Identities of representations of Lie algebras and *-polynomial identities

1999 ◽  
Vol 48 (1) ◽  
pp. 153-162 ◽  
Author(s):  
Tsetska Rashkova ◽  
Vesselin Drensky
Keyword(s):  
Lie Algebras ◽  
Polynomial Identities ◽  
Representations Of Lie Algebras

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.

scholarly journals On representations of lie algebras for quantized hamiltonians

1997 ◽  
Vol 266 ◽  
pp. 69-79 ◽  
Author(s):  
L.A-M. Hanna ◽  
M.E. Khalifa ◽  
S.S. Hassan
Keyword(s):  
Lie Algebras ◽  
Representations Of Lie Algebras

Representations Subduced on an Ideal of a Lie Algebra

1962 ◽  
Vol 14 ◽  
pp. 293-303 ◽  
Author(s):  
B. Noonan
Keyword(s):  
Irreducible Representation ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Kronecker Product ◽  
Point Of View ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Representations Of Lie Algebras ◽  
The Ideal

This paper considers the properties of the representation of a Lie algebra when restricted to an ideal, the subduced* representation of the ideal. This point of view leads to new forms for irreducible representations of Lie algebras, once the concept of matrices of invariance is developed. This concept permits us to show that irreducible representations of a Lie algebra, over an algebraically closed field, can be expressed as a Lie-Kronecker product whose factors are associated with the representation subduced on an ideal. Conversely, if one has such factors, it is shown that they can be put together to give an irreducible representation of the Lie algebra. A valuable guide to this work was supplied by a paper of Clifford (1).

scholarly journals On Real Irreducible Representations of Lie Algebras

1959 ◽  
Vol 14 ◽  
pp. 59-83 ◽  
Author(s):  
Nagayoshi Iwahori
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Number Field ◽  
Irreducible Representations ◽  
The Real ◽  
Representations Of Lie Algebras ◽  
Real Matrices ◽  
Real Number Field

Let us consider the following two problems:Problem A. Let g be a given Lie algebra over the real number field R. Then find all real, irreducible representations of g.Problem B. Let n be a given positive integer. Then find all irreducible subalgebras of the Lie algebra ôí(w, R) of all real matrices of degree n.

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