Download LIE-ADMISSIBLE ALGEBRAS AND THE VIRASORO

J. Korean Math. Soc. 33 (1996), No. 4, 231128 p. 1123
LIE-ADMISSIBLE ALGEBRAS AND THE VIRASORO ALGEBRA
HYO CHUL MYUNG∗
ABSTRACT. Third power-associative, Lie-admissible products on the Virasoro
algebra are determined in terms of linear functionals and bilinear forms.
1. Introduction
Let A be an (nonassociative) algebra with multiplication x y over a field
F, and denote by A− the algebra with multiplication [x, y] = x y − yx
defined on the vector space A. If A− is a Lie algebra, then A is called
Lie-admissible. Lie-admissible algebras arise in various topics, including
geometry of invariant affine connections on Lie groups and classical and
quantum mechanics(see [2,5,6,7] and references therein). The main approach
to the structure theory has been to determine all Lie-admissible products on a
prescribed Lie algebra under certain conditions. Flexibility or, more generally,
third power-associativity is commonly used in the study of Lie-admissible
algebras [1,2,4,5,7].
An algebra A is termed flexible if A satisfies the flexible law
(1)
(x y)x = x(yx)
for all x, y ∈ A. More generally, A is called third power-associative if
(2)
x2x = x x2
for all x ∈ A. Much of the structure theory for Lie-admissible algebras has
focused on finite-dimensional algebras [1,5]. The primary concern of this
note is to determine all third power-associative, Lie-admissible products on
the Virasoro algebra.
Received May 14, 1996. Revised July 15, 1996.
1991 AMS Subject Classification: 17B68, 17D25.
Key words: Lie-admissible algebra, Virasoro algebra.
∗ Partially supported by KOSEF Grant 96-K1-0101-01-01
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2. Main section
Let L be a Lie algebra with multiplication [x, y] over F of characteristic 6=
2. A multiplication x y defined on the vector space L is said to be compatible
with L if x y − yx = [x, y] for all x, y ∈ L; i.e., L − = L. If x y is compatible
with L, then
1
(3)
x y = [x, y] + x ◦ y
2
1
for x, y ∈ L, where x ◦ y = 2 (x y + yx). In particular, L with product x y
is Lie-admissible. A central problem in the study of Lie-admissible algebras
is to determine all compatible multiplications defined on Lie algebras. This
problem has been resolved for finite-dimensional third power-associative Lieadmissible algebras A with A− semisimple over an algebraically closed field
of characteristic 0 [1,5], whereas this is open for the infinite-dimensional
case. In this note, we consider this problem for the Virasoro algebra, which is
important in theoretical physics and provides a basic example of graded Lie
algebras [3].
Let W be the (full) Witt algebra over F with multiplication
(4)
[ei , ej ] = ( j − i )ei+ j , i, j ∈ Z,
where {ei : i ∈ Z} is a basis of W. The Virasoro algebra V over F(of
characteristic 0) is a one-dimensional extension V = W ⊕ Fc of W with
multiplication
1
[ei , ej ] = ( j − i )ei+ j + (i 3 − i )δi+ j,0 c,
(5)
12
[c, V] = [V, c] = 0, i, j ∈ Z.
The following is our principle result.
THEOREM. A multiplication x y defined on V is third power-associa tive
and compatible with V if and only if it is given by
1
x y = [x, y] + τ (x)y + τ (y)x + σ (x, y)c,
2
(6)
cx = xc = αx + λ(x)c, c2 = βc, x, y ∈ W,
where λ, τ : W → F are linear functionals, σ : W × W → F is a symmetric
bilinear form and α, β ∈ F are fixed scalars. Moreover, the multiplication
(6) is flexible if and only if λ = τ = 0, α = 0 and σ is identically zero on
W × W.
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Proof. Assume that x y is third power-associative and compatible with V.
In view of (3), it suffices to determine the commutative product x ◦ y on V.
Since x y − yx = [x, y], (2) expresses as [x, x ◦ x] = 0 for x ∈ V which is
linearized to the identity
2[x, x ◦ y] + [y, x ◦ x] = 0, x, y ∈ V.
(7)
1 (i 3 − i )δ
If φ(ei , ej ) = 12
i+ j,0 for i, j ∈ Z, then φ is an F-valued 2-cocycle
of W. Since the product x ◦ y is commutative, we can let
ei ◦ e j =
(8)
X
γikj ek + σ (ei , ej )c, i, j ∈ Z
k∈Z
with γikj = γ jki ∈ F for a symmetric bilinear form σ : W × W → F.
Letting x = y = ei in (7), one has from (5) and (8)
0 = [ei , ei ◦ ei ] =
X
γiik (k − i )ei+k +
k
X
γiik φ(ei , ek )c,
k
and hence
γiik = 0, i 6= k, i, k ∈ Z.
If x = ei and y = ej in (7), then, as above,
2
X
γikj [ei , ek ] = γiii [ei , ej ],
k
2
X
γikj (k − i )ei+k + 2
k
X
γikj φ(ei , ek )c = γiii ( j − i )ei+ j + γiii φ(ei , ej )c,
k
which imply
γikj = 0, i 6= k 6= j,
j
j
2γi j = γiii = 2γj i , i 6= j
for i, j, k ∈ Z. Therefore, (8) becomes
ei ◦ ej =
1 i
1 j
γii ej + γ j j ei + σ (ei , ej )c,
2
2
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Hyo Chul Myung
and the first relation of (6) follows from this with the linear functional τ :
W → F defined by τ (ei ) = 12 γiii (i ∈ Z).
Next, we linearize (7) to the identity
(9)
[u, v ◦ w] + [w, u ◦ v] + [v, w ◦ u] = 0
for u, v, w ∈ V. Since 2[ei , c ◦ ei ] = [ei ◦ ei , c] = 0, there exist linear
functionals µ, λ : W → F such that
cei = ei c = c ◦ ei = ei ◦ c = µ(ei )ei + λ(ei )c.
If u = ej , v = c and w = ei in (9), then it follows that
µ(ei − ej )[(i − j )ei+ j + φ(ei , ej )c] = 0
and hence µ(ei ) = µ(ej ) for i 6= j . Since [ei , c ◦ c] = 0 by (7) for all i ∈ Z,
cc = c ◦ c = βc for some β. Letting µ(ei ) = α ∈ F for all i ∈ Z, we
have the last relations of (6). Since third power-associativity is equivalent
to (9), it is easy to see that any multiplication on V given by (6) is third
power-associative.
Notice that flexibility (1) is equivalent to the identity
(10)
u ◦ [v, u] = [u ◦ v, u], u, v ∈ V.
Assume that V is flexible under the product in (6). If u = x and v = y in
(10), then
τ (x)[y, x] + τ ([y, x])x + σ (x, [y, x])c = τ (x)[y, x]
and hence
(11)
τ ([y, x]) = σ (x, [y, x]) = 0
for all x, y ∈ W. Since [W, W] = W, τ = 0 and the linearization of
σ (x, [y, x]) = 0 gives the invariance of σ :
(12)
σ ([x, y], z) = σ (x, [y, z]), x, y, z ∈ W.
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We now show that σ is identically zero on W × W. If i 6= 0, then by (4)
and (12)
σ ([ei , e−i ], e0 ) = −2i σ (e0 , e0 )
= σ (ei , [e−i , e0 ]) = i σ (ei , e−i )
and hence σ (ei , e−i ) = −2σ (e0 , e0 ). For a j 6= 0 with i + j 6= 0, one has by
(4) and (12)
σ ([ei , ej ], e−(i+ j ) ) = −2( j − i )σ (e0 , e0 )
= σ (ei , [ej , e−(i+ j ) ]) = −(i + 2 j )σ (ei , e−i )
= 2(i + 2 j )σ (e0 , e0 ),
which implies σ (e0 , e0 ) = 0 and so, by the above σ (ei , e−i ) = −2σ (e0 ,
e0 ) = 0 for all i ∈ Z. Assume i + j 6= 0. By (4) and (12), σ ([e0 , ei ], ej )
= i σ (ei , ej ) = −σ ([ei , e0 ], ej ) = −σ (ei , [e0 , ej ]) = − j σ (ei , ej ), and thus
σ (ei , ej ) = 0. This shows that σ (ei , ej ) = 0 for all i, j ∈ Z.
The linearization of (10) implies the identity
(13)
[u, v ◦ w] = v ◦ [u, w] + [u, v] ◦ w
for all u, v, w ∈ V. To verify λ = 0 and α = 0, for any i ∈ Z, choose j ∈ Z
1
such that i + j 6= 0 and j 6= 0, ±1, so 12
( j 3 − j ) 6= 0. If u = ej , v = ei and
w = e− j in (13), then
0 = [ej , ei ◦ e− j ] = ei ◦ [ej , e− j ] + [ej , ei ] ◦ e− j
1 3
1 3
( j − j )ei ◦ c =
( j − j )(αei + λ(ei )c),
=
12
12
and hence λ(ei ) = α = 0 for all i ∈ Z.
Conversely, it is easy to see that the multiplication (6) with σ = λ = τ = 0
and α = 0 satisfies (10) and thus is flexible. As a corollary to the proof of the Theorem, we have :
COROLLARY. Let W be the Witt algebra given by (4) over a field F of
characteristic 0. A multiplication x y defined on W is third power-associative
and compatible with W if and only if it is given by
1
(14)
x y = [x, y] + τ (x)y + τ (y)x, x, y ∈ W
2
for a linear functional τ : W → F . The multiplication (14) is flexible if and
only if τ = 0, i.e., x y = 12 [x, y] for all x, y ∈ W.
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Hyo Chul Myung
Proof. This follows from the same calculation as in the proof of the Theorem. Multiplications similar to (6) and (14) have been obtained for matrix
algebras, octonion algebras and finite-dimensional simple Lie algebras over
an algebraically closed field of characteristic 0 [1,4,5]. It is reasonable to pose
the conjecture that the Theorem holds for any affine Kac-Moody algebra.
It was brought to our attention by Georgia Benkart that λ = 0 and σ = 0
for the flexible case in the Theorem.
References
1. G. M. Benkart, Power-associative Lie-admissible algebras, J. Algebra 90 (1984), 37-58.
2. A. Elduque and H. C. Myung, Mutations of Alternative Algebras, Math. Appl. Vol. 278,
Kluwer Acad. Publ., Boston, 1994.
3. V. Kac, Infinite Dimensional Lie Algebras, 3rd ed., Cambridge Univ. Press, New York, 1990.
4. H. C. Myung, Power-associative products on octonions, Comm. Algebra 13 (1985), 16811697.
5. H. C. Myung, Malcev-Admissible Algebras, Prog. Math. Vol. 64, Birkhäuser, Boston, 1986.
6. H. C. Myung, Non-Unital Composition Algebras, Lecture Notes Ser. no. 22, Global Analysis
Research Center, Seoul, 1994.
7. S. Okubo, Introduction to Octonion and Other Non-Associative Algebras in Physics, Cambridge Univ. Press, New York, 1995.
Department of Mathematics
KAIST
Taejon 305-701, Korea
E-mail: [email protected]