Proc. Indian Acad. Sci. (Math. Sci.) Vol. 126, No. 3, August 2016, pp. 341–352.
DOI 10.1007/s12044-016-0292-5
Diamond lemma for the group graded quasi-algebras
MAMTA BALODI1 , HUA-LIN HUANG2
and SHIV DATT KUMAR1,∗
1 Department of Mathematics, Motilal Nehru National Institute of Technology,
Allahabad 211 004, India
2 School of Mathematics, Shandong University, Jinan 250100, China
* Corresponding author.
E-mail: [email protected]; [email protected]; [email protected]
MS received 20 May 2014; revised 7 June 2015
Abstract. Let G be a group. We prove that every expression in a G-graded quasialgebra can be reduced to a unique irreducible form and the irreducible words form a
basis for the quasi-algebra. The result obtained is applied to some interesting classes of
group graded quasi-algebras like generalized octonions.
Keywords.
Diamond lemma; quasi-algebra; octonions.
2010 Mathematics Subject Classification.
16S15, 17A30, 16T05.
1. Introduction
The term quasi-algebra was introduced in [2] as an algebra in a monoidal category. Since
the associativity constraints in these categories are allowed to be nontrivial, the class
of quasi-algebras contains various important examples of non-associative algebras like
the octonions and other Cayley algebras [2]. The need for non-associative geometry for
non-commutative differential forms was shown in [3], where it has been proved that all
differential form algebras on the standard q-deformation quantum groups, if they are to
be bicovariant and to have classical dimensions, must be non-associative. This shows that
one should seriously consider non-associative algebras. Quasi-associative algebras play
an important role in the theory of nonassociative algebras.
Given a group G and a 3-cocycle : G × G × G −→ k ∗ , G-graded quasi-algebras
satisfy
(a · b) · c = (|a|, |b|, |c|) a · (b · c)
for the homogeneous elements a, b, c of degrees |a|, |b| and |c| respectively. If is trivial,
then the quasi-algebra is nothing but a graded associative algebra.
The diamond lemma is a well-known reduction method used in algebra. The original
diamond lemma was stated in graph theory by Newman [9], which says that given a directed
graph, if it satisfies the descending chain condition (DCC), i.e. any directed path from a
vertex has finite length and the diamond condition – namely, any two distinct edges starting
from a vertex in the graph may be extended to directed paths that end up at a common vertex, then every connected component of the graph has a unique maximal vertex. Bergman
in [4] proved an analogue of the Newman’s diamond lemma for associative algebras.
c Indian Academy of Sciences
341
342
Mamta Balodi et al.
In this paper, we prove the diamond lemma for the group graded quasi-algebras
and to make the result more accessible, we apply it on some interesting group graded
quasi-algebras like octonion algebra and generalized octonions. Explicitly, we prove the
following: Let F = kX be a free group graded quasi-algebra on a set X over a field k,
where X is the set of all monomials on X. We generalize the notion of partial ordering,
reduction system S and ambiguities [4] for the group graded quasi-algebras in a suitable
way and prove that
(1) every expression in F reduces to a unique irreducible one.
(2) the set of irreducible words (under the DCC and diamond conditions) precisely gives
the basis for R = F/I , where I is the ideal generated by the elements Wσ − fσ ,
(Wσ , fσ ) ∈ S.
The first step in the classification project of quasi-quantum groups [5, 6] is to say
whether the subalgebra (in a graded quiver Majid algebra) generated by the paths having
unit of the group as the source vertex, is finite dimensional or not. This subalgebra forms
a G-graded quasi-algebra. As mentioned above, the version of diamond lemma, provided
in this paper, works as a tool to find a basis for group graded quasi-algebras. We hope that
this will help in the classification of quasi-quantum groups.
The paper is organized as follows: in §2, we give the ideas of monoidal categories and
group graded quasi-algebras. In §3, we define the reduction system and ambiguities for
these algebras. In §4, we prove the main result and in §5, we find the basis for some group
graded quasi-algebras. Throughout the paper, k denotes the field of complex numbers.
2. Preliminaries
DEFINITION 2.1
A monoidal category is a collection (C, ⊗, a, 1, l, r), where
(1)
(2)
(3)
(4)
C is a category,
⊗ : C × C −→ C is a bifunctor called the tensor product bifunctor,
1 ∈ C is the unit object, and
aV ,W,U : (V ⊗ W ) ⊗ U −→ V ⊗ (W ⊗ U ), lV : 1 ⊗ V −→ V and rV : V ⊗ 1 −→ V
are natural isomorphisms
such that the following diagrams commute:
Diamond lemma for the group graded quasi-algebras
343
More about monoidal categories can be found in [7].
DEFINITION 2.2
Let (H, , ε) be a coalgebra. A comodule over H is a vector space V with a map β :
V −→ V ⊗ H , such that
(id ⊗ ) ◦ β = (β ⊗ id) ◦ β,
(id ⊗ ε) ◦ β = id,
where id is the identity map on H .
Let (H, φ) be a dual quasi-Hopf algebra [2], φ being the reassociator of H . The
category of comodules (H, φ) forms a monoidal category with associativity constraint as
aV ,W,U ((v ⊗ w) ⊗ u) = v0 ⊗ (w0 ⊗ u0 )φ(v1 , w1 , u1 )
for v ∈ V , w ∈ W and u ∈ U . Here we use the Sweedler’s notation [10] for the image of
an element v ∈ V under the map β, i.e. β(v) = v0 ⊗ v1 , with summation sign omitted.
DEFINITION 2.3
Let H be a dual quasi-Hopf algebra. A H -comodule quasi-algebra A is an algebra in the
category of H -comodules i.e. a H -comodule with a product map such that
(a · b) · c = (a1 · b1 ) · c1 φ(a2 , b2 , c2 ),
β(a · b) = β(a)β(b),
for all a, b ∈ A, where the last expression uses the tensor product algebra in A ⊗ H .
DEFINITION 2.4
Let G be a group with identity element u. A normalized 3-cocycle on G is a map :
G × G × G −→ k ∗ which satisfies for all e, f, g, h ∈ G,
(e, f, g)(e, f g, h)(f, g, h) = (ef, g, h)(e, f, gh),
(g, u, h) = 1.
Let H = kG be the group algebra of the group G and a 3-cocycle on G. Then
(kG, ) can be understood as a dual quasi-Hopf algebra (see [2]). A (kG, )-comodule
V becomes a G-graded space with β(v) = v ⊗ gv , on homogeneous element v of degree
gv . In the monoidal category of (kG, )-comodules, the notion of an algebra becomes
a G-graded quasi-algebra. Hence the definition of a kG-comodule quasi-algebra can be
given as follows.
DEFINITION 2.5 [2]
Let G be a group and a 3-cocycle on G. A G-graded quasi-algebra A is a G-graded
vector space with a product, preserving the total degree and associative in the sense
(a · b) · c = a · (b · c)(|a|, |b|, |c|)
of homogeneous degree.
∀ a, b, c ∈ A,
344
Mamta Balodi et al.
Example 2.6. Let F be a 2-cochain on a group G, i.e. a map F : G × G −→ k ∗ satisfying
F (u, a) = 1 = F (a, u) for any a ∈ G, where u is the identity element of G. Then the
group algebra kG twisted by defining the new product on it as
g.h = F (g, h)gh,
∀ g, h ∈ G,
forms a G-graded quasi-algebra, denoted by kF G, whose 3-cocycle is = ∂F , which
means
(g, h, f ) =
F (h, f )F (g, hf )
.
F (g, h)F (gh, f )
3. Reductions and ambiguities
In this section, we define the notions of reduction systems and ambiguities for group
graded quasi-algebras.
Let X be any set and X denote the set of words on X. Suppose F = kX is the
free G-graded
quasi-algebra on X. Then any nonzero element f ∈ F is of the form
f = ni=1 ai mi , where ai ∈ k ∗ and mi = (· · · ((mi1i mi2i )mi3i ) · · · )miri ∈ X.
3.1 Reduction system
For a free G-graded quasi-algebra F, we define a reduction system to be the set
S = {σ = (Wσ , fσ ) | Wσ ∈ X, fσ ∈ F}.
For any σ ∈ S, and A, B ∈ X, we define a reduction, the linear operator
R(A.σ ).B : F −→ F,
that sends the monomials (A · Wσ ) · B and A · (Wσ · B) to (A · fσ ) · B and A · (fσ · B),
respectively and fixes all other monomials.
Let f ∈ F. If the coefficients of (A · Wσ ) · B and A · (Wσ · B) in f are 0, then
R(A·σ )·B (f ) = f and we say R(A·σ )·B is trivial on f . If every reduction under S is trivial
on f , then we say f is irreducible (under S). We have the following result.
Lemma 3.1. All irreducible elements of F form a subspace.
Proof. Let f, g ∈ F be any irreducible elements and R(A·σ )·B be any reduction. Then we
have R(A·σ )·B (f ) = f and R(A·σ )·B (g) = g so that R(A·σ )·B (αf + βg) = αf + βg for
any α, β ∈ k.
This subspace of all irreducible elements will be denoted by Firr .
Let f ∈ F. As in [4], if there is a finite sequence of reductions R1 , . . . , Rt under S
such that Rt · · · R1 (f ) = d and d ∈ Firr , then we say d is a reduced form of f under S.
DEFINITION 3.2 [4]
An element f ∈ F is said to be reduction-finite, if for every infinite sequence of reductions R1 , R2 , . . . , there exists a natural number i such that ∀j > i, Rj +1 is trivial on
Rj · · · R2 R1 (f ).
Diamond lemma for the group graded quasi-algebras
345
Lemma 3.3 [4]. All reduction-finite elements form a subspace of F and if f ∈ F is
reduction-finite, then f has a reduced form.
Proof. Let f , g ∈ F be reduction-finite elements and R1 , R2 , . . . be any infinite sequence
of reductions. Since f is reduction-finite, there exists a natural number i such that for
all j > i , Rj +1 is trivial on Rj · · · R2 R1 (f ). Similarly, there exists i such that for all
j > i , Rj +1 is trivial on Rj · · · R2 R1 (g). Taking i = max (i , i ), it follows that for all
j > i, Rj +1 is trivial on Rj · · · R2 R1 (αf + βg) for any α, β ∈ k.
DEFINITION 3.4 [4]
An element f ∈ F is said to be reduction-unique if it is reduction-finite and has a unique
reduced form.
This unique reduced value will be denoted by RS (f ).
Lemma 3.5.
(1) Reduction-unique elements form a subspace of F.
(2) Let a, b, c ∈ F, if for all monomials A, B, C occurring in a, b, c respectively, (A·B)·
C is reduction-unique, then for any finite composition of reductions R, (a · R(b)) · c
is reduction-unique, and RS ((a · R(b)) · c) = RS ((a · b) · c).
Proof. Similar to the proof of Lemma 1.1 in [4].
Remark 3.6. Since reduction-unique elements form a subspace of F, we can regard RS
as a k-linear map from this subspace to Firr .
3.2 Ambiguities
A 5-tuple (σ, τ, A, B, C), where σ, τ ∈ S, A, B, C ∈ X\{1}, such that
Wσ = A · B
and
Wτ = B · C,
is an overlap ambiguity of S. It is called resolvable if there exist compositions of
reductions R and R such that
R(fσ · C) = (|A|, |B|, |C|) R (A · fτ ).
A 5-tuple (σ, τ, A, B, C), σ = τ ∈ S, A, B, C ∈ X, is an inclusion ambiguity if
Wσ = B
and
Wτ = (A · B) · C (or A · (B · C)).
It is resolvable if (A·fσ )·C (or A·(fσ ·C)) and fτ can be reduced to a common expression.
DEFINITION 3.7
A monomial partial order on X is a partial order ≤ such that
B ≤ B ⇒ (A · B) · C ≤ (A · B ) · C
We write A < B if and only if A ≤ B and A = B.
and
A · (B · C) ≤ A · (B · C).
346
Mamta Balodi et al.
The partial order ≤ is said to satisfy descending chain condition (DCC) if there is no
infinite properly descending chain in X with respect to ≤.
DEFINITION 3.8
Given a reduction system S, if for all σ = (Wσ , fσ ) ∈ S, fσ is a linear combination of
monomials < Wσ , then ≤ is said to be compatible with S.
Lemma 3.9 Let ≤ be a monomial partial order on X, compatible with the reduction
system S. If ≤ satisfies DCC on X, then every element of F is reduction-finite.
Proof. Since reduction-finite elements form a subspace of F, we only need to show that
every monomial is reduction-finite. Suppose that
U = {D ∈ X| D is not reduction-finite} = ∅.
Since the partial order ≤ satisfies DCC, there is a minimal element D0 in U . As D0
is not reduction-finite, there exists an infinite sequence of reductions R1 , R2 , . . ., such
that for all i ∈ N, there exists j (i) > i, Rj +1 is not trivial on Rj · · · R2 R1 (D0 ). If
Rl (D0 ) = D0 , for all 1 ≤ l ≤ j + 1,
Rj +1 (Rj · · · R2 R1 (D0 )) = D0 = Rj · · · R2 R1 (D0 ),
which means, there is some R(A·σ )·B which is not trivial on D0 i.e.
R(A·σ )·B (D0 ) = (A · fσ ) · B (or A · (fσ · B)).
Using the compatibility of ≤ with S and minimality of D0 , we see that (A · fσ ) · B
(or A · (fσ · B)) is a linear combination of reduction-finite monomials <D0 and hence,
is also reduction-finite. But then D0 itself would be reduction-finite, which leads to a
contradiction.
Let ≤ be a monomial partial order on X, compatible with the reduction system S. For
any A ∈ X, we define a subspace of F,
IA = span of {(B · (Wσ − fσ )) · C | (B · Wσ ) · C < A}.
Remark 3.10. In the quasi-algebra F, we have I(A·B)·C = IA·(B·C) .
DEFINITION 3.11
An overlap ambiguity (σ, τ, A, B, C) is said to be resolvable relative to ≤, if
fσ · C − αA · fτ ∈ I(A·B)·C ,
where α = (|A|, |B|, |C|).
DEFINITION 3.12
An inclusion ambiguity (σ, τ, A, B, C) is said to be resolvable relative to ≤, if
(A · fσ ) · C − fτ ∈ I(A·B)·C .
Diamond lemma for the group graded quasi-algebras
347
(A·σ )·B
Remark 3.13. We denote by f −→ g, the fact that
R(A·σ )·B (f ) = g
and
R(A·σ )·B is nontrivial on f.
This implies
f − g = c(A · (Wσ − fσ )) · B,
where c ∈ k ∗ .
4. Main theorem
Now we are ready to prove the diamond lemma for group graded quasi-algebras.
Theorem 4.1. Let S be a reduction system for a free G-graded quasi-algebra F, ≤ be a
monomial partial order on X, compatible with S and satisfies DCC, then the following
are equivalent:
(i)
(ii)
(iii)
(iv)
All ambiguities of S are resolvable.
All ambiguities of S are resolvable relative to ≤.
All elements of F are reduction-unique under S.
As vector spaces
F = Firr ⊕ I,
where I is the two-sided ideal of F generated by {Wσ − fσ | σ ∈ S}.
Proof. Since every element of F is reduction-finite, it has a reduced form.
(iii) ⇒ (iv): Under the assumption of (iii), RS becomes an onto map from F to Firr .
Hence, we only need to show ker(RS ) = I . Let f ∈ I , then
p
f =
ci [Ai · (Wσi − fσi )] · Bi
i=1
so that
RS (f ) =
p
i=1
ci RS [{Ai · (Wσi − fσi )} · Bi ].
But
RS [{Ai · (Wσi − fσi )} · Bi ] = RS [(Ai · Wσi ) · Bi ] − RS [(Ai · fσi ) · Bi ] = 0,
by Lemma 3.5. Hence f ∈ ker(RS ) and I ⊆ ker(RS ). By Remark 3.13, it is easy to show
that ker(RS ) ⊆ I .
(iv) ⇒ (iii): Let f ∈ F. Suppose f1 and f2 are two different reduced values of f . Then
f −→ f10 −→ f11 −→ · · · −→ f1p = f1
which implies
f − f1 =
p
i=1
ci [Ai · (Wσi − fσi )] · Bi ∈ I.
348
Mamta Balodi et al.
Similarly f − f2 ∈ I and hence f1 − f2 ∈ I . Also f1 − f2 ∈ Firr . Therefore f1 = f2
and f is reduction-unique.
(iii) ⇒ (i): Let (σ, τ, A, B, C) be any overlap ambiguity of S i.e., Wσ = A · B and
Wτ = B · C. Since all elements of F are reduction-unique under S,
fσ · C −→ f1 −→ f2 −→ · · · −→ fp = RS ((A · B) · C),
A · fτ −→ g1 −→ g2 −→ · · · −→ gq = RS (A · (B · C)),
i.e., there exist compositions of reductions R and R such that
R(fσ · C) = RS ((A · B) · C)
and
R (A · fτ ) = RS (A · (B · C))
which imply
R(fσ · C) = (|A|, |B|, |C|) R (A · fτ ).
Hence the ambiguity is resolvable.
(i) ⇒ (ii): Let (σ, τ, A, B, C) be a resolvable overlap ambiguity of S. Then there exist
compositions of reductions R and R such that
R(fσ · C) = (|A|, |B|, |C|) R (A · fτ )
i.e.,
fσ · C = f10 −→ f11 −→ · · · −→ f1p = f0
and
αA · fτ = f20 −→ f21 −→ · · · −→ f2q = f0 ,
where α = (|A|, |B|, |C|). The above expressions imply
p
fσ · C − f0 =
ci [Ai · (Wσi − fσi )] · Bi
i=1
and
αA · fτ − f0 =
q
j =1
cj [Aj · (Wσj − fσj )] · Bj .
Since fσ · C is a linear combination of monomials < Wσ · C = (A · B) · C and A · fτ is a
linear combination of monomials < A · Wτ = A · (B · C), each (Ai · Wσi ) · Bi < (A · B) · C
and each (Aj · Wσj ) · Bj < A · (B · C). Therefore
fσ · C − αA · fτ ∈ I(A·B)·C (= IA·(B·C) )
i.e., the ambiguity is resolvable relative to ≤. Similarly we can prove for inclusion ambiguity also.
(ii) ⇒ (iii): As the reduction-unique elements of F form a subspace, it suffices to prove
that all monomials in X are reduction-unique.
Diamond lemma for the group graded quasi-algebras
349
Let M be any monomial in X. Inductively, we assume that all monomials <M are
reduction-unique. Then the subspace spanned by such monomials is contained in the
domain of RS . Since by Lemma 3.5,
RS ((B · (Wσ − fσ )) · C) = 0
for all (B · (Wσ − fσ )) · C ∈ IM , IM ⊆ ker(RS ).
Now suppose R(A1 ·σ )·B1 and R(A2 ·τ )·B2 are any two reductions, each acting nontrivially
on M, i.e
R(A1 ·σ )·B1 (M) = (A1 · fσ ) · B1
R(A2 ·τ )·B2 (M) = (A2 · fτ ) · B2 .
and
Using the compatibility of ≤ with the reduction system S, we see that (A1 · fσ ) · B1 and
(A2 · fτ ) · B2 are linear combination of monomials <M and hence, are reduction-unique.
If we show that
RS (R(A1 ·σ )·B1 (M)) = RS (R(A2 ·σ )·B2 (M)),
(1)
then M would also be reduction-unique.
Without loss of generality, we assume that deg(A1 ) ≤ deg(A2 ). If
M = (A1 · Wσ ) · B1 = (A2 · Wτ ) · B2 ,
where (σ, τ, A, B, C) is an inclusion ambiguity of S, then A1 = A, B1 = B, A2 =
1, B2 = 1 and therefore
R(A1 ·σ )·B1 (M) = (A · fσ ) · C
and
R(A2 ·σ )·B2 (M) = fτ .
Since the ambiguity is resolvable relative to ≤,
(A · fσ ) · C − fτ ∈ I(A·B)·C = IM
and IM is annihilated by RS , thus (1) is proved. This completes the proof.
5. Applications
As an application of the main result, we consider the following examples:
Example 5.1 Consider the group Z4 with the 3-cocycle defined as
1,
y + z < 4,
(x, y, z) =
y + z ≥ 4.
(−1)x ,
Let A be a Z4 -graded quasi-algebra generated by the set X = {1, w} with the 3-cocycle
defined above such that
w 2 · w 2 = −1.
The only ambiguity is (w 2 · w 2 ) · w 2 , which is resolvable trivially. Since
((w · w) · w) · w =
=
=
=
((w2 ) · w) · w
(2, 1, 1)w2 · (w · w)
w 2 · (w · w)
w2 · w2 .
(using the definition of A)
(using the definition of )
350
Mamta Balodi et al.
Therefore the irreducible words are 1, w, w2 and (w · w) · w and hence according to
Theorem 4.1, the set
{1, w, w2 , (w · w) · w}
forms a basis for A (the Tesseranion algebra [11]).
Example 5.2 The octonion algebra is generated by X = {1, a, b, c} such that
a 2 = b2 = c2 = −1,
ab + ba = ac + ca = bc + cb = 0.
(2)
Define a partial ordering on X as 1 < a < b < c. The grading group is G = Z2 × Z2 ×
Z2 with degrees of the homogeneous elements a, b, c; (1, 0, 0), (0, 1, 0) and (0, 0, 1),
respectively. The 3-cocycle on G is given by
(x, y, z) = (−1)|x y z|
for all x = (x1 , x2 , x3 ), y = (y1 , y2 , y3 ), z = (z1 , z2 , z3 ) in G, where |x y z| denotes the
determinant of the matrix whose columns are the vectors x, y and z respectively (see [1]).
Also note that (x, y, z) = 1 or − 1.
We write (2) as
a 2 = −1,
b2 = −1,
ca = −ac,
c2 = −1,
ba = −ab,
cb = −bc
in order to make the reduction system compatible with the partial ordering. The ambiguities are
(aa)a, (bb)b, (cc)c, (ba)a, (ca)a, (bb)a, (cc)b.
It is easy to check that the above ambiguities are resolvable. Therefore, according to
Theorem 4.1, the monomials not containing aa, bb, cc, ba, ca and cb as sub strings will
form a basis. Furthermore, (bc)a = (ac)b = −(ab)c, hence the set
{1, a, b, c, ab, bc, ac, (ab)c}
is a basis for the octonion algebra.
Now we discuss the graded quasi-algebras On and Mn , introduced in [8]. The grading
group for both the algebras is (Z2 )n . Consider the 3-cocycle on (Z2 )n which takes the
triplets (ei , ej , ek ) to −1, for i = j = k and all other elements to 1. Here
ei = (0, . . . , 0, 1, 0, . . . , 0),
where 1 stands at i-th position.
Example 5.3 (Generalized octonions On ). The algebra On was introduced in [8]. It has n
generators u1 , u2 , . . . , un such that
ui 2 = −1
and
ui · uj = −uj · ui .
The degree of the homogeneous element ui is ei , 1 ≤ i ≤ n. Therefore, in On , we have
(ui · uj ) · uk = (ei , ej , ek ) ui · (uj · uk ) = −ui · (uj · uk )
Diamond lemma for the group graded quasi-algebras
351
for i = j = k, otherwise they are equal. Now, define a partial ordering on the generators
as
ui < uj
iff i < j.
We assume
ui1 < ui2 < · · · < uin ,
(3)
where i1 < i2 < · · · < in . We may rewrite the reduction system as
uir 2 = −1,
uis · uir = −uir · uis ,
r < s.
Therefore the ambiguities are
(uir · uir ) · uir ,
(uis · uis ) · uir ,
and
(uis · uir ) · uir ,
(4)
where r < s. Now we check for the resolvability of the above ambiguities.
As (uir · uir ) · uir = −uir and, also (eir , eir , eir )uir · (uir · uir ) = −uir . Hence the
ambiguity (uir · uir ) · uir is resolvable. Further (uis · uis ) · uir = −uir and
(eis , eis , eir ) uis · (uis · uir ) =
=
=
=
=
−uis · (uir · uis )
−(eis , eir , eis )(uis · uir ) · uis
(uir · uis ) · uis
(eir , eis , eis ) uir · (uis · uis )
−uir ,
which shows that the ambiguity (uis · uis ) · uir is also resolvable. At last, we have the
following ambiguity:
(uis · uir ) · uir =
=
=
=
−(uir · uis ) · uir
−(eir , eis , eir ) uir · (uis · uir )
uir · (uir · uis )
−uis
which is same as (eis , eir , eir ) uis · (uir · uir ) so that the ambiguity is resolvable. Hence
the irreducible monomials
1, ui1 , . . . , uin , uir · uis , (uir · uis ) · uit , . . . , (. . . ((ui1 · ui2 )·) . . .) · uin ,
where r < s < t, form a basis for On .
The presentation of Mn is exactly the same as On , except the generators of Mn commute. Let m1 , m2 , . . . , mn denote the generators of Mn . Then, as computed for On , the
monomials
1, mi1 , . . . , min , mir · mis , (mir · mis ) · mit , . . . , (. . . ((mi1 · mi2 )·) . . .) · min ,
form the basis for Mn , where r < s < t.
352
Mamta Balodi et al.
The algebra Op,q [8] is generated by n elements w1 , w2 , . . . , wn , where n = p + q,
such that
1
if 1 ≤ i ≤ p,
2
wi =
−1
if p + 1 ≤ i ≤ n,
wi · wj = −wj · wi
for all i = j . It is a (Z2 )n graded quasi-algebra. The 3-cocycle on (Z2 )n is same as
defined earlier. The set
{1, wi1 , . . . , win , wir · wis , (wir · wis ) · wit , . . . , (. . . ((wi1 · wi2 )·) . . .) · win }
form a basis for Op,q .
Acknowledgements
The first author sincerely thanks the Council of Scientific and Industrial Research (CSIR),
Government of India for research fellowship (JRF and SRF: 09/1032(0002)/2010-EMRI). The third author thanks Department of Science and Technology, Government of India
for research project grant (SR/S4/MS:642/10-1).
References
[1] Akrami S E and Majid S, Braided cyclic cocycles and non-associative geometry, J. Math.
Phys. 45 (2004) 38–83
[2] Albuquerque H and Majid S, Quasialgebra structure of the octonions, J. Algebra 220
(1999) 188–224
[3] Beggs E J and Majid S, Semiclassical differential structures, Pac. J. Math. 224 (2006)
1–44
[4] Bergman G M, The diamond lemma for ring theory, Adv. Math. 29 (1978) 178–218
[5] Huang H, Quiver approaches to quasi-Hopf algebras, J. Math. Phys. 50 (2009) 043501
9pp
[6] Huang H, From projective representations to quasi-quantum groups, Sci. China Math. 55
(2012) 2067–2080
[7] Kassel C, Quantum groups, Graduate Texts in Math. (1995) (New York: Springer-Verlag)
p. 155
[8] Morier-Genoud S and Ovsienko V, A series of algebras generalizing the octonions and
Hurwitz–Radon identity, Comm. Math. Phys. 306 (2011) 83–118
[9] Newman M H A, On theories with a combinatorial definition of ‘equivalence’, Ann.
Math. 43 (1942) 223–243
[10] Sweedler M E, Hopf Algebras (1969) (Benjamin: New York)
[11] Wills-Toro L A, Classification of some graded not necessarily associative division
algebras I, Comm. Algebra 42(12) (2014) 5019–5049
C OMMUNICATING E DITOR: B V Rajarama Bhat