Let * be the natural involution on a group algebra FG induced by setting 9-+ g-1, for all 9 E G. ... more Let * be the natural involution on a group algebra FG induced by setting 9-+ g-1, for all 9 E G. Here we survey on the results recently obtained on the Lie nilpotence of the symmetric and skew elements of FG under *.
Proceedings of the American Mathematical Society, 1997
Let F G FG be the group algebra of a torsion group over an infinite field F F . Let U U be the gr... more Let F G FG be the group algebra of a torsion group over an infinite field F F . Let U U be the group of units of F G FG . We prove that if U U satisfies a group identity, then F G FG satisfies a polynomial identity. This confirms a conjecture of Brian Hartley.
Proceedings of the American Mathematical Society, 2021
Star-fundamental algebras are special finite dimensional algebras with involution ∗ * over an alg... more Star-fundamental algebras are special finite dimensional algebras with involution ∗ * over an algebraically closed field of characteristic zero defined in terms of multialternating ∗ * -polynomials. We prove that the upper-block matrix algebras with involution introduced in Di Vincenzo and La Scala [J. Algebra 317 (2007), pp. 642–657] are star-fundamental. Moreover, any finite dimensional algebra with involution contains a subalgebra mapping homomorphically onto one of such algebras. We also give a characterization of star-fundamental algebras through the representation theory of the symmetric group.
Let G be a finite group and A a G-graded algebra over a field F of characteristic zero. We charac... more Let G be a finite group and A a G-graded algebra over a field F of characteristic zero. We characterize the T G-ideals Id G (A) of graded identities of A such that the multiplicities m λ in the graded cocharacter of A are bounded by one. We do so by exhibiting a set of identities of the T G-ideal. As a consequence we characterize the varieties of G-graded algebras whose lattice of subvarieties is distributive.
Transactions of the American Mathematical Society, 2009
Let R be a special simple Jordan algebra over a field of characteristic zero. We exhibit a noncom... more Let R be a special simple Jordan algebra over a field of characteristic zero. We exhibit a noncommutative Jordan polynomial f multialternating on disjoint sets of variables which is not a polynomial identity of R. We then study the growth of the polynomial identities of the Jordan algebra R through an analysis of its sequence of Jordan codimensions. By exploiting the basic properties of the polynomial f , we are able to compute the exponential rate of growth of the sequence of Jordan codimensions of R and prove that it equals the dimension of the Jordan algebra over its center. We also show that for any finite dimensional special Jordan algebra, such an exponential rate of growth cannot be strictly between 1 and 2.
... of index 2. Proof. Suppose that gh = hg and assume first that hgh −1 g = ghgh −1 . Since by L... more ... of index 2. Proof. Suppose that gh = hg and assume first that hgh −1 g = ghgh −1 . Since by Lemma 4, gh −1 = hg −1 and hg = g −1 h −1 , we obtain h2 = hhg −1 g = hgh −1 g = ghgh −1 = gg −1 h −1 h −1 = h −2 , and h4 = 1 ...
Let R be an algebra with involution over a field F of characteristic different from 2 and let be ... more Let R be an algebra with involution over a field F of characteristic different from 2 and let be a multilinear -polynomial over F which, when evaluated in R, takes nilpotent values. We prove that if R has no nil right ideals and , then f is a -polynomial identity for R.
Let G be a finite group and A a G-graded algebra over a field of characteristic zero. When A is a... more Let G be a finite group and A a G-graded algebra over a field of characteristic zero. When A is a PI-algebra, the graded codimensions of A are exponentially bounded and one can study the corresponding graded cocharacters via the representation theory of products of symmetric groups. Here we characterize in two different ways when the corresponding multiplicities are bounded by a constant.
The Lie algebra sl 2 = sl 2 (K) of 2 × 2 traceless matrices over a field K has only three nontriv... more The Lie algebra sl 2 = sl 2 (K) of 2 × 2 traceless matrices over a field K has only three nontrivial G-gradings when G is a group, the ones induced by G = Z 2 , Z 2 × Z 2 and Z. Here we prove that when char(K) = 0, the variety var G (sl 2) of G-graded Lie algebras generated by sl 2 , is a minimal variety of exponential growth, and in case G = Z 2 × Z 2 or Z, var G (sl 2) has almost polynomial growth.
Let * be an involution of a group G extended linearly to the group algebra K G. We prove that if ... more Let * be an involution of a group G extended linearly to the group algebra K G. We prove that if G contains no 2-elements and K is a field of characteristic p = 2, then the *-symmetric elements of K G are Lie nilpotent (Lie n-Engel) if and only if K G is Lie nilpotent (Lie n-Engel).
Let F be an infinite field of characteristic different from 2, G a group and * an involution of G... more Let F be an infinite field of characteristic different from 2, G a group and * an involution of G extended by linearity to an involution of the group algebra F G. Here we completely characterize the torsion groups G for which the *-symmetric units of F G satisfy a group identity. When * is the classical involution induced from g → g −1 , g ∈ G, this result was obtained in [A. Giambruno, S.K. Sehgal, A. Valenti, Symmetric units and group identities, Manuscripta Math. 96 (1998) 443-461].
Let U be the group of units of the group algebra FG of a group G over a field F. Suppose that eit... more Let U be the group of units of the group algebra FG of a group G over a field F. Suppose that either F is infinite or G has an element of infinite order. We characterize groups G so that U satisfies a group identity. Under the assumption that G modulo the torsion elements is nilpotent this gives a complete classification of such groups. For torsion groups this problem has already been settled in recent years. @ 2000AcademicPress .
Let * be an involution of a group algebra F G induced by an involution of the group G. For char F... more Let * be an involution of a group algebra F G induced by an involution of the group G. For char F = 2, we classify the groups G with no 2-elements and with no nonabelian dihedral groups involved whose Lie algebra of *-skew elements is nilpotent.
International Journal of Algebra and Computation, 1998
Let ℚAn be the group algebra of the alternating group over the rationals. By exploiting the theor... more Let ℚAn be the group algebra of the alternating group over the rationals. By exploiting the theory of Young tableaux, we give an explicit description of the minimal central idempotents of ℚAn. As an application we construct finitely many generators for a subgroup of finite index in the centre of the group of units of ℚAn.
Page 1. COMMUNICATIONS IN ALGEBRA, 22(5), 1685-1701 (1994) RINGS WITH ALGEBRAIC n-ENGEL ELEMENTS ... more Page 1. COMMUNICATIONS IN ALGEBRA, 22(5), 1685-1701 (1994) RINGS WITH ALGEBRAIC n-ENGEL ELEMENTS Departamento di Maternatica ed Applicazioni Universita di Palcrlno via Archirafi 34 90123 - Palesrno - Italia giaml~runo~ipal~~at.cles.it ...
Let * be the natural involution on a group algebra FG induced by setting 9-+ g-1, for all 9 E G. ... more Let * be the natural involution on a group algebra FG induced by setting 9-+ g-1, for all 9 E G. Here we survey on the results recently obtained on the Lie nilpotence of the symmetric and skew elements of FG under *.
Proceedings of the American Mathematical Society, 1997
Let F G FG be the group algebra of a torsion group over an infinite field F F . Let U U be the gr... more Let F G FG be the group algebra of a torsion group over an infinite field F F . Let U U be the group of units of F G FG . We prove that if U U satisfies a group identity, then F G FG satisfies a polynomial identity. This confirms a conjecture of Brian Hartley.
Proceedings of the American Mathematical Society, 2021
Star-fundamental algebras are special finite dimensional algebras with involution ∗ * over an alg... more Star-fundamental algebras are special finite dimensional algebras with involution ∗ * over an algebraically closed field of characteristic zero defined in terms of multialternating ∗ * -polynomials. We prove that the upper-block matrix algebras with involution introduced in Di Vincenzo and La Scala [J. Algebra 317 (2007), pp. 642–657] are star-fundamental. Moreover, any finite dimensional algebra with involution contains a subalgebra mapping homomorphically onto one of such algebras. We also give a characterization of star-fundamental algebras through the representation theory of the symmetric group.
Let G be a finite group and A a G-graded algebra over a field F of characteristic zero. We charac... more Let G be a finite group and A a G-graded algebra over a field F of characteristic zero. We characterize the T G-ideals Id G (A) of graded identities of A such that the multiplicities m λ in the graded cocharacter of A are bounded by one. We do so by exhibiting a set of identities of the T G-ideal. As a consequence we characterize the varieties of G-graded algebras whose lattice of subvarieties is distributive.
Transactions of the American Mathematical Society, 2009
Let R be a special simple Jordan algebra over a field of characteristic zero. We exhibit a noncom... more Let R be a special simple Jordan algebra over a field of characteristic zero. We exhibit a noncommutative Jordan polynomial f multialternating on disjoint sets of variables which is not a polynomial identity of R. We then study the growth of the polynomial identities of the Jordan algebra R through an analysis of its sequence of Jordan codimensions. By exploiting the basic properties of the polynomial f , we are able to compute the exponential rate of growth of the sequence of Jordan codimensions of R and prove that it equals the dimension of the Jordan algebra over its center. We also show that for any finite dimensional special Jordan algebra, such an exponential rate of growth cannot be strictly between 1 and 2.
... of index 2. Proof. Suppose that gh = hg and assume first that hgh −1 g = ghgh −1 . Since by L... more ... of index 2. Proof. Suppose that gh = hg and assume first that hgh −1 g = ghgh −1 . Since by Lemma 4, gh −1 = hg −1 and hg = g −1 h −1 , we obtain h2 = hhg −1 g = hgh −1 g = ghgh −1 = gg −1 h −1 h −1 = h −2 , and h4 = 1 ...
Let R be an algebra with involution over a field F of characteristic different from 2 and let be ... more Let R be an algebra with involution over a field F of characteristic different from 2 and let be a multilinear -polynomial over F which, when evaluated in R, takes nilpotent values. We prove that if R has no nil right ideals and , then f is a -polynomial identity for R.
Let G be a finite group and A a G-graded algebra over a field of characteristic zero. When A is a... more Let G be a finite group and A a G-graded algebra over a field of characteristic zero. When A is a PI-algebra, the graded codimensions of A are exponentially bounded and one can study the corresponding graded cocharacters via the representation theory of products of symmetric groups. Here we characterize in two different ways when the corresponding multiplicities are bounded by a constant.
The Lie algebra sl 2 = sl 2 (K) of 2 × 2 traceless matrices over a field K has only three nontriv... more The Lie algebra sl 2 = sl 2 (K) of 2 × 2 traceless matrices over a field K has only three nontrivial G-gradings when G is a group, the ones induced by G = Z 2 , Z 2 × Z 2 and Z. Here we prove that when char(K) = 0, the variety var G (sl 2) of G-graded Lie algebras generated by sl 2 , is a minimal variety of exponential growth, and in case G = Z 2 × Z 2 or Z, var G (sl 2) has almost polynomial growth.
Let * be an involution of a group G extended linearly to the group algebra K G. We prove that if ... more Let * be an involution of a group G extended linearly to the group algebra K G. We prove that if G contains no 2-elements and K is a field of characteristic p = 2, then the *-symmetric elements of K G are Lie nilpotent (Lie n-Engel) if and only if K G is Lie nilpotent (Lie n-Engel).
Let F be an infinite field of characteristic different from 2, G a group and * an involution of G... more Let F be an infinite field of characteristic different from 2, G a group and * an involution of G extended by linearity to an involution of the group algebra F G. Here we completely characterize the torsion groups G for which the *-symmetric units of F G satisfy a group identity. When * is the classical involution induced from g → g −1 , g ∈ G, this result was obtained in [A. Giambruno, S.K. Sehgal, A. Valenti, Symmetric units and group identities, Manuscripta Math. 96 (1998) 443-461].
Let U be the group of units of the group algebra FG of a group G over a field F. Suppose that eit... more Let U be the group of units of the group algebra FG of a group G over a field F. Suppose that either F is infinite or G has an element of infinite order. We characterize groups G so that U satisfies a group identity. Under the assumption that G modulo the torsion elements is nilpotent this gives a complete classification of such groups. For torsion groups this problem has already been settled in recent years. @ 2000AcademicPress .
Let * be an involution of a group algebra F G induced by an involution of the group G. For char F... more Let * be an involution of a group algebra F G induced by an involution of the group G. For char F = 2, we classify the groups G with no 2-elements and with no nonabelian dihedral groups involved whose Lie algebra of *-skew elements is nilpotent.
International Journal of Algebra and Computation, 1998
Let ℚAn be the group algebra of the alternating group over the rationals. By exploiting the theor... more Let ℚAn be the group algebra of the alternating group over the rationals. By exploiting the theory of Young tableaux, we give an explicit description of the minimal central idempotents of ℚAn. As an application we construct finitely many generators for a subgroup of finite index in the centre of the group of units of ℚAn.
Page 1. COMMUNICATIONS IN ALGEBRA, 22(5), 1685-1701 (1994) RINGS WITH ALGEBRAIC n-ENGEL ELEMENTS ... more Page 1. COMMUNICATIONS IN ALGEBRA, 22(5), 1685-1701 (1994) RINGS WITH ALGEBRAIC n-ENGEL ELEMENTS Departamento di Maternatica ed Applicazioni Universita di Palcrlno via Archirafi 34 90123 - Palesrno - Italia giaml~runo~ipal~~at.cles.it ...
Uploads
Papers by A. Giambruno