In this paper we investigate the Lie structure of the derived Lie superalgebra K,K, with K the set of skew elements of a semiprime associative superalgebra A with superinvolution. We show that if U ...is a Lie ideal of K,K, then either there exists an ideal J of A such that the Lie ideal J∩K,K is nonzero and contained in U, or A is a subdirect sum of A′, A″, where the image of U in A′ is central, and A″ is a subdirect product of orders in simple superalgebras, each at most 16-dimensional over its center.
We investigate the ε-Lie structure of K and K,K; here K denotes the skew-symmetric elements of an (ε,G)-Lie color algebra (obtained from an associated algebra A) with an ε-involution. The ...relationship with the (associative) ideals of A is also explored.
In this paper some results on the Lie structure of prime superalgebras are discussed. We prove that, with the exception of some special cases, for a prime superalgebra A over a ring of scalars Φ with ...1/2∈Φ, if L is a Lie ideal of A and W is a subalgebra of A such that W,L⊆W, then either L⊆Z or W⊆Z. Likewise, if V is a submodule of A and V,L⊆V, then either V⊆Z or L⊆Z or there exists an ideal of A,M, such that 0≠M,A⊆V. This work extends to prime superalgebras some results of I.N. Herstein, C. Lanski and S. Montgomery on prime algebras.
We study semiprime superalgebras with superinvolution whose symmetric elements are not zero divisors, and semiprime superalgebras with superinvolution, with nonzero odd part, whose skewsymmetric ...elements are not zero divisors. We prove that, in both cases, such superalgebras are a domain or the subdirect sum of a domain and its opposite.
We study semiprime superalgebras with superinvolution, under certain additional regularity conditions. More precisely, we assume regularity conditions either on every nonzero homogeneous symmetric ...element, or on every nonzero homogeneous skewsymmetric element.
Maximal subalgebras of Jordan superalgebras Elduque, Alberto; Laliena, Jesús; Sacristán, Sara
Journal of pure and applied algebra,
11/2008, Volume:
212, Issue:
11
Journal Article
Peer reviewed
Open access
The maximal subalgebras of the finite-dimensional simple special Jordan superalgebras over an algebraically closed field of characteristic 0 are studied. This is a continuation of a previous paper by ...the same authors about maximal subalgebras of simple associative superalgebras, which is instrumental here.
In this note we emphasise the relationship between the structure of an associative superalgebra with superinvolution and the structure of the Lie substructure of skewsymmetric elements. More ...explicitly, we show that if A is a semiprime associative superalgebra with superinvolution and K is the Lie superalgebra of skewsymmetric elements satisfying K
2
, K
2
= 0, then A is a subdirect product of orders in simple superalgebras each at most 4-dimensional over its center.
In this paper we investigate the Lie structure of the Lie superalgebra
K of skew elements of a semiprime associative superalgebra
A with superinvolution. We show that if
U is a Lie ideal of
K, then ...either there exists an ideal
J of
A such that the Lie ideal
J
∩
K
,
K
is nonzero and contained in
U, or
A is a subdirect sum of
A
′
,
A
″
, where the image of
U in
A
′
is central, and
A
″
is a subdirect product of orders in simple superalgebras, each at most 16-dimensional over its center.
The Jacobson Coordinatization Theorem describes the structure of unitary Jordan algebras containing the algebra \(H_n(F)\) of symmetric nxn matrices over a field F with the same identity element, for ...\(n\geq 3\). In this paper we extend the Jacobson Coordinatization Theorem for n=2. Specifically, we prove that if J is a unitary Jordan algebra containing the Jordan matrix algebra \(H_2(F)\) with the same identity element, then J has a form \(J=H_2(F)\otimes A_0+k\otimes A_1\), where \(A=A_0+A_1\) is a \(Z_2\)-graded Jordan algebra with a partial odd Leibniz bracket {,} an \(k=e_{12}-e_{21}\in M_2(F)\) with the multiplication given by \((a\otimes b)(c\otimes d)=ac\otimes bd + a,c\otimes \{b,d\},\) the commutator a,c is taken in \(M_2(F)\).
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