scholarly journals Polynomial identities with involution, superinvolutions and the Grassmann envelope

2017 ◽  
Vol 145 (5) ◽  
pp. 1843-1857 ◽  
Author(s):  
Eli Aljadeff ◽  
Antonio Giambruno ◽  
Yakov Karasik
Keyword(s):  
Polynomial Identities ◽  
Grassmann Envelope

Codimension growth for weak polynomial identities, and non-integrality of the PI exponent

2020 ◽  
Vol 63 (4) ◽  
pp. 929-949
Author(s):  
David Levi da Silva Macedo ◽  
Plamen Koshlukov
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Crucial Role ◽  
Close Relation ◽  
Polynomial Identities ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Representations Of Lie Algebras ◽  
Grassmann Envelope

Let K be a field of characteristic zero. In this paper, we study the polynomial identities of representations of Lie algebras, also called weak identities, or identities of pairs. These identities are determined by pairs of the form (A, L) where A is an associative enveloping algebra for the Lie algebra L. Then a weak identity of (A, L) (or an identity for the representation of L associated to A) is an associative polynomial which vanishes when evaluated on elements of L⊆ A. One of the most influential results in the area of PI algebras was the theory developed by Kemer. A crucial role in it was played by the construction of the Grassmann envelope of an associative algebra and the close relation of the identities of the algebra and its Grassmann envelope. Here we consider varieties of pairs. We prove that under some restrictions one can develop a theory similar to that of Kemer's in the study of identities of representations of Lie algebras. As a consequence, we establish that in the case when K is algebraically closed, if a variety of pairs does not contain pairs corresponding to representations of sl2(K), and if the variety is generated by a pair where the associative algebra is PI then it is soluble. As another consequence of the methods used to obtain the above result, and applying ideas from papers by Giambruno and Zaicev, we were able to construct a pair (A, L) such that its PI exponent (if it exists) cannot be an integer. We recall that the PI exponent exists and is an integer whenever A is an associative (a theorem by Giambruno and Zaicev), or a finite-dimensional Lie algebra (Zaicev). Gordienko also proved that the PI exponent exists and is an integer for finite-dimensional representations of Lie algebras.

On Polynomial Identities of Jordan Pairs of Rectangular Matrices

1997 ◽  
Vol 260 (1-3) ◽  
pp. 257-271 ◽  
Author(s):  
A Iltyakov
Keyword(s):  
Polynomial Identities

scholarly journals Polynomial identities related to special Schubert varieties

2021 ◽  
Author(s):  
Francesca Cioffi ◽  
Davide Franco ◽  
Carmine Sessa
Keyword(s):  
Arbitrary Number ◽  
Schubert Variety ◽  
Decomposition Theorem ◽  
Point Of View ◽  
Explicit Description ◽  
Intersection Cohomology ◽  
Schubert Varieties ◽  
Polynomial Identities ◽  
Poincaré Polynomial

AbstractLet $$\mathcal S$$ S be a single condition Schubert variety with an arbitrary number of strata. Recently, an explicit description of the summands involved in the decomposition theorem applied to such a variety has been obtained in a paper of the second author. Starting from this result, we provide an explicit description of the Poincaré polynomial of the intersection cohomology of $$\mathcal S$$ S by means of the Poincaré polynomials of its strata, obtaining interesting polynomial identities relating Poincaré polynomials of several Grassmannians, both by a local and by a global point of view. We also present a symbolic study of a particular case of these identities.

scholarly journals Graded Polynomial Identities of the Jordan Superalgebra of a Bilinear Form

1996 ◽  
Vol 184 (1) ◽  
pp. 255-296 ◽  
Author(s):  
Sergei Yu. Vasilovsky
Keyword(s):  
Bilinear Form ◽  
Polynomial Identities ◽  
Jordan Superalgebra ◽  
Graded Polynomial Identities

scholarly journals Star-polynomial identities: Computing the exponential growth of the codimensions

2017 ◽  
Vol 469 ◽  
pp. 302-322 ◽  
Author(s):  
A. Giambruno ◽  
C. Polcino Milies ◽  
A. Valenti
Keyword(s):  
Exponential Growth ◽  
Polynomial Identities

scholarly journals Eulerian Polynomial Identities and Algebras Satisfying a Standard Identity

1994 ◽  
Vol 169 (3) ◽  
pp. 913-928 ◽  
Author(s):  
M. Domokos
Keyword(s):  
Polynomial Identities ◽  
Standard Identity ◽  
Eulerian Polynomial

scholarly journals A bijection which implies Melzer's polynomial identities: the? 1,1 (p,p+1) case

1996 ◽  
Vol 36 (2) ◽  
pp. 145-155 ◽  
Author(s):  
Omar Foda ◽  
S. Ole Warnaar
Keyword(s):  
Polynomial Identities

scholarly journals Rings with involution and polynomial identities

1969 ◽  
Vol 11 (2) ◽  
pp. 186-194 ◽  
Author(s):  
Wallace S Martindale
Keyword(s):  
Polynomial Identities ◽  
Rings With Involution

Algebras with polynomial identities

1999 ◽  
Vol 16 (9) ◽  
Author(s):  
Plamen Koshlukov
Keyword(s):  
Polynomial Identities
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