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A new series of central elements is found in the free alternative algebra. More exactly, let $Alt[X]$ and $SMalc[X]\subset Alt[X]$ be the free alternative algebra and the free special Malcev algebra over a field of characteristic 0 on a... more
A new series of central elements is found in the free alternative algebra. More exactly, let $Alt[X]$ and $SMalc[X]\subset Alt[X]$ be the free alternative algebra and the free special Malcev algebra over a field of characteristic 0 on a set of free generators $X$, and let $f(x,y,x_1,\ldots,x_n)\in SMalc[X]$ be a multilinear element which is trivial in the free associative algebra. Then the element $u_n=u_n(x,x_1,\ldots,x_n)=f(x^2,x,x_1,\ldots,x_n)-f(x,x^2,x_1,\ldots,x_n)$ lies in the center of the algebra $Alt[X]$. The elements $u_n(x,x_1,\ldots,x_n)$ are uniquely defined up to a scalar for a given $n$, and they are skew-symmetric on the variables $x_1,\ldots,x_n$. Moreover, $u_n=0$ for $n=4m+2,\,4m+3$. and $u_n\neq 0$ for $n=4m,4m+1$. The ideals generated by the elements $u_{4m},\,u_{4m+1}$ lie in the associative center of the algebra $Alt[X]$ and have trivial multiplication.
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Correction to the title of the book!! Ivan Shestakov has belatedly revealed that Pascual Jordan came from an old Spanish family who emigrated to Germany during the Napoleanic wars. The original family name was Jorda (pronounced... more
Correction to the title of the book!! Ivan Shestakov has belatedly revealed that Pascual Jordan came from an old Spanish family who emigrated to Germany during the Napoleanic wars. The original family name was Jorda (pronounced “Hchh-orda”, as the Spanish J is aspirated like ch in the Scottish “loch” or the German “ich”), and the first-born son was traditionally named Pascual. The family name was eventually Germanicized to a proper Jordan (“Yorr-dahn”), but by rights the algebras should be known as Jorda algebras. I think it is too late to repair the damage, and abuse language (but preserve vocal chords) by calling them Jordan algebras instead of Hchhor-dan algebras.
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This volume contains contributions from the conference on 'Algebras, Representations and Applications' (Maresias, Brazil, August 26 - September 1, 2007), in honor of Ivan Shestakov's 60th birthday. This book will be of... more
This volume contains contributions from the conference on 'Algebras, Representations and Applications' (Maresias, Brazil, August 26 - September 1, 2007), in honor of Ivan Shestakov's 60th birthday. This book will be of interest to graduate students and researchers working in the theory of Lie and Jordan algebras and superalgebras and their representations, Hopf algebras, Poisson algebras, Quantum Groups, Group Rings and other topics.
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In this article we investigate further a notion of noncommutative transcendence degree, the lower transcendence degree, introduced by J. J Zhang in 1998, with important connections to the classical Gelfand-Kirillov transcendence degree,... more
In this article we investigate further a notion of noncommutative transcendence degree, the lower transcendence degree, introduced by J. J Zhang in 1998, with important connections to the classical Gelfand-Kirillov transcendence degree, noncommutative projective algebraic geometry and many open problems in ring theory. We compute the value of this invariant for many different algebras, showing that they are in fact LD-stable, which reduces the question of finding this invariant to the computation of the Gelfand-Kirillov dimension. We show that the lower transcendence degree behaves well with respect to taking invariants of the rings under consideration by actions of finite groups, and that it is Morita invariant, demonstrating that it has good theoretical properties lacking in other notions of noncommutative transcendence degree. Finally, many applications appear through the text.
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A finite dimensional Jordan algebra \(J\) over a field \({\bf k}\) is called \textit{basic} if the quotient algebra \(J/{\rm Rad} J\) is isomorphic to a direct sum of copies of \({\bf k}\). We describe all basic Jordan algebras \(J\) with... more
A finite dimensional Jordan algebra \(J\) over a field \({\bf k}\) is called \textit{basic} if the quotient algebra \(J/{\rm Rad} J\) is isomorphic to a direct sum of copies of \({\bf k}\). We describe all basic Jordan algebras \(J\) with \(({\rm Rad} J)^2=0\) of finite and tame representation type over an algebraically closed field of characteristic 0.
We prove that if a field k is infinite, char(k)=0 and k has not nontrivial automorphisms then automorphic equivalence of representations of Lie algebras coincide with geometric equivalence. We achieve our result by consideration of... more
We prove that if a field k is infinite, char(k)=0 and k has not nontrivial automorphisms then automorphic equivalence of representations of Lie algebras coincide with geometric equivalence. We achieve our result by consideration of 1-sorted objects. We suppose that our method can be perspective in the further researches.
v1 = ∂1 + t0(∂2 + t1(∂3 + t2(∂4 + t3(∂5 + t4(∂6 + · · · ))))); v2 = ∂2 + t1(∂3 + t2(∂4 + t3(∂5 + t4(∂6 + · · · )))). Let L = Liep(v1, v2) ⊂ DerR be the restricted Lie algebra generated by these derivations. We establish the following... more
v1 = ∂1 + t0(∂2 + t1(∂3 + t2(∂4 + t3(∂5 + t4(∂6 + · · · ))))); v2 = ∂2 + t1(∂3 + t2(∂4 + t3(∂5 + t4(∂6 + · · · )))). Let L = Liep(v1, v2) ⊂ DerR be the restricted Lie algebra generated by these derivations. We establish the following properties of this algebra in case p = 2, 3. a) L has a polynomial growth with Gelfand-Kirillov dimension ln p/ ln((1+ √ 5)/2). b) the associative envelope A = Alg(v1, v2) of L has Gelfand-Kirillov dimension 2 ln p/ ln((1+ √ 5)/2). c) L has a nil-p-mapping. d) L , A and the augmentation ideal of the restricted enveloping algebra u = u0(L) are direct sums of two locally nilpotent subalgebras. The question whether u is a nil-algebra remains open. e) the restricted enveloping algebra u(L) is of intermediate growth. These properties resemble those of Grigorchuk and Gupta-Sidki groups. Mathematics Subject Classification 2000: 17B05, 17B50, 17B66, 16P90, 11B39.
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Over an arbitrary ring of scalars, we build a Jordan algebra J having two elements x,y∈J such that x◦y=0, but whose U-operators do not commute. This shows that nondegeneracy is a necessary condition in the main theorem of Commuting... more
Over an arbitrary ring of scalars, we build a Jordan algebra J having two elements x,y∈J such that x◦y=0, but whose U-operators do not commute. This shows that nondegeneracy is a necessary condition in the main theorem of Commuting U-Operators in Jordan Algebras by J. A Anquela, T. Cortés and H. P. Petersson (Trans. Amer. Math. Soc. 366 (2014), 5877–5902). As a consequence, we obtain examples of Jordan systems over arbitrary rings of scalars that cannot be imbedded in nondegenerate systems.
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Abstract We classify the central simple finite-dimensional noncommutative Jordan superalgebras over an algebraically closed field of characteristic p > 2 . The case of characteristic 0 was considered by the authors in the previous... more
Abstract We classify the central simple finite-dimensional noncommutative Jordan superalgebras over an algebraically closed field of characteristic p > 2 . The case of characteristic 0 was considered by the authors in the previous paper [21] . In particular, we describe Leibniz brackets on all finite dimensional central simple Jordan superalgebras except mixed (nor vector neither Poisson) Kantor doubles of the supercommutative superalgebra B ( m , n ) .
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In this paper we describe all group gradings by a finite Abelian group G of several types of simple Jordan and Lie algebras over an algebraically closed field F of characteristic zero. 2004 Published by Elsevier Inc.
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Let F be a field of characteristic different of 2 and let M1|1(F)(+) denote the Jordan superalgebra of 2 × 2 matrices over the field F. The aim of this paper is to classify irreducible (unital and one-sided) Jordan bimodules over the... more
Let F be a field of characteristic different of 2 and let M1|1(F)(+) denote the Jordan superalgebra of 2 × 2 matrices over the field F. The aim of this paper is to classify irreducible (unital and one-sided) Jordan bimodules over the Jordan superalgebra M1|1(F)(+).