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Papers by Richard Block

Research paper thumbnail of Differentiably simple algebras

Bulletin of the American Mathematical Society, 1968

Research paper thumbnail of Generalized dual coalgebras of algebras, with applications to cofree coalgebras

Journal of Pure and Applied Algebra, 1985

Research paper thumbnail of Determination of the irreducible divided power Hopf algebras

Research paper thumbnail of Commutative Hopf algebras, lie coalgebras, and divided powers

Research paper thumbnail of On Torsion-Free Abelian Groups and

It is known that many of the classes of simple Lie algebras of prime characteristic of nonclassic... more It is known that many of the classes of simple Lie algebras of prime characteristic of nonclassical type have simple infinite-dimensional analogues of characteristic zero (see, for example, [4, p. 518]). We consider here analogues of those algebras which are defined by a modification of the definition of a group algebra. Thus we consider analogues of the Zassenhaus algebras as generalized by Albert and Frank in [l]. The algebras considered are defined as follows. Let G be a nonzero abelian group, F a field, g an additive mapping from G to F, and fan alternate biadditive mapping from GXG to F. We index a basis of an algebra over F by G, denoting by ua the basis element corresponding to a in G, and define multiplication by

Research paper thumbnail of The lie algebras with a quotient trace form

Illinois Journal of Mathematics, 1965

Research paper thumbnail of The Lie algebras with a nondegenerate trace form

Illinois Journal of Mathematics, 1964

Research paper thumbnail of Modules over differential polynomial rings

Bulletin of the American Mathematical Society, 1973

Research paper thumbnail of On Torsion-Free Abelian Groups and Lie Algebras

Proceedings of the American Mathematical Society, 1958

Research paper thumbnail of Irreducible representations of Lie algebra extensions

Bulletin of the American Mathematical Society, 1974

Research paper thumbnail of The irreducible representations of the Weyl algebra A1

Lecture Notes in Mathematics, 1979

Research paper thumbnail of On Lie algebras of rank one

Transactions of the American Mathematical Society, 1964

Research paper thumbnail of On the Mills-Seligman axioms for Lie algebras of classical type

Transactions of the American Mathematical Society, 1966

Research paper thumbnail of New Simple Lie Algebras of Prime Characteristic

Transactions of the American Mathematical Society, 1958

Research paper thumbnail of Noncommutative Jordan Algebras with Commutators Satisfying an Alternativity Condition

Proceedings of the National Academy of Sciences, 1970

The theorems of this paper show that the main results in the structure and representation theory ... more The theorems of this paper show that the main results in the structure and representation theory of Jordan algebras and of alternative algebras are valid for a larger class of algebras defined by simple identities which obviously hold in the Jordan and alternative cass. A new unification of the Jordan and associative theories is also achieved.

Research paper thumbnail of The restricted simple Lie algebras are of classical or Cartan type

Proceedings of the National Academy of Sciences, 1984

The classification of the finite-dimensional restricted simple Lie algebras over an algebraically... more The classification of the finite-dimensional restricted simple Lie algebras over an algebraically closed field F of prime characteristic p > 7 is announced. Such an algebra is either of classical type (an analogue over F of a finite-dimensional simple Lie algebra over the complex numbers) or of Cartan type (an analogue over F of one of the infinite Lie algebras of Cartan over the complex numbers).

Research paper thumbnail of On nilpotency in the Ado-Harish-Chandra theorem on Lie algebra representations

Proceedings of the American Mathematical Society, 1986

Let L L be a finite-dimensional Lie algebra, over an arbitrary field, and regard L L as embedded ... more Let L L be a finite-dimensional Lie algebra, over an arbitrary field, and regard L L as embedded in its enveloping algebra UL. Theorem. If K K is an ideal of L L and K K is nilpotent of class q q , then for any r r there exists a finite-dimensional representation ρ \rho of L L which vanishes on all products (in UL) of ≥ q r + 1 \geq qr + 1 elements of K K and is faithful on the subspace of UL spanned by all products of ≤ r \leq r elements of L L . This result sharpens (with respect to nilpotency) the Ado-Iwasawa theorem on the existence of faithful representations and the Harish-Chandra theorem on the existence of representations separating points of UL.

Research paper thumbnail of Transitive groups of collineations on certain designs

Pacific Journal of Mathematics, 1965

Research paper thumbnail of On the orbits of collineation groups

Mathematische Zeitschrift, 1971

Research paper thumbnail of On automorphism groups of block designs

Journal of Combinatorial Theory, 1968

Research paper thumbnail of Differentiably simple algebras

Bulletin of the American Mathematical Society, 1968

Research paper thumbnail of Generalized dual coalgebras of algebras, with applications to cofree coalgebras

Journal of Pure and Applied Algebra, 1985

Research paper thumbnail of Determination of the irreducible divided power Hopf algebras

Research paper thumbnail of Commutative Hopf algebras, lie coalgebras, and divided powers

Research paper thumbnail of On Torsion-Free Abelian Groups and

It is known that many of the classes of simple Lie algebras of prime characteristic of nonclassic... more It is known that many of the classes of simple Lie algebras of prime characteristic of nonclassical type have simple infinite-dimensional analogues of characteristic zero (see, for example, [4, p. 518]). We consider here analogues of those algebras which are defined by a modification of the definition of a group algebra. Thus we consider analogues of the Zassenhaus algebras as generalized by Albert and Frank in [l]. The algebras considered are defined as follows. Let G be a nonzero abelian group, F a field, g an additive mapping from G to F, and fan alternate biadditive mapping from GXG to F. We index a basis of an algebra over F by G, denoting by ua the basis element corresponding to a in G, and define multiplication by

Research paper thumbnail of The lie algebras with a quotient trace form

Illinois Journal of Mathematics, 1965

Research paper thumbnail of The Lie algebras with a nondegenerate trace form

Illinois Journal of Mathematics, 1964

Research paper thumbnail of Modules over differential polynomial rings

Bulletin of the American Mathematical Society, 1973

Research paper thumbnail of On Torsion-Free Abelian Groups and Lie Algebras

Proceedings of the American Mathematical Society, 1958

Research paper thumbnail of Irreducible representations of Lie algebra extensions

Bulletin of the American Mathematical Society, 1974

Research paper thumbnail of The irreducible representations of the Weyl algebra A1

Lecture Notes in Mathematics, 1979

Research paper thumbnail of On Lie algebras of rank one

Transactions of the American Mathematical Society, 1964

Research paper thumbnail of On the Mills-Seligman axioms for Lie algebras of classical type

Transactions of the American Mathematical Society, 1966

Research paper thumbnail of New Simple Lie Algebras of Prime Characteristic

Transactions of the American Mathematical Society, 1958

Research paper thumbnail of Noncommutative Jordan Algebras with Commutators Satisfying an Alternativity Condition

Proceedings of the National Academy of Sciences, 1970

The theorems of this paper show that the main results in the structure and representation theory ... more The theorems of this paper show that the main results in the structure and representation theory of Jordan algebras and of alternative algebras are valid for a larger class of algebras defined by simple identities which obviously hold in the Jordan and alternative cass. A new unification of the Jordan and associative theories is also achieved.

Research paper thumbnail of The restricted simple Lie algebras are of classical or Cartan type

Proceedings of the National Academy of Sciences, 1984

The classification of the finite-dimensional restricted simple Lie algebras over an algebraically... more The classification of the finite-dimensional restricted simple Lie algebras over an algebraically closed field F of prime characteristic p > 7 is announced. Such an algebra is either of classical type (an analogue over F of a finite-dimensional simple Lie algebra over the complex numbers) or of Cartan type (an analogue over F of one of the infinite Lie algebras of Cartan over the complex numbers).

Research paper thumbnail of On nilpotency in the Ado-Harish-Chandra theorem on Lie algebra representations

Proceedings of the American Mathematical Society, 1986

Let L L be a finite-dimensional Lie algebra, over an arbitrary field, and regard L L as embedded ... more Let L L be a finite-dimensional Lie algebra, over an arbitrary field, and regard L L as embedded in its enveloping algebra UL. Theorem. If K K is an ideal of L L and K K is nilpotent of class q q , then for any r r there exists a finite-dimensional representation ρ \rho of L L which vanishes on all products (in UL) of ≥ q r + 1 \geq qr + 1 elements of K K and is faithful on the subspace of UL spanned by all products of ≤ r \leq r elements of L L . This result sharpens (with respect to nilpotency) the Ado-Iwasawa theorem on the existence of faithful representations and the Harish-Chandra theorem on the existence of representations separating points of UL.

Research paper thumbnail of Transitive groups of collineations on certain designs

Pacific Journal of Mathematics, 1965

Research paper thumbnail of On the orbits of collineation groups

Mathematische Zeitschrift, 1971

Research paper thumbnail of On automorphism groups of block designs

Journal of Combinatorial Theory, 1968

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