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A. S. Dzhumadil'daev Institute of Mathematics Almaty, Kazakhstan askar56@hotmail.com

Commutative co cycles and algebras with antisymmetric identities
Algebras with one of the following identities are considered: [[t1 , t2 ], t3 ] + [[t2 , t3 ], t1 ] + [[t3 , t1 ], t2 ] = 0 (Lie-Admissible), [t1 , t2 ]t3 + [t2 , t3 ]t1 + [t3 , t1 ]t2 = 0 (0-Lie-Admissible, shortly 0-Alia), {[t1 , t2 ], t3 } + {[t2 , t3 ], t1 } + {[t3 , t1 ], t2 } = 0 (1-Lie-admissible, shortly 1-Alia), where [t1 , t2 ] = t1 t2 - t2 t1 and {t1 , t2 } = t1 t2 + t2 t1 . For an algebra A = (A, ) with multiplication denote by A(q) an algebra with vector space A and multiplication a q b = a b + q b a. Theorem 1. Any algebra with a skew-symmetric identity of degree 3 is (anti)-isomorphic to one of the fol lowing algebras: · Lie-admissible algebra · 0-Alia algebra · 1-Alia algebra · algebra of the form A
(q )

for some left-Alia algebra A and q K , such that q 2 = 0, 1.

Any right (left) Alia algebra is anti-isomorphic to its opposite algebra, left (right) Alia Algebra. For anti-commutative algebra (A, ) call a bilinear map : A в A A commutative cocycle, if (a b, c) + (b c, a) + (c a, b) = 0, (a, b) = (b, a), for any a, b, c A. Algebra with identities [a, b] c + [b, c] a + [c, a] b = 0 a [b, c] + b [c, a] + c [a, b] = 0 is called two-sided Alia. Theorem 2. For any anti-commutative algebra (A, ) with commutative cocycle an algebra (A, ), where a b = a b + (a, b), is 1-Alia. Conversely, any 1-Alia algebra is isomorphic to algebra of a form (A, ) for some anti-commutative algebra A and some commutative cocycle . Moreover, if (A, ) is Lie algebra with commutative cocycle , then (A, ) is two-sided Alia and, conversely, any two-sided Alia algebra is isomorphic to algebra of a form (A, ) for some Lie algebra A and commutative cocycle . Theorem 3. Let L be classical Lie algebra over a field of characteristic p = 2. Then it has non-trivial commutative cocycles only in the fol lowing cases L = sl2 or p = 3. Standard construction of q -Alia algebras. Let (U, ·) be associative commutative algebra with linear maps f , g : U U . Denote by Aq (U, ·, f , g ) an algebra defined on a vector space U by the rule a b = a · f (b) + g (a · b) - q f (a) · b. Then Aq (U, ·, f , g ) is q -Alia. Example. (C[x], ) under multiplication a b = (a) 2 (b) is 1-Alia and simple. Example. (C[x], ), where a b = 3 (a)b + 4 2 (a) (b) + 5 (a) 2 (b) + 2a 3 (b), is 0-Alia and simple. It is exceptional 0-Alia algebra. Example. Let (i,j ) be symmetric matrix. Then (C[x1 , . . . , xn ], ), where a b = i,j (i (a)j (b) + i j (a)b/2) is 0-Alia. It is simple iff the matrix (i,j ) is non-degenerate. 1


Example. Let m be positive integer and A = (C[x], ) an algebra with multiplication a b = a m (b) - q m (a)b + q m (ab) Then A is q -Alia and simple. Let sk be standard skew-symmetric polynomial, sk =
S y m
k

sig n t

(1)

· · · t

(k)

.

For a skew-symmetric polynomial f an anti-commutative algebra (A, ) is called f -Lie if it satisfies the identity f = 0. Call it minimal f -Lie if f = 0 is minimal identity that does not follow from anti-commutativity identity. For example, any Lie algebra is s4 -Lie. There exist interesting examples of simple minimal s4 -Lie algebras. Theorem 4. Let U be an associative commutative algebra with derivations D1 , D2 . Then (U, D1 D2 ) is s4 -Lie. This algebra is Lie if differential system {D1 , D2 } is in involution. Theorem 5. Let U be an associative commutative algebra with derivation D. Then (U, id D2 ) is s4 -Lie. Example. Algebra with base {ei , i -1} and multiplication ei ej = (i - j )(i + j + 3)e
i+j

is minimal s4 -Lie and simple. Theorem 6. Let A be sd -Lie, where d = 3 or d = 4. If f is a skew-symmetric polynomial of degree d, then A is f -Lie.

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