The prime spectrum of algebras of quadratic growth

We study prime algebras of quadratic growth. Our first result is that if $A$ is a prime monomial algebra of quadratic growth then $A$ has finitely many prime ideals $P$ such that $A/P$ has GK dimension one. This shows that prime monomial algebras of quadratic growth have bounded matrix images. We next show that a prime graded algebra of quadratic growth has the property that the intersection of the nonzero prime ideals $P$ such that $A/P$ has GK dimension 2 is non-empty, provided there is at least one such ideal. From this we conclude that a prime monomial algebra of quadratic growth is either primitive or has nonzero locally nilpotent Jacobson radical. Finally, we show that there exists a prime monomial algebra $A$ of GK dimension two with unbounded matrix images and thus the quadratic growth hypothesis is necessary to conclude that there are only finitely many prime ideals such that $A/P$ has GK dimension 1.Comment: 23 page

On the Lattice of Strong Radicals

AbstractIt is shown that the class of all strong radicals containing the prime radical is not a sublattice of the lattice of all radicals. This gives a negative answer to some questions of Sands and Puczylowski

Group gradings on finitary simple Lie algebras

We classify, up to isomorphism, all gradings by an arbitrary abelian group on simple finitary Lie algebras of linear transformations (special linear, orthogonal and symplectic) on infinite-dimensional vector spaces over an algebraically closed field of characteristic different from 2.Comment: Several typographical errors have been correcte

On Herstein's Lie Map Conjectures, II

AbstractThe theory of functional identities is used to study derivations of Lie algebras arising from associative algebras. Definitive results are obtained modulo algebras of “low dimension.” In particular, Lie derivations of [K,K]/([K,K]∩Z), where K is the Lie algebra of skew elements of a prime algebra with involution and Z is its center, are described. This solves the last remaining open problem of Herstein on Lie derivations. For a simple algebra with involution the Lie algebra of all derivations of [K,K]/([K,K]∩Z) is thoroughly analyzed. Maps that act as derivations on arbitrary fixed polynomials are also discussed, and in particular a solution is given for Herstein's question concerning maps of K which act like a derivation on xm, m being a fixed odd integer

On graded polynomial identities with an antiautomorphism

AbstractLet G be a commutative monoid with cancellation and let R be a strongly G-graded associative algebra with finite G-grading and with antiautomorphism. Suppose that R satisfies a graded polynomial identity with antiautomorphism. We show that R is a PI algebra

Classification of group gradings on simple Lie algebras of types A, B, C and D

For a given abelian group G, we classify the isomorphism classes of G-gradings on the simple Lie algebras of types A_n (n >= 1), B_n (n >= 2), C_n (n >= 3) and D_n (n > 4), in terms of numerical and group-theoretical invariants. The ground field is assumed to be algebraically closed of characteristic different from 2.Comment: 20 pages, no figure

Golod-Shafarevich algebras, free subalgebras and Noetherian images

It is shown that Golod-Shaferevich algebras of a reduced number of defining relations contain noncommutative free subalgebras in two generators, and that these algebras can be homomorphically mapped onto prime, Noetherian algebras with linear growth. It is also shown that Golod-Shafarevich algebras of a reduced number of relations cannot be nil

Inner Ideals of Simple Locally Finite Lie Algebras

Inner ideals of simple locally finite dimensional Lie algebras over an algebraically closed field of characteristic 0 are described. In particular, it is shown that a simple locally finite dimensional Lie algebra has a non-zero proper inner ideal if and only if it is of diagonal type. Regular inner ideals of diagonal type Lie algebras are characterized in terms of left and right ideals of the enveloping algebra. Regular inner ideals of finitary simple Lie algebras are described

Lie Isomorphisms in Prime Rings with Involution

AbstractLet R and R′ be prime rings with involutions of the first kind and with respective Lie subrings of skew elements K and K′. Furthermore assume (RC : C) ≠ 1, 4, 9, 16, 25, 64, where C is the extended centroid of R. It is shown that any Lie isomorphism of K onto K′ can be extended uniquely to an associative isomorphism of 〈K〉 onto 〈K′〉, where 〈K〉 and 〈K′〉 are respectively the associative subrings generated by K and K′