Download PARADIGM OF QUASI-LIE AND QUASI-HOM

PARADIGM OF QUASI-LIE AND QUASI-HOM-LIE ALGEBRAS AND
QUASI-DEFORMATIONS
Sergei Silvestrov
Centre for Mathematical Sciences, Lund University, Box 118, SE-221 00 Lund, Sweden
e-mail: [email protected]
Abstract
Main constructions and examples of quasi-deformations of Lie algebras via twisted derivations leading to quasi-Lie algebras are reviewed. Among quasi-Lie algebras presented are
quasi-Hom-Lie deformations of sl2 enveloping algebra, color Lie algebras and quasi-HomLie and Hom-Lie algebras deformations of infinite-dimensional Lie algebras of Witt and
Virasoro type.
Keywords: quasi-Lie algebra, quasi-Hom-Lie algebra, Hom-Lie algebra
MSC2000: 17A30, 17A45, 17A36, 17B75, 17B37, 17B68, 17B66
INTRODUCTION
The area of quantum deformations (or q-deformations) of Lie algebras began a period of rapid
expansion around 1985 when Drinfel’d [13] and Jimbo [23] independently considered deformations of U(g), the universal enveloping algebra of a Lie algebra g, motivated, among other
things, by their applications to quantum integrable systems, the Yang–Baxter equation and
quantum inverse scattering methods [25]. This method developed by L.D. Faddeev and his
collaborators gave rise to a great interest in quantum groups and Hopf algebras. Since then
several other versions of (q-) deformed Lie algebras have appeared, especially in physical contexts such as string theory. The main objects for these deformations were infinite-dimensional
algebras, primarily the Heisenberg algebras (oscillator algebras) and the Virasoro algebra, see
[4, 8, 9, 10, 18, 19, 21] and the references therein. For more details why these algebras are
important in physics, see [4, 8, 9, 10, 11, 12, 16, 17], for instance.
An important question considered for these algebras is whether they obey some deformed
(twisted) versions of skew-symmetry or Jacobi identity. At the same time, there is a well-known
direct generalization of Lie algebras and Lie superalgebras to general commutative grading
groups, the class of color Lie algebras. In these algebras generalized skew-symmetry and Jacobi type identities, graded by a commutative group and twisted by a scalar bi-character, hold.
A remarkable and not yet fully exploited and understood feature of color Lie algebras and in
particular Lie superalgebras is that they often appear simultaneously with usual Lie algebras in
various parametric (deformation) families of algebras for initial, final or other important special values of the deformation parameters. In [18, 27, 28], the authors introduced the classes
This work was partially supported by the Swedish Foundation for International Cooperation in Research and
Higher Education (STINT), the Crafoord foundation, The Royal Physiographic Society in Lund, The Swedish
Royal Academy of Sciences, Lund University and the Marie Curie Research Training Network ”Liegrits”.
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of quasi-Lie algebras and their subclasses of quasi-Hom-Lie algebras and Hom-Lie algebras
enlarging the class of Lie algebra. These classes of algebras are tailored in a way suitable for
simultaneous treatment, on the one hand, of the Lie algebras, Lie superalgebras and the general
color Lie algebras and on the other hand the deformations arising in connection with calculi of
twisted, discretized or deformed derivatives and corresponding generalizations, discretizations
and deformations of vector fields and differential calculus. It has been shown in [18, 27, 28, 29]
that the class of quasi-Hom-Lie algebras contains as a subclass on the one hand the color Lie
algebras and in particular Lie superalgebras and Lie algebras, and on the other hand various
known and new single and multi-parameter families of algebras obtained using twisted derivations and constituting deformations and quasi-deformations of universal enveloping algebras of
Lie and color Lie algebras and of algebras of vector-fields. The main feature of quasi-Lie algebras, quasi-Hom-Lie algebras and Hom-Lie algebras is that the skew-symmetry and the Jacobi
identity are twisted by several deforming twisting maps and also in quasi-Lie and quasi-HomLie algebras the Jacobi identity in general contains six twisted triple bracket terms.
1
DEFINITIONS AND NOTATIONS
Throughout this article we let F for simplicity be a field of characteristic zero and L F (L) the set
of linear maps of the linear space L over the field F. Most of the results are valid for arbitrary
field F at some places with exception of characteristic 2.
Definition 1. (Larsson, Silvestrov [28]) Let L be a linear space. A quasi-Lie algebra is a
tuple (L, h·, ·i, α, β, θ, ω) where h·, ·i : L × L is a bilinear bracket product, α, β are linear space
morphisms and θ, ω : Dθ , Dω ⊆ L × L → LF (L) are such that
• hx, yi = ω(x, y)hy, xi,
for (x, y) ∈ Dω ;
• x,y,z θ(z, x)(hα(x), hy, zii + βhx, hy, zii) = 0,
for (z, x), (y, z), (x, y) ∈ Dθ ,
where x,y,z denotes cyclic summation with respect to x, y, z.
Remark 2. Note that (ω(x, y)ω(y, x) − id)hx, yi = 0, if (x, y), (y, x) ∈ Dω , which follows from the
computation hx, yi = ω(x, y)hy, xi = ω(x, y)ω(y, x)hx, yi.
Specializing to ω = θ and additionally imposing the condition hα(x), α(y)i = β ◦ αhx, yi defines
the class of quasi-hom-Lie algebras introduced in [27]; furthermore, if β = id and ω ≡ −id we
get the class of hom-Lie algebras [18] and additionally if α = id we get the class of Lie algebras.
Note that with this definition quasi-hom-Lie algebras include not only hom-Lie algebras as a
subclass, but also color Lie algebras and in particular Lie super-algebras [27].
Example 3. Let Γ be an abelian group, and let L be a Γ-graded linear space over F. A color Lie
algebra or Γ-graded ε-Lie algebra (see [5, 6, 35, 36, 37, 38]) is Γ-graded linear space L with a
bilinear multiplication h·, ·i satisfying
• hx, yi = −ε(γx , γy )hy, xi
• ε(γz , γx )hx, hy, zii + ε(γx , γy )hy, hz, xii + ε(γy , γz )hz, hx, yii = 0,
for x ∈ Lγx , y ∈ Lγy and z ∈ Lγz . The map ε : Γ × Γ → F∗ , called a commutation factor, is a
bi-character on Γ with a symmetry condition, namely a map satisfying
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• ε(γx , γy )ε(γy , γx ) = 1
• ε(γx + γy , γz ) = ε(γx , γz )ε(γy , γz ), and ε(γx , γy + γz ) = ε(γx , γy )ε(γx , γz ).
Notice that color Lie algebras include both Lie algebras (with Γ = {e} and ε = 1) and Lie superalgebras (with Γ = Z2 and ε(γx , γy ) = (−1)γx γy ). Color Lie algebras are quasi-Lie algebras
or more exactly are quasi-hom-Lie-algebras). This can be seen by grading L by Γ in the definition of quasi-Lie algebras (see also [30]), and putting α= β = id and θ(x, y)v = ω(x, y)v =
−ε(γx , γy )v for v ∈ L, where (x, y) ∈ Dω = Dθ = ∪γ∈Γ Lγ × ∪γ∈Γ Lγ and γx , γy ∈ Γ are the
graded degrees of x and y. The ω-symmetry and the quasi-Jacobi identity give the respective
identities in the definition of a color Lie algebra.
If L as a linear space over F has a basis {e1 , . . . , en } and (e j , ek ) ∈ Dω ∩ Dθ for any j, k ∈
{1, . . . , n}, then the product is defined completely by its action on the basis
n
he j , ek i =
∑ clj,k el ,
clj,k ∈ F
for
j, k, l ∈ {1, . . . , n},
l=1
and as in the case of Lie algebras, ω-symmetry and quasi-Jacobi identity can be rewritten in
terms of algebraic constraints for the structure constants {clj,k }:
• (ω-symmetry)
∑ns=1 csk, j ω(e j , ek )l,s = clj,k
• (quasi-Jacobi)
j,k,l ∑ni,s,r=1 csk,l (αi, j cri,s + cij,s βr,i )θ(el , e j ) p,r = 0,
for j, k, l, p ∈ {1, . . . , n}, where αi, j , βr,i , θ(el , e j ) p,r are matrix elements of the matrices for the
twisting operators α, β, ω(e j , ek ), θ(el , e j ) in the basis {e1 , . . . , en }. This means that we have a
multi-parameter family of varieties. The equations for the varieties of Lie algebras, Lie superalgebras and color Lie algebras are obtained for the special choices of parameters corresponding
to the respective choices of twisting maps. A very important and nice problem would be to
study the (geometric) moduli for these algebras.
Example 4. (Qasi-Leibniz algebras) The Jacobi identity for Lie algebras can be rewritten, using
skew-symmetry, as the Leibniz rule for the map x 7→ hx, zi, namely
hhx, yi, zi = hhx, zi, yi + hx, hy, zii.
(1)
Keeping this identity but dropping the requirement that the bracket is skew-symmetric, one gets
the class of Leibniz algebras. The similar consideration can be repeated in the case of quasi-Lie
algebras. The corresponding form of the quasi-Jacobi identity is
θ(y, z) ω(α(z), hx, yi)hhx, yi, α(z)i + β ◦ ω(z, hx, yi)hhx, yi, zi
= −θ(x, y) ω(α(y), hz, xi)hω(z, x)hx, zi, α(y)i + β ◦ ω(y, hz, xi)hω(z, x)hx, zi, yii
−θ(z, x) hα(x), hy, zii + βhx, hy, zii
for (z, x), (x, y), (y, z) ∈ Dθ , (α(z), hx, yi), (α(y), hz, xi), (y, hz, xi), (z, x) ∈ Dω . When α = β = id
and θ ≡ ω ≡ − id, one recovers (1) as should be for Lie and Leibniz algebras. So, if we keep
the quasi-Jacobi identity in that Leibniz-like form, but drop the ω-symmetry axiom, we get a
generalization of Leibniz algebras which we call quasi-Leibniz algebras.
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2
QUASI-LIE ALGEBRAS, TWISTED DERIVATIONS AND DEFORMATIONS
In this section we survey a quasi-deformation method devised in [18, 27, 28] and leading to
hom-Lie and quasi-hom-Lie algebras associated to discrete analogues of vector fields built via
twisted derivations. We also demonstrate application of the method to sl2 (F) resulting in parametric families of algebras, the quasi-hom-Lie and hom-Lie quasi-deformations of sl2 (F), considered in [29]. One of the main features of this quasi-deformation method is that the deformed
algebras come endowed with a twisted Jacobi identity thus yielding typically not a Lie algebra
but a hom-Lie or a more general quasi-Hom-Lie algebra introduced in [18, 27, 28]. Also, when
this deformation scheme has been applied to sl2 (F), by choosing parameters suitably, one can
deform sl2 (F) into the Heisenberg Lie algebra and some other three-dimensional Lie algebras in
addition to more exotic types of algebras, this being in stark contrast to the classical deformation
schemes where sl2 (F) is typically rigid.
The idea to deform algebraic, analytic and geometric structures within the appropriate category
is obviously not new. The first modern appearance is often attributed to Kodaira and Spencer
and deformations of complex structures on complex manifolds. This was, however, soon extended and generalized in an algebraic-homological setting by Gerstenhaber, Grothendieck and
Schlessinger. Nowadays deformation-theoretic ideas penetrate most aspects of both mathematics and physics and cut to the core of theoretical and computational problems. In the case of
Lie algebras, quantum deformations (or q-deformations) and quantum groups associated to Lie
algebras have been investigated for over twenty years, still growing richer by the minute. We
note that the deformed objects in the above cases seldom, if ever, belong to the original category of Lie algebras. However, typically, one retains the undeformed objects in the appropriate
limit. The deformations we describe here do not necessarily have this property, and therefore,
following [29], we refer to these as quasi-deformations.
Given a Lie algebra and its representation on a commutative, associative algebra with unity,
our quasi-deformation scheme can be described as the procedure of substituting the original
operators generating the representation by their deformed (σ-twisted) versions, then describing
in a special way a new deformed bracket multiplication for the new operators and then pulling
this new structure back from the representation to the algebra level, and considering the resulting
algebras as the quasi-deformation of the original one. So what is actually changed, or deformed,
is the selected representation and the bracket product in its image. A special feature of the
algebras obtained after the pull back of that specially constructed bracket product is that they
provide examples of hom-Lie and quasi-Hom-Lie algebras.
The basic initial undeformed object for the examples of Hom-Lie and quasi-Hom-Lie quasideformations reviewed in this section is the classical sl2 (F), the simple Lie algebra generated
as a linear space by elements H, E and F with relations
hH, Ei = 2E,
hH, Fi = −2F,
hE, Fi = H.
(2)
The basic starting point is the following standard representation of sl2 (F) by first order differential operators acting on a linear space of functions in one variable: E 7→ ∂, H 7→ −2t∂,
F 7→ −t 2 ∂. Firstly, this representation is ”deformed” (generalized) to first order operators
acting on an algebra A , where ∂ is replaced by ∂σ , a σ-derivation on A . In [18] it was
shown that, for a σ-derivation ∂σ on a commutative associative algebra A with unity, the
rank one module A · ∂σ admits the structure of a C-algebra with a σ-deformed commutator
ha · ∂σ , b · ∂σ i = σ(a) · ∂σ (b · ∂σ ) − σ(b) · ∂σ (a · ∂σ ) as a multiplication, satisfying a generalized
twisted six-term Jacobi identity of a quasi-Hom-Lie algebra. The article [29] builds on this
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and provides elaborated examples of quasi-deformed sl2 (F) by the above outlined method. The
result yields a natural quasi-deformation of sl2 (F) that is a quasi-Hom-Lie algebra in general.
By choosing different base algebras A we obviously get different algebra structures on A · ∂σ .
So in a way we have two different deformation ”parameters”, namely σ and A . (Of course there
is some dependence between σ and A .)
Quasi-Hom-Lie and Hom-Lie algebras associated with σ-derivations
Let us now recall the main definitions and results from [18] and [27] needed in this section.
Throughout this section we let A be a commutative, associative F-algebra with unity 1. Furthermore σ will always denote an endomorphism on A . Then by a twisted derivation or σ-derivation
on A we mean an F-linear map ∂σ : A → A such that a σ-twisted Leibniz rule holds:
∂σ (ab) = ∂σ (a)b + σ(a)∂σ (b).
(3)
In the paper [18] the notion of a hom-Lie algebra as a deformed version of a Lie algebra was
introduced, motivated by some of the examples of deformations of the Witt and Virasoro algebras constructed using σ-derivations. The Hom-Lie algebras are a subclass of quasi-Hom-Lie
algebras, a subclass of quasi-Lie algebras introduced in [28]. We let Dσ (A ) denote the set of
σ-derivations on A .
The following theorem from [18], that introduces an F-algebra structure on A · ∂σ , makes it into
a quasi-hom-Lie algebra and is of central importance for the method.
Fix a homomorphism σ : A → A , an element ∂σ ∈ Dσ (A ), and an element δ ∈ A . The following
two conditions play important role:
σ(Ann(∂σ )) ⊆ Ann(∂σ ),
∂σ (σ(a)) = δσ(∂σ (a)), for a ∈ A ,
(4)
(5)
where Ann(∂σ ) := {a ∈ A | a · ∂σ = 0}. Let A · ∂σ = {a · ∂σ | a ∈ A } denote the cyclic left
A -submodule of Dσ (A ) generated by ∂σ and extend σ to A · ∂σ by σ(a · ∂σ ) = σ(a) · ∂σ .
Theorem 5. (Hartwig, Larsson, Silvestrov, [18]) If (4) holds, that is σ(Ann(∂σ )) ⊆ Ann(∂σ ),
then the map h·, ·iσ defined by setting
ha · ∂σ , b · ∂σ iσ = (σ(a) · ∂σ ) ◦ (b · ∂σ ) − (σ(b) · ∂σ ) ◦ (a · ∂σ ),
(6)
for a, b ∈ A and where ◦ denotes composition of maps, is a well-defined F-algebra product on
the F-linear space A · ∂σ . It satisfies the following identities for all a, b, c ∈ A :
ha · ∂σ , b · ∂σ iσ = (σ(a)∂σ (b) − σ(b)∂σ (a)) · ∂σ ,
ha · ∂σ , b · ∂σ iσ = −hb · ∂σ , a · ∂σ iσ ,
(7)
(8)
and if, in addition, (5) holds, that is ∂σ (σ(a)) = δσ(∂σ (a)) for an element δ ∈ A and all a ∈ A ,
then the deformed six-term Jacobi identity is satisfied
a,b,c hσ(a) · ∂σ , hb · ∂σ , c · ∂σ iσ iσ + δ · ha · ∂σ , hb · ∂σ , c · ∂σ iσ iσ = 0
for all a, b, c ∈ A .
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(9)
For a detailed proof of this result see [18]. The algebra (A · ∂σ , h·, ·iσ ) is by this theorem a
quasi-Hom-Lie algebra with α = σ, β = δ and ω ≡ −idA ·∂σ .
Choose ∂σ ∈ Dσ (A ) and t ∈ A . By Theorem 5, there is a skew-symmetric algebra structure,
given by (7), on the F-subspace and A -module A · ∂σ of elements a · ∂σ with a ∈ A (usually
denoted simply by a∂σ ). When this algebra structure is restricted to the F-linear subspace
S := LinSpanF {e = ∂σ , h = −2t∂σ , f = −t 2 ∂σ } = LinSpanF {e, h, f } of A · ∂σ without, at this
point, assuming closure, and provided σ(1) = 1 and ∂σ (1) = 0, the relation (7) yields
hh, f i = 2σ(t)∂σ (t)t∂σ , hh, ei = 2∂σ (t)∂σ , he, f i = −(σ(t) + t)∂σ (t)∂σ .
(10)
Remark 6. If σ = id and ∂σ (t) = 1, then we retain the classical sl2 (F) with relations (2).
When A = F[t] is a polynomial algebra, σ(1) = 1 and ∂σ (1) = 0, to have closure of (10), the
polynomials σ(t) = q(t) and ∂σ (t) = p(t) must obey deg(∂σ (t)σ(t)) = deg(p(t)q(t)) ≤ 1 thus
falling into three cases
Case 1 σ(t) = q(t) = q0 + q1t, q1 6= 0, p(t) = p0 ,
Case 2 σ(t) = q(t) = q0 , q0 6= 0, p(t) = p0 + p1t,
Case 3 σ(t) = q(t) = 0, p(t) = p0 + p1t + · · · + pnt n .
Remark 7. If we allow σ(t) = q(t) and ∂σ (t) = p(t) where p, q are arbitrary polynomials in
t then we get a deformation of sl2 (F) which does not preserve dimension; that is, brackets of
the basis elements e, f , h are not simply linear combinations in these elements but include more
new ”basis” elements. This phenomena could possibly be interesting to study further.
By (7), in the Case 1 the relations (10) become
hh, f i : −2q0 e f + q1 h f + q20 eh − q0 q1 h2 − q21 f h = −q0 p0 h − 2q1 p0 f
2
hh, ei : −2q0 e + q1 he − eh = 2p0 e
(11a)
(11b)
q1 + 1
p0 h.
(11c)
2
The associative algebra with three abstract generators e, h and f and defining relations (11a),
(11b) and (11c) can be seen as a multi-parameter deformation of U(sl2 (F)). Notice that changing the role of h and f in (11a) does not correspond to changing h and f in (7). This means that,
in a sense, the skew-symmetry of (7) is ”hidden” in (11a). If, for instance, h f , hi is calculated
from (7) one sees that indeed h f , hi = −hh, f i as one would expect, and one gets exactly minus the left-hand-side of (11a). The condition (5) holds for δ = q1 . So by Theorem 5 we now
have a deformed Jacobi identity x,y,z (hσ(x), hy, zii + q1 hx, hy, zii) = 0 on A · ∂σ = F[t] · ∂σ . By
defining α(x) := q−1
1 σ(x) this can be re-written as the Jacobi-like relation for a hom-Lie algebra
he, f i : e f + q20 e2 − q0 q1 he − q21 f e = −q0 p0 e +
x,y,z h(α + id)(x), hy, zii = 0.
(12)
Note that this twisted Jacobi identity follows directly from the general Theorem 5. This shows
that one does not have to make any lengthy trial computations with different more or less ad
hoc assumptions on the form of the twisted Jacobi identity.
In the Case 2, the relations (10) combined with (7) become
hh, f i :
−2e f + q0 eh = −p0 h − 2p1 f
hh, ei :
−2q0 e2 − eh = 2p0 e − p1 h
q0 p1 + p0
e f + q20 e2 = −q0 p0 e +
h + p1 f ,
2
he, f i :
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It is obvious that we cannot recover the classical sl2 (F) from this deformation by specifying
suitable parameters since, in a sense, the commutators ”collapsed” because of the structure of
σ. It is easy to deduce that δ = 0 in this case and so
x,y,z hσ(x), hy, zii = 0
(14)
on A · ∂σ making it into a hom-Lie algebra.
In the Case 3, it follows from (10) that deg(p(t)) ≤ 1. The relations (10), combined with (7),
become 0 = 0,−eh = 2p0 e − p1 h, e f = 12 p0 h + p1 f . Thus we get actually an algebra with three
generators and two relations. In this case (5) becomes 0 = ∂σ (σ(t k )) = δ · σ(∂σ (t k )), valid for
all k ∈ Z≥0 for δ = 0 giving again (14) and thus a hom-Lie algebra.
Returning to Case 1 and to (11a), (11b) and (11c), for q0 = 0 and q := q1 6= 0 we obtain a
deformation of sl2 (F) corresponding to replacing the ordinary derivation operator with p0 Dq ,
f (t)
for q 6= 1 and
the constant multiple of the Jackson q-derivative defined as Dq : f (t) 7→ f (qt)−
qt−t
0
as the usual derivative f (t) for q = 1. The relations (11a), (11b) and (11c) specialize to
h f − q f h = −2p0 f ,
he − q−1 eh = 2q−1 p0 e,
e f − q2 f e =
q+1
p0 h.
2
(15)
By taking q = 1 and p0 = 1, one obtains the usual commutation relations for sl2 (F), corresponding to the ordinary derivation operator ∂. By taking p0 = q0 = 0 in (15) we get an “abelianized”
version. Note, however that the operator representation we started with collapses in this case
since ∂σ = 0. For special values of parameters these quasi-deformations contain also color
Lie algebras and Lie superalgebras. When q = −1 and p0 = 0, the endomorphism σ becomes
σ(t) = −t and the Jackson q-derivative is thus given by f 7→ ( f (t) − f (−t))/2t. The defining
relations (15) then become h f + f h = 0, he + eh = 0, e f − f e = 0. This is a color-commutative
Lie algebra graded by Z22 with graded decomposition V = V(0,0) ⊕ V(1,0) ⊕ V(0,1) ⊕ V(1,1) =
F{0} ⊕ Fy ⊕ F{0} ⊕ Fx ⊕ Fz .
We denote the ”lifting” of the right-hand-side of (11a), (11b) (11c) for p0 = 1, q0 = 0, to an
abstract skew-symmetric algebra with products hh, f i = −2q f , hh, ei = 2e, he, f i = q+1
2 h, by
q sl2 (F) and call it informally the ”Jackson sl2 (F)”. The algebra q sl2 (F) is a Hom-Lie algebra
with α = q−1 σ acting on the generators e, f , h as α(e) = q−1 e, α(h) = h and α( f ) = q f , and
with the Hom-Lie Jacobi identity given by (12). It is not clear à priori since q sl2 (F) is not a
subalgebra of A · ∂σ as it is defined abstractly without reference to a particular representation. It
is in principle possible that the twisted Jacobi identity is a result of some extra relations available
in that representation as σ-derivations. It is not difficult to verify however that indeed q sl2 (F)
is itself a hom-Lie algebra [29]. When p0 6= 1 we get a natural one-parameter deformation of
q sl2 (F). The algebra F{e, f , h}/(15), with p0 = 1 in (15), can be thought of as an analogue
of the universal enveloping algebra for q sl2 (F). For q = 1 it is indeed the universal enveloping
algebra of the Lie algebra sl2 (F). We denote by Uq the algebra F{e, f , h}/(15). When p0 6= 1
we similarly get a one-parameter deformation of Uq . Note also that unlike the undeformed
U(sl2 (F)), the one-parameter deformation of Uq (with q 6= 1) has non-trivial one-dimensional
representations. When q → 1, in which case we “approach sl2 (F)”, the representation explodes
to infinity and thus do not converge to a representation of sl2 (F). Some of these algebras
appeared in the literature before in totally different contexts as examples of almost quadratic
algebras, iterated Ore extensions, noetherian domains of Gel’fand–Kirillov dimension less then
or equal three, Auslander-regular algebras, Artin–Schelter regular algebras, Koszul algebras,
algebras in non-commutative projective algebraic geometry [1, 2, 3, 14, 32, 33, 35, 39] as well
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as in a mathematical physics literature in connection to quantum groups and quantum algebras
and their applications (see [29] for more detailed discussion and references).
In [26, 29], the quasi-deformations of sl2 (F) with base algebra F[t]/(t N ) where considered.
Some unusual interesting multi-parameter families of hom-Lie algebras, quasi-Hom-Lie algebras and associative envelopes arise in this way. For example in one of the cases for N = 3 one
gets the algebras defined by relations
hh, f i :
hh, ei :
q1 h f + 2q2 f 2 − q21 f h = 0
q1 he + 2q2 f e − eh = −p1 h − 2p2 f
he, f i :
e f − q21 f e = p1 (q1 + 1) f .
(16)
Once again, we cannot recover sl2 (F) from this deformation by choosing suitable values of the
parameters since he, f i can never be h, hh, ei can never be 2e and hh, f i is identically zero.
Take p1 = p2 = 0, q1 = 1 and ε := −2q2 in (16). Then the first relation in (16) becomes
h f − f h = ε f 2 which is the defining relation for the Jordanian quantum plane or the ε-deformed
quantum plane (see [7]).
If F = C, q1 = 1, p1 = 1, a = 2p2 and q2 = 0, then (16) turn into the relations for the universal
enveloping algebra of the three-dimensional Lie algebra g:
h f − f h = 0,
he − eh = −h − a f ,
ef − fe = 2f.
This Lie algebra is solvable and g(2) = 0 where g(1) = hg, gi and g(i) = hg(i−1) , g(i−1) i for every
a ∈ C. When instead p1 = 0 and p2 = −1/2, we get h f − f h = 0, he − eh = f , e f − f e = 0,
the relations for the Heisenberg Lie algebra. Putting instead p1 = p2 = 0 we get the threedimensional polynomial algebra in three commuting variables. So, in a sense, the class of
algebras with three generators and defining relations (16), being obtained via a special twisting
of sl2 (F), is a multi-parameter deformation of the polynomial algebra in three commuting variables, of the Heisenberg Lie algebra and of the above solvable Lie algebra. The specific form
of the twisted Jacobi identity can be obtained again from Theorem 5, but it is more complicated
and both Hom-Lie and quasi-Hom-Lie algebras appear for different cases of parameters [29].
Quasi-Hom-Lie and Hom-Lie algebras of Witt and Virasoro type
Application of Theorem 5 to Lie algebras of vector fields leads immediately to parametric families of infinite-dimensional quasi-hom-Lie and hom-Lie algebras. These algebras are precisely
the objects behind the discrete analogues of the vector fields obtained when derivations are
replaced by their various discrete counterparts.
Example 8. Let A be the unique factorization domain F[t,t −1 ], the Laurent polynomials in t
over the field F. Then the space Dσ (A ) can be generated by a single element D as a left A module, that is, Dσ (A ) = A · D (Theorem 1 in [18]). When σ(t) = qt with q 6= 0 and q 6= 1, one
can take D as t times the Jackson q-derivative
D=
id −σ
1−q
:
f (t) 7→
f (t) − f (qt)
.
1−q
Then (4) and (5) are satisfied with δ = 1, and Theorem 5 yields a quasi-Lie (and even a homL
Lie) algebra structure on Dσ (A ) as follows. The F-linear space Dσ (A ) = n∈Z F · dn , with
dn = −t n D can be equipped with the skew-symmetric bracket h·, ·iσ defined on generators by
(6) as
hdn , dm iσ = qn dn dm − qm dm dn = hdn , dm iσ = ({n}q − {m}q )dn+m ,
172
where {n}q = (qn − 1)/(q − 1) for q 6= 1 and {n}1 = n. This bracket satisfies the σ-deformed
Jacobi-identity
(qn + 1)hdn , hdl , dm iσ iσ + (ql + 1)hdl , hdm , dn iσ iσ + (qm + 1)hdm , hdn , dl iσ iσ = 0.
Obviously this can be viewed as a q-deformed Witt algebra. It can be also shown that there is
a central extension Virq of this deformation in the category of hom-Lie algebras [18], therefore
being a natural q-deformation of the Virasoro algebra. The algebra Virq is spanned by elements
{dn | n ∈ Z} ∪ {c} where c is central, i.e., hVirq , ci = hc, Virq i = 0. The bracket of dn , dm is
computed according to
hdn , dm i = ({n}q − {m}q )dm+n + δn+m,0
q−n
{n − 1}q {n}q {n + 1}q c.
6(1 + qn )
Note that when q = 1 we retain the classical Virasoro algebra
hdn , dm i = (n − m)dm+n + δn+m,0
1
{n − 1}{n}{n + 1}c.
12
from conformal field and string theories.
Example 9. We let A = F[t,t −1 ] as before and σ be some non-zero endomorphism such that
σ(t) = qt s . The element D = (id − σ)/(α−1 · t k ) generates a cyclic A -submodule M of Dσ (A ).
The element δ can be computed to be δ = qkt (s−1)k . Put dn = −t n D. The F-linear space M =
⊕i∈Z F · di allows the structure of an algebra (by Theorem 5) with bracket defined on generators
(by (6)) as
hdn , dm iσ = qn dns dm − qm dms dn
and satisfying relations
hdn , dm iσ = αqm dms+n−k − αqn dns+m−k ,
with s ∈ Z and α ∈ F. Note that when s 6= 1, the expression for the bracket of dn and dm involves
four generators. The σ-deformed Jacobi identity becomes
n,m,l qn hdns , hdm , dl iσ iσ + qkt (s−1)k hdn , hdm , dl iσ iσ = 0.
(17)
This defines a quasi-Lie algebra which is also a quasi-hom-Lie algebra but not a hom-Lie algebra. For the detailed computations, see [18]. A direct, but tedious computation, shows that the
algebra with abstract generators {dn | n ∈ Z} ∪ {t} and relations
qn dns dm − qm dms dn = αqm dms+n−k − αqn dns+m−k ,
t · dn = dn+1 , ∀n, m ∈ Z
satisfies (17).
It is interesting to observe that, when q = 1, k = 0 and s = 1, we get a commutative algebra with
countable number of generators instead of the Witt algebra. So, this is a quasi-deformation of
the Witt algebra.
173
Example 10. By the bold font we denote the vectors k = (k1 , k2 , . . . , kn ), ki ∈ Z. Consider the
±1
±1 ∼ C[z1 ,...,zn ,u1 ,...,un ]
algebra A = C[z±1
1 , z2 , . . . , zn ] = (z1 u1 −1,...,zn un −1) of Laurent polynomials in z1 , z2 , . . . , zn . Let
S
S
σ(zi ) = qzi z1i,1 · · · zni,n for 1 ≤ i ≤ n and qzi ∈ F∗ . Notice that σ is determined by an integral
matrix S = [Si, j ] and the vector of F-numbers (qz1 , . . . , qzk ). A common divisor on (id −σ)(A )
−G1
∗
n
n
is g = Q−1 z1G1 · · · zG
· · · z−G
n for Q ∈ F and so ∂σ = Qz1
n (id −σ) generates an A -submodule
Gn δ 1
δn
1
M of Dσ (A ). For these ∂σ and σ by formula (5) one gets that δ = qG
z1 · · · qzn z1 · · · zn , where
δk = G1 S1,k + G2 S2,k + · · · + Gk (Sk,k − 1) + · · · + Gn Sn,k for all k = 1, . . . , n, that is (δ1 , . . . , δn ) =
→
−
→
−
G (S − I) with G = (G1 , . . . , Gn ). Let wr (l) = ∑ni=1 li Si,r for r = 1, . . . , n. Now, σ(zl11 · · · zlnn ) =
l S1,1 +···+ln Sn,1
σ(z1 )l1 · · · σ(zn )ln = qlz11 · · · qlznn z11
l S1,n +···+ln Sn,n
w (l)
w (l)
= qlz11 · · · qlznn z1 1 · · · zn n and thus
w (l)−l1
w (l)−ln
∂σ (zl11 · · · zlnn ) = −Qzl11 −G1 · · · zlnn −Gn qlz11 · · · qlznn z1 1
· · · zn n
− 1 . Consider F-linear space
L
l1
ln
l∈Zn F · dl spanned by dl = dl1 ,...,ln = −z1 · · · zn ∂σ . It can be endowed with a bracket defined
on generators (by (6)) as
· · · zn1
hdk , dl iσ = qkz11 · · · qkznn dw1 (k),...,wn (k) dl − qlz11 · · · qlznn dw1 (l),...,wn (l) dk =
Qqlz11 · · · qlznn dw1 (l)+k1 −G1 ,...,wn (l)+kn −Gn − Qqkz11 · · · qkznn dw1 (k)+l1 −G1 ,...,wn (k)+ln −Gn .
The bracket satisfies the σ-deformed Jacobi identity
k,l,h
Gn δ 1
δn
1
qkz11 · · · qkznn hdw1 (k),...,wn (k) , hdl , dh iσ iσ + qG
z1 · · · qzn z1 · · · zn hdk , hdl , dh iσ iσ
= 0.
→
−
→
−
→
−
Moreover, if G ∈ Zn satisfies G S = G , then a hom-Lie algebra is obtained.
Example 11. Let f ∈ F[t] and define σ( f )(t) = f (t +h), h ∈ F∗ and ∂σ ( f )(t) = σ( f )(t)− f (t) =
f (t + h) − f (t). This operator is easily checked to be a σ-derivation. We have that F[t] is a
L
UFD (Unique Factorization Domain) and S := Dσ (F[t]) = n∈Z≥0 F · dn , where dn = −t n ∂σ .
By Theorem 5 we compute the bracket and thus the commutation relations satisfied by the
generators {dk }:
n m n n− j
m m−k
hdn , dm iS = ∑
h d j dm − ∑
h
dk dn =
j=0 j
k=0 k
m+n−1 h
n
m i n+m−k
=
−
h
dk ,
∑
k−m
k−n
k=min(m,n)
assuming ms = 0 for s < 0. In the present case we have δ = 1, and for α = σ : a∂σ 7→ σ(a)∂σ
we note that
n n n i
n i
n
n−i
α(dn ) = σ(−t )∂σ = ∑
h (−t ∂σ ) = ∑
h dn−i .
i=0 i
i=0 i
According to Theorem 5, this gives us that (S, h·, ·iS , σ) is a hom-Lie algebra with deformed
Jacobi identity
n n i
n,m,l ∑
h dn−i + dn , hdm , dl i = 0.
i=0 i
The algebra thus obtained is a natural Witt-type algebra associated to an additive shift difference
operator ∂σ , frequently used in many parts of mathematics, science and engineering.
174
Remark 12. An interesting direction with many potential applications is to construct and study
deformed Virasoro-type algebras obtained as central extensions of this hom-Lie algebra, as
well as more general quasi-Lie algebras derived from Theorem 5, within the classes of homLie algebras, quasi-hom Lie algebras and quasi-Lie algebras. Another possibly fruitful line of
investigation would be to study general extensions of one quasi-Lie algebra by another. For
the construction in this spirit for Lie algebras see [22] where it is remarked that a certain short
exact sequence can be seen as a central extension of Vir by an infinite-dimensional Heisenberg
algebra h∞ .
Remark 13. Some developments in expanding theory of Quasi-Lie and in particular quasi-HomLie and Hom-Lie algebras are beyond the scope of this artice. In [34] we provided a different
way for constructing Hom-Lie algebras by extending the fundamental construction of Lie algebras from associative algebras via commutator bracket multiplication. To this end we defined
the notion of Hom-associative algebras generalizing associative algebras to a situation where associativity law is twisted, and showed that the commutator product defined using the multiplication in a Hom-associative algebra leads naturally to Hom-Lie algebras. We have introduced also
Hom-Lie-admissible algebras and more general G-Hom-associative algebras with subclasses of
Hom-Vinberg and pre-Hom-Lie algebras, generalizing to the twisted situation Lie-admissible
algebras, G-associative algebras, Vinberg and pre-Lie algebras respectively, and showed that
for these classes of algebras the operation of taking commutator leads to Hom-Lie algebras as
well. In another forthcoming article by the same authors the classes of bialgebras, co-algebras
and Hopf algebras are extended to accommodate Hom-associative and Hom-Lie algebras and
corresponding quasi-deformations of Lie algebras.
REFERENCES
[1] M. Artin, W. Schelter, Graded Algebras of Global Dimension 3, Adv. Math. 66 (1987),
171–216.
[2] A. D. Bell, S. P. Smith, Some 3-dimensional Skew Polynomial Rings, Preprint, 1997.
[3] J.-E. Björk, E. K. Ekström, Filtered Auslander-Gorenstein rings, in “Operator algebras,
unitary representations, enveloping algebras, and invariant theory (Paris, 1989)” Progr.
Math. 92, Birkhäuser, Boston, MA, 1990, 425–448.
[4] M. Chaichian, P. Kulish, J. Lukierski, q-deformed Jacobi identity, q-oscillators and qdeformed infinite-dimensional algebras, Phys. Lett. B 237 (1990), 401–406.
[5] X.-W. Chen, T. Petit, F. Van Oystaeyen, Note on the cohomology of color Hopf and Lie
algebras, J. Algebra, 299 (2006), 419-442.
[6] X.-W. Chen, S. D. Silvestrov, F. Van Oystaeyen, Representations and cocycle twists of
color lie Algebras, Algebr. Represent. Theory 9 (2006), 633–650.
[7] S. Cho, J. Madore, K. S. Park, Noncommutative Geometry of the h-deformed Quantum
Plane, J. Phys. A 31 (1998), 2639–2654.
[8] T. L. Curtright, C. K. Zachos, Deforming maps for quantum algebras, Phys. Lett. B 243
(1990), 237–244.
175
[9] E. V. Damaskinsky, P. P. Kulish, Deformed oscillators and their applications (Russian),
Zap. Nauch. Semin. LOMI 189 (1991), 37–74; Engl. transl. in J. Soviet Math. 62 (1992),
2963–2986.
[10] C. Daskaloyannis, Generalized Deformed Virasoro Algebras, Modern Phys. Lett. A 7
(1992), 809–816.
[11] P. Deligne, P. Etingof, D. S. Freed, L. C. Jeffrey, D. Kazhdan, J. W. Morgan, D. Morrison,
E. Witten, (Eds) “Quantum Fields and Strings: A Course for Mathematicians”, 2 vol.,
Amer. Math. Soc., Providence, 1999.
[12] P. Di Francesco, P. Mathieu, D. Sénéchal, “Conformal Field Theory”, Springer, New York,
1997.
[13] V. G. Drinfel’d, Hopf algebras and the quantum Yang-Baxter equation, Soviet Math. Doklady 32 (1985), 254–258.
[14] E. K. Ekström, The Auslander condition on graded and filtered noetherian rings,
in “Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, 39ème Année (Paris,
1987/1988)” Lect. Notes Math. 1404, Springer, Berlin, 1989, 220–245.
[15] I. Frenkel, J. Lepowsky, A. Meurman, “Vertex Operator Algebras and the Monster”, Academic Press, Boston, 1988.
[16] J. Fuchs, “Affine Lie Algebras and Quantum Groups”, Cambridge Univ. Press, Cambridge,
1992.
[17] J. Fuchs, Lectures on Conformal Field Theory and Kac-Moody Algebras, in “Conformal
field theories and integrable models (Budapest, 1996)”, Lect. Notes Phys. 498, Springer,
Berlin, 1997, 1–54.
[18] J. T. Hartwig, D. Larsson, S. D. Silvestrov, Deformations of Lie algebras using σderivations, J. Algebra 295 (2006), 314–361.
[19] L. Hellström, S. D. Silvestrov, “Commuting Elements in q-Deformed Heisenberg Algebras”, World Scientific, River Edge, 2000.
[20] N. Hu, q-Witt algebras, q-Lie algebras, q-holomorph structure and representations, Algebra Colloq. 6 (1999), 51–70.
[21] H. P. Jakobsen, Matrix Chain Models and their q-deformations, Preprint Mittag-Leffler
Institute, Report No 23, 2003/2004, 15 pp.
[22] H. P. Jakobsen, H. C.-W. Lee, Matrix Chain Models and Kac-Moody algebras, Contemp.
Math. 343 (2004), 147–165.
[23] M. Jimbo, A q-difference analogue of U(g) and the Yang–Baxter equation, Lett. Math.
Phys. 10 (1985), 63–69.
[24] C. Kassel, ”Quantum Groups”, Graduate Texts Math. 155, Springer, Berlin, 1995.
176
[25] P. P. Kulish, N. Y. Reshetikhin, Quantum Linear Problem for the Sine-Gordon Equation
and Higher Representations, Zap. Nauchn. Sem. Leninggrad. Otdel. Mat. Inst. Steklov 101
(1981) 101–110, Engl. transl. in J. Soviet Math 23 (1983) no 4.
[26] D. Larsson, G. Sigurdsson, S. D. Silvestrov, On some almost quadratic algebras coming
from twisted derivations. J. Nonlinear Math. Phys. 13 (2006), suppl. 1, 76–86.
[27] D. Larsson, S. D. Silvestrov, Quasi-hom-Lie algebras, Central Extensions and 2-cocyclelike identities, J. Algebra 288 (2005), 321–344.
[28] D. Larsson, S. D. Silvestrov, Quasi-Lie algebras, Contemp. Math. 391 (2005), 241-248.
[29] D. Larsson, S. D. Silvestrov, Quasi-deformations of sl2 (F) using twisted derivations,
Comm. Algebra, in press, arXiv:math.RA/0506172.
[30] D. Larsson, S. Silvestrov, Graded quasi-Lie algebras, Czechoslovak J. Phys., 55 (2005),
1473–1478.
[31] L. Le Bruyn, Conformal sl2 Enveloping Algebras, Comm. Algebra 23 (1995), 1325–1362.
[32] L. Le Bruyn, S. P. Smith, M. Van den Bergh, Central Extensions of Three-Dimensional
Artin–Schelter Regular Algebras, Math. Z. 222 (1996), 171–212.
[33] L. Le Bruyn, M. Van den Bergh, On Quantum Spaces of Lie Algebras, Proc. Amer. Math.
Soc. 119 (1993), 407–414.
[34] A. Makhlouf, S. D. Silvestrov, Hom-algebra structures, J. Lie Theory Appl., in press,
arXiv:math.RA/0601766.
[35] D. Piontkovski, S. D. Silvestrov, Cohomology of 3-dimensional color Lie algebras, J.
Algebra, in press, DOI: doi:10.1016/j.jalgebra.2006.11.008; arXiv:math.KT/0508573.
[36] V. Rittenberg, D. Wyler, Generalized superalgebras, Nucl. Phys. B 13 (1978), 189–202.
[37] M. Scheunert, Generalized Lie algebras, J. Math. Phys. 20 (1979), 712–720.
[38] S. D. Silvestrov, On the classification of 3-dimensional colored Lie algebras, in “Quantum
Groups and Quantum Spaces”, Banach Center Publications 40 (1997) 159–170.
[39] F. Van Oystaeyen, “Algebraic Geometry for Associative Algebras”, Dekker, New York,
2000.
177
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