure.
Theorem 50 provides a matching Jacobi identity, as well as an explicit form for the defining interior bracket. We calculate said bracket in the cases where that is most interesting.
4.2.1
Two (q, −q)−deformations
First deformation. As an intuitive first approach, let σ : t → qt, τ : t → −qt. We observe the following properties:
2. τ is invertible.
The first property ensures Dσ ,τ(A ) is a free A −module with rank 1 and generator D=τ − σ
r : f (t) 7→
(τ − σ )( f (t)) r
where r is a gcd((τ − σ )(A )). Now, any element in gcd((τ − σ)(A )) divides (τ − σ)(t) = −2qt, which is a unit when q 6= 0 ⇒ r has the form −2ctk for some k ∈ Z. We choose k= 0, c = −q and 2q is, indeed, a gcd((τ − σ )(A )).
Therefore, D =τ − σ
2q is a generator.
Invertibility on τ implies we can define a bracket in the following way:
h f · D, g · Diσ ,τ = ((σ ◦ τ−1)( f ) · D) ◦ (τ−1(g) · τ−1◦ D) − ((σ ◦ τ−1)(g) · D) ◦ (τ−1( f ) · τ−1◦ D) = = ((σ ◦ τ−1)( f ) · (D ◦ τ−1)(g) − (σ ◦ τ−1)(g) · (D ◦ τ−1)( f )) · D.
Skew-symmetry comes immediately.
In order to find a Hom-Jacobi identity, we wonder if there is a function δ :A → A such that (D ◦ τ−1) ◦ (σ ◦ τ−1) = δ ((σ ◦ τ−1) ◦ (D ◦ τ−1)).
We can choose δ = (σ ◦ τ −1)(r)
r = −1. This is a multiple of the unit in C - the bracket defined above, by Theorem 50, verifies the following quasi-Jacobi identity:
f,g,h h(σ ◦ τ
−1)( f ) · D, hg · D, h · Di
σ ,τiσ ,τ+ δ h f · D, hg · D, h · Diσ ,τiσ ,τ = 0 By replacing terms we get:
f,g,h(h−g · D, (−g(D ◦ τ −1)(h) + h · (D ◦ τ−1)(g)) · Di σ ,τ− h f · D, (−g(D ◦ τ−1)(h)+ +h(D ◦ τ−1)(g)) · Diσ ,τ) = f,g,h 2hα( f ), hg, hiσ ,τiσ ,τ with α = −id,
which is twice a Hom-Jacobi identity. We gather these calculations in a proposition: Proposition 59. Let σ : t → qt, τ : t → −qt . The following bracket:
h f · D, g · Di := ((σ ◦ τ−1)( f )D(g) − (σ ◦ τ−1)(g)D( f )) · D
provides the space A · D = Dσ ,τ(A ) with a Hom-Lie algebra structure (A · D,h·,·i,α) verifying the Hom-Jacobi identity
f,g,h2hα( f ), hg, hii = 0 for α = −2id. This α can be seen as σ ◦ τ−1+ δ id
Second deformation. For the second deformation, consider δ = −id as above; we ob-serve the following:
1. (D ◦ σ )( f (t)) = f(q 2t) − f (−q2t) 2q = −(σ ◦ D)( f (t)). 2. (D ◦ τ)( f (t)) = f(−q 2t) − f (q2t) 2q = −(τ ◦ D)( f (t)). 3. τ ◦ σ = f (−q2t) = σ ◦ τ.
As well as the intuitive σ −bracket defined as follows:
h f · D, g · Di = (σ ( f )D(g) − σ (g)D( f )) · D This bracket verifies the following Hom-Jacobi identity:
f,g,hh(τ − σ )( f ) · D, hg · D, h · Dii = 0.
So it defines a Hom-Lie algebra structure (A · D,h·,·i,τ − σ) on A · D.
These two cases are dual in a sense: the bracket defined in the latter corresponds to the one defined in the former composed with τ. Considering τ(A ) = A , the two Hom-Lie algebras defined this way are indeed isomorphic through τ. Moreover,A is commutative, associative and unital, and thus σ and τ are interchangeable - Either σ or τ act as isomorphisms between these two Hom-Lie algebras.
4.2.2
Considerations on σ and τ
When a 6= ±1, mappings of the form σ : t → qtaare not invertible. This condition prevents the construction above to be possible in the general case - one can consider a more general rule for σ and τ that allows a similar construction to exist.
Let 0 6= c ∈ C. This commutes with all elements in A . It is already known that any general σ , τ won’t satisfy the desired properties - We search for some conditions on them based on their action on generators of the form tn· D.
c(σ ( f g) − σ ( f )σ (g)) = c(τ( f g) − τ( f )τ(g)) c σ (tn+l− q(tn)aq(tl)a = cτ (tn+l) − p(tn)bp(tl)b c q(ta(n+l)− q2ta(n+l) = cptb(n+l)− p2tb(n+l) c(q − q2)ta(n+l)= c(p − p2)tb(n+l)
Since c ∈ C, it is clearly a unit and then p− p2 q− q2 = t (a−b)(n+l)⇒ (a − b)(n + l) = 0 → a = b ⇒ p− p 2 q− q2 = 1 → p ∈ {q, 1 − q}. If p = q then τ = σ and this provides that 0 is indeed a (σ , σ )−derivation. Let p = 1 − q. Then,A · D is a free A −module of rank 1 with generator
D= τ − σ r : t →
(1 − 2q)ta r ,
where r is a gcd((τ − σ )(A )). Any element in gcd((τ − σ)(A )) must divide (τ − σ)(t) = (1 − 2q)t, which is a unit when4 q6= 1/2 ⇒ r has the form (1 − 2q)ctk for some c ∈ C\{0}, k ∈ Z. We can pick c = 1, z = 0 so (1 − 2q) is, indeed, a gcd((τ − σ )(A )).
So D = (1 − 2q)t a
1 − 2q = t
ais a generator forA · D. Now we can define the following bracket:
h f · D, g · Di := (σ ( f )D(g) − σ (g)D( f )) · D
Note that D and σ commute (as q commutes with both), and therefore htn· D,tl· Di = 0 for any n, l ∈ Z. Naturally, now every possible Hom-Jacobi identity holds up, as this cyclic sum
n,m,lhα(t
n), htm· D,tl· Dii = 0
for any α :A → A . Particularly, α = id works perfectly fine and therefore we have recovered a Lie algebra with this bracket.
4.2.3
Another level of q−deformation
Consider σ : t → qta, τ : t → pta, a ∈ Z\{0}.
Both σ and τ are endomorphisms, allowing the (σ , τ)−derivation D =τ − σ
r , which acts on generators as
D(tn) = (p
n− qn)tan p− q
Note that σ and τ commute. Thus, we can define the following bracket: h f · D, g · Di := (σ ( f )D(g) − σ (g)D( f )) · D.
4If q = 1
If we can consider a partitionA = L n∈Z dn= L n∈Z −tan·D, then hd n, dmi = [n] qn − [m] qm dn+m which would allow to recover the commutation relations for the q−deformed Witt algebra. The partition obviously doesn’t work unless a = ±1.5 However, one can pick a subalge-bra Aa = C [ta,t−a] in which the commutation relations known for the q−deformation we have already seen - Not in vain, the mapping ϕ :A → Aagiven by ϕ(t) = t±ais indeed an isomorphism.
This construction is, thus, trivial unless σ and τ are either the same or τ = kσ for some complex number k. The case where a = 1 has been already considered, as it is the canonical (p, q)−deformation of the Witt algebra.
4.2.4
A twice non-linearly deformed Witt algebra
When considering more general cases for σ and τ, Hom-Lie algebra properties are not al-ways kept - the remaining structure can be loaded with a wider quasi-Jacobi identity and imbue our algebra with a quasi-Lie algebra structure in a parallel fashion.
Let σ : t → qta, τ : t → t−1. A is a UFD, so A · D is a free A −module of rank 1, where D=τ − σ r : f (t) → f(t−1) − f (qta) t−1− qta . Now, D(tn) = t −n− qntan t−1− qta = t −n+1(1 + qta+ · · · + (qta)n−1) ⇒ t−1− qta divides t−n− qntan = (τ − σ )(tn) ⇒ r := t−1− qtadivides (τ − σ )( f (t)) and thus is a gcd((τ − σ )(A )). t−1− qtais not a unit but τ is invertible and thus we can compute δ as
δ = σ (τ −1(r)) r = σ (t − q−1t−a) t−1− qta = qt− t−a t−1− qta = −t 1−a.
Note that, if a = 1 then δ = −1. This is a multiple of idC, and thus we can define a Hom-Lie algebra structure overA · D.
Invertibility of τ can be used to force an interior bracket overA · D:
h f · D, c · Di = ((σ ◦ τ−1)( f )(D ◦ τ−1)(g) − (σ ◦ τ−1)(g)(D ◦ τ−1)( f )) · D
5If that was the case, all generators are of the form −tan, with n, a ∈ Z and a fix. But 1 is not a multiple of
a, and thus −t · D is not a generator, implying L
n∈Z
− tan· D ⊂ L
n∈Z
Let f = −tn, g = −tl, we search for a further relation between these generators: h−tn· D, −tl· Di = q−nt−ant l− q−lt−al t−1− qta − q −lt−altn− q−nt−an t−1− qta · D = = q−l t−1− qta(−t l−an) − q−n t−1− qta(−t n−al) · D
This relation between the generators dn= −tn· D allows to rewrite this bracket as hdn, dli =
q−l
t−1− qtadl−an− q−n
t−1− qtadn−al
This bracket (see Theorem 50) verifies the following quasi-Jacobi identity:
f gh hσ (τ
−1( f )) · D, hg · D, h · Dii + δ h f · D, hg · D, h · Dii = 0.
Morphisms σ and τ provide a quasi-Lie algebra structure on A · D, with a twisting morphism α = σ ◦ τ−1+ δ id : dn→ q−nd−an− t1−adn, β = δ = −t1−a, θ = idA and ω = −idA. This is guaranteed by Theorem 50
4.2.5
A quasi (p, q)-deformation
The condition above states a really tight condition for morphisms of the form t → qta in order to generate a Hom-Lie algebra. We let loose of the relation between σ and τ aiming for a more general algebraic structure that can be generated by them.
Let σ : t → qtafor q ∈ C\{0} and a ∈ Z\{1}6, τ : t → pt, p ∈ C\{0}.
Theorem 60. LetA = C[t,t−1]. The C−linear space Dσ ,τ(A ) =
M
n∈Z C · dn
with D= − σ −τ
qta−pt, dn= −tn·D, σ (t) = qtaand τ(t) = pt can be equipped with the bracket multiplication
h·, ·iσ :Dσ ,τ(A ) × Dσ ,τ(A ) −→ Dσ ,τ(A ) defined in terms of its generators as
hdm, dliσ = sign(l − m)cqypY−1 Y−y−1
∑
k=0 q p k tb−1+Y +a·y+k(a−1)This bracket satisfies skew-symmetry
hdm, dliσ = −hdl, dmiσ and a σ −deformed Jacobi identity
n,m,lhσ (dn ), hdm, dliσiσ+ δahdn, hdm, dliσiσ = 0 Where hdn, hdm, dliσiσ = = sign(l − m)cqypY−1 Y−y−1
∑
k=0 q p k S1qz1pZ1−1 Z1−z1−1∑
s=0 q p s tb−1+Z1+az1+s(a−1) ! hσ (dn), hdm, dliσiσ = = sign(l − m)cqy+npY−1 Y−y−1∑
k=0 q p k SaqzapZa−1 Za−za−1∑
s=0 q p stb−1+Za+aza+s(a−1) ! with δa= p−1 a−1 ∑ k=0 q p k−a
t(k+1)(a−1)and the following coefficients: Y := max(m, l).
y:= min(m, l).
Z1:= max(n, b−1+Y +ay+k(a−1)). S1:= sign(b − 1 + Y + ay + k(a − 1) − n). z1:= min(n, b − 1 +Y + ay + k(a − 1)). Za := max(an, b − 1 + Y + ay + k(a − 1)). za := min(an, b − 1 + Y + ay + k(a − 1)). Sa:= sign(b − 1 + Y + ay + k(a − 1) − an).
This defines a quasi-Hom-Lie algebra structure onA .
Indeed, we define a (σ , τ)−derivation as D : t → σ − τ
gcd((σ − τ){A }). Due toA being a UFD, (σ − τ)(t) = qta− pt is a perfectly good gcd((σ − τ){A }) up to a unit ctb−1for some b∈ Z and c ∈ C. Thus, D can be seen as the derivation that acts on generators tnas
D(tn) = ct−b+1(qt
a)n− (pt)n qta− pt
This (σ , τ)−derivation allows to introduce the natural interior bracket: htn· D,tl· Diσ =σ (tn)D(tl) − D(tn)σ (tl)
· D.
This bracket can be computed directly as ctb· (qta)n(qt a)l− (pt)l qta− pt − ct b(qta)l(qta)n− (pt)n qta− pt = = ctb(qt a)n (qta)l− (pt)l − (qta)l((qta)n− (pt)n) qta− pt = ct b(qta)l(pt)n− (qta)n(pt)l qta− pt = = ctb(qta)l(pt)l(pt) n−l− (qta)n−l qta− pt = −ct b(qta)l(pt)l(qta)n−l− (pt)n−l qta− pt
The minus sign is related to l − n.Assume n < l and compute the bracket using skew-symmetry:
hdn, dliσ = −hdl, dniσ = ct
b(qta)n(pt)n(qta)l−n− (pt)l−n qta− pt Now, let Y := max(n, l), y := min(n, l).
hdn, dliσ = ct b(qta)y(pt)y(qta)Y−y− (pt)Y−y qta− pt = = sign(l − n)cqypY−1 Y−y−1
∑
k=0 q p k tb−1+Y +ay+k(a−1)The fact that Y − y − 1 > 0 provides good definition for the sum, which takes out depen-dence on the signs of n and l.
The Witt algebraA is a UFD, and that together with σ(τ(t)) = qpta= τ(σ (t)) allows for an element δaof the form
δa= σ (r) r = σ (qta− pt) qta− pt = p −1a−1
∑
k=0 q p k−a t(k+1)(a−1)This space is endorsed with a quasi-Jacobi identity as follows:
n,m,lhhσ (dn
), hdm, dliσiσ+ δahdn, hdm, dliσiσ = 0
This quasi-Jacobi identity can be computed and presented explicitly: in order to do that, we define the coefficients
