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 CONSTRUCTIVE NONCOMMMUTATIVE INVARIANT THEORY

2021 ◽  
Author(s):  
MÁTYÁS DOMOKOS ◽  
VESSELIN DRENSKY
Keyword(s):  
Lie Algebra ◽  
Associative Algebra ◽  
Invariant Theory ◽  
Reductive Group ◽  
Symmetric Tensor ◽  
Free Associative Algebra ◽  
Tensor Algebra ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Degree Bounds

AbstractThe problem of finding generators of the subalgebra of invariants under the action of a group of automorphisms of a finite-dimensional Lie algebra on its universal enveloping algebra is reduced to finding homogeneous generators of the same group acting on the symmetric tensor algebra of the Lie algebra. This process is applied to prove a constructive Hilbert–Nagata Theorem (including degree bounds) for the algebra of invariants in a Lie nilpotent relatively free associative algebra endowed with an action induced by a representation of a reductive group.

scholarly journals A functorial version of a construction of Hochschild and Mostow for representations of Lie algebras

1978 ◽  
Vol 18 (1) ◽  
pp. 95-98 ◽  
Author(s):  
William H. Wilson
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Dimensional ◽  
Representations Of Lie Algebras ◽  
Restriction Functor

Let be a Lie algebra, a complemented ideal of , and W an -module. Hochschild and Mostow have described the construction of a -module “induced” from W, which is finite-dimensional provided W is finite-dimensional and satisfies a nilpotent action condition. This note describes a modification of their construction which is functorial and a weak adjoint to the restriction functor from –modules to -modules.

scholarly journals MAD SUBALGEBRAS AND LIE SUBALGEBRAS OF AN ENVELOPING ALGEBRA

2010 ◽  
Vol 82 (3) ◽  
pp. 401-423
Author(s):  
XIN TANG
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Two Dimensional ◽  
Base Field ◽  
Inner Derivation ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Characterization Theorems

AbstractLet 𝒰(𝔯(1)) denote the enveloping algebra of the two-dimensional nonabelian Lie algebra 𝔯(1) over a base field 𝕂. We study the maximal abelian ad-nilpotent (mad) associative subalgebras and finite-dimensional Lie subalgebras of 𝒰(𝔯(1)). We first prove that the set of noncentral elements of 𝒰(𝔯(1)) admits the Dixmier partition, 𝒰(𝔯(1))−𝕂=⋃ 5i=1Δi, and establish characterization theorems for elements in Δi, i=1,3,4. Then we determine the elements in Δi, i=1,3 , and describe the eigenvalues for the inner derivation ad Bx,x∈Δi, i=3,4 . We also derive other useful results for elements in Δi, i=2,3,4,5 . As an application, we find all framed mad subalgebras of 𝒰(𝔯(1)) and determine all finite-dimensional nonabelian Lie algebras that can be realized as Lie subalgebras of 𝒰(𝔯(1)) . We also study the realizations of the Lie algebra 𝔯(1) in 𝒰(𝔯(1)) in detail.

Standard Representations of Simple Lie Algebras

1968 ◽  
Vol 20 ◽  
pp. 344-361 ◽  
Author(s):  
I. Z. Bouwer
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Partial Solution ◽  
Irreducible Representations ◽  
Complex Numbers ◽  
Dominant Weight ◽  
Infinite Dimensional ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Algebraic Representations

Let L be any simple finite-dimensional Lie algebra (defined over the field K of complex numbers). Cartan's theory of weights is used to define sets of (algebraic) representations of L that can be characterized in terms of left ideals of the universal enveloping algebra of L. These representations, called standard, generalize irreducible representations that possess a dominant weight. The newly obtained representations are all infinite-dimensional. Their study is initiated here by obtaining a partial solution to the problem of characterizing them by means of sequences of elements in K.

scholarly journals NOTE ON VERMA BASES FOR REPRESENTATIONS OF SIMPLE LIE ALGEBRAS

2013 ◽  
Vol 53 (5) ◽  
pp. 450-456 ◽  
Author(s):  
Severin Pošta ◽  
Miloslav Havlíček
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Alternate Proof ◽  
Simple Lie Algebras ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
The One

We discuss the construction of the Verma basis of the enveloping algebra and of finite dimensional representations of the Lie algebra An. We give an alternate proof of so-called Verma inequalities to the one given in [1] by P. Littelmann.

scholarly journals Wronskian Envelope of a Lie Algebra

Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Laurent Poinsot
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Commutative Algebra ◽  
Differential Algebra ◽  
Free Lie Algebra ◽  
Sufficient Condition ◽  
Enveloping Algebra ◽  
Lie Bracket ◽  
Differential Algebras

The famous Poincaré-Birkhoff-Witt theorem states that a Lie algebra, free as a module, embeds into its associative envelope—its universal enveloping algebra—as a sub-Lie algebra for the usual commutator Lie bracket. However, there is another functorial way—less known—to associate a Lie algebra to an associative algebra and inversely. Any commutative algebra equipped with a derivation , that is, a commutative differential algebra, admits a Wronskian bracket under which it becomes a Lie algebra. Conversely, to any Lie algebra a commutative differential algebra is universally associated, its Wronskian envelope, in a way similar to the associative envelope. This contribution is the beginning of an investigation of these relations between Lie algebras and differential algebras which is parallel to the classical theory. In particular, we give a sufficient condition under which a Lie algebra may be embedded into its Wronskian envelope, and we present the construction of the free Lie algebra with this property.

scholarly journals PLAIN REPRESENTATIONS OF LIE ALGEBRAS

2001 ◽  
Vol 63 (3) ◽  
pp. 571-591 ◽  
Author(s):  
A. A. BARANOV ◽  
A. E. ZALESSKII
Keyword(s):  
Representation Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
General Problem ◽  
Associative Algebras ◽  
Ground Field ◽  
Finite Dimensional ◽  
Representations Of Lie Algebras ◽  
Completely Reducible ◽  
Finite Dimensional Lie Algebras

In this paper we study representations of finite dimensional Lie algebras. In this case representations are not necessarily completely reducible. As the general problem is known to be of enormous complexity, we restrict ourselves to representations that behave particularly well on Levi subalgebras. We call such representations plain (Definition 1.1). Informally, we show that the theory of plain representations of a given Lie algebra L is equivalent to representation theory of finitely many finite dimensional associative algebras, also non-semisimple. The sense of this is to distinguish representations of Lie algebras that are of complexity comparable with that of representations of associative algebras. Non-plain representations are intrinsically much more complex than plain ones. We view our work as a step toward understanding this complexity phenomenon.We restrict ourselves also to perfect Lie algebras L, that is, such that L = [L, L]. In our main results we assume that L is perfect and [sfr ][lfr ]2-free (which means that L has no quotient isomorphic to [sfr ][lfr ]2). The ground field [ ] is always assumed to be algebraically closed and of characteristic 0.

Codimension growth of central polynomials of Lie algebras

2020 ◽  
Vol 32 (1) ◽  
pp. 201-206
Author(s):  
Antonio Giambruno ◽  
Mikhail Zaicev
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Adjoint Representation ◽  
Closed Field ◽  
Polynomial Identities ◽  
Algebraically Closed Field ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Linearly Independent

AbstractLet L be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic zero and let I be the T-ideal of polynomial identities of the adjoint representation of L. We prove that the number of multilinear central polynomials in n variables, linearly independent modulo I, grows exponentially like {(\dim L)^{n}}.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

scholarly journals PERIODIC AUTOMORPHISMS, COMPATIBLE POISSON BRACKETS, AND GAUDIN SUBALGEBRAS

2021 ◽  
Author(s):  
DMITRI I. PANYUSHEV ◽  
OKSANA S. YAKIMOVA
Keyword(s):  
Poisson Bracket ◽  
Lie Algebra ◽  
Invariant Theory ◽  
Cyclic Permutation ◽  
Poisson Brackets ◽  
Enveloping Algebra ◽  
Order Automorphism ◽  
Finite Dimensional ◽  
Compatible Poisson Brackets ◽  
Commutative Subalgebras

AbstractLet 𝔮 be a finite-dimensional Lie algebra. The symmetric algebra (𝔮) is equipped with the standard Lie–Poisson bracket. In this paper, we elaborate on a surprising observation that one naturally associates the second compatible Poisson bracket on (𝔮) to any finite order automorphism ϑ of 𝔮. We study related Poisson-commutative subalgebras (𝔮; ϑ) of 𝒮(𝔮) and associated Lie algebra contractions of 𝔮. To obtain substantial results, we have to assume that 𝔮 = 𝔤 is semisimple. Then we can use Vinberg’s theory of ϑ-groups and the machinery of Invariant Theory.If 𝔤 = 𝔥⊕⋯⊕𝔥 (sum of k copies), where 𝔥 is simple, and ϑ is the cyclic permutation, then we prove that the corresponding Poisson-commutative subalgebra (𝔮; ϑ) is polynomial and maximal. Furthermore, we quantise this (𝔤; ϑ) using a Gaudin subalgebra in the enveloping algebra 𝒰(𝔤).

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