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 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).

Homotopy skein modules of orientable 3-manifolds

1990 ◽  
Vol 108 (3) ◽  
pp. 475-488 ◽  
Author(s):  
Jim Hoste ◽  
Jósef H. Przytycki
Keyword(s):  
Lie Algebra ◽  
Symmetric Tensor ◽  
Orientable Surface ◽  
Universal Enveloping Algebra ◽  
Tensor Algebra ◽  
Algebra Structure ◽  
Skein Modules ◽  
Enveloping Algebra ◽  
Skein Module ◽  
Link Homotopy

AbstractWe define the homotopy skein module of an arbitrary orientable 3-manifold M. This module is similar to the ordinary skein module defined by the second author but is more appropriate when considering oriented links in M up to link homotopy rather than isotopy. We compute the homotopy skein module of M = F × I for any orientable surface F and show that it is free. In the case where M = F × I the homotopy skein module may be given an algebra structure and we show that as an algebra it is isomorphic to the universal enveloping algebra of the Goldman–Wolpert Lie algebra of F. We show, also in this case, that the homotopy skein module is a quantization of the symmetric tensor algebra associated to the Goldman–Wolpert Lie algebra.

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 A note on the zeroth products of Frenkel–Jing operators

2017 ◽  
Vol 16 (03) ◽  
pp. 1750053 ◽  
Author(s):  
Slaven Kožić
Keyword(s):  
Lie Algebra ◽  
Vertex Operator ◽  
Vertex Operator Algebra ◽  
Vertex Algebra ◽  
Infinite Dimensional ◽  
Enveloping Algebra ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Quantum Vertex ◽  
Algebra Theory

Let [Formula: see text] be an untwisted affine Kac–Moody Lie algebra. The top of every irreducible highest weight integrable [Formula: see text]-module is the finite-dimensional irreducible [Formula: see text]-module, where the action of the simple Lie algebra [Formula: see text] is given by zeroth products arising from the underlying vertex operator algebra theory. Motivated by this fact, we consider zeroth products of level [Formula: see text] Frenkel–Jing operators corresponding to Drinfeld realization of the quantum affine algebra [Formula: see text]. By applying these products, which originate from the quantum vertex algebra theory developed by Li, on the extension of Koyama vertex operator [Formula: see text], we obtain an infinite-dimensional vector space [Formula: see text]. Next, we introduce an associative algebra [Formula: see text], a certain quantum analogue of the universal enveloping algebra [Formula: see text], and construct some infinite-dimensional [Formula: see text]-modules [Formula: see text] corresponding to the finite-dimensional irreducible [Formula: see text]-modules [Formula: see text]. We show that the space [Formula: see text] carries a structure of an [Formula: see text]-module and, furthermore, we prove that the [Formula: see text]-module [Formula: see text] is isomorphic to the [Formula: see text]-module [Formula: see text].

On the Garsia Lie Idempotent

2005 ◽  
Vol 48 (3) ◽  
pp. 445-454 ◽  
Author(s):  
Frédéric Patras ◽  
Christophe Reutenauer ◽  
Manfred Schocker
Keyword(s):  
Symmetric Group ◽  
Lie Algebra ◽  
Associative Algebra ◽  
Group Algebra ◽  
Orthogonal Projection ◽  
Gram Matrix ◽  
Free Associative Algebra ◽  
Free Lie Algebra ◽  
Rational Group ◽  
Lie Idempotent

AbstractThe orthogonal projection of the free associative algebra onto the free Lie algebra is afforded by an idempotent in the rational group algebra of the symmetric group Sn, in each homogenous degree n. We give various characterizations of this Lie idempotent and show that it is uniquely determined by a certain unit in the group algebra of Sn−1. The inverse of this unit, or, equivalently, the Gram matrix of the orthogonal projection, is described explicitly. We also show that the Garsia Lie idempotent is not constant on descent classes (in fact, not even on coplactic classes) in Sn.

scholarly journals Canonical bases and standard monomials

1998 ◽  
Vol 41 (3) ◽  
pp. 611-623
Author(s):  
R. J. Marsh
Keyword(s):  
Lie Algebra ◽  
Canonical Basis ◽  
Monomial Basis ◽  
Fundamental Weight ◽  
Enveloping Algebra ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Standard Monomials ◽  
Standard Monomial ◽  
Quantized Enveloping Algebra

Let U be the quantized enveloping algebra associated to a simple Lie algebra g by Drinfel'd and Jimbo. Let λ be a classical fundamental weight for g, and ⋯(λ) the irreducible, finite-dimensional type 1 highest weight U-module with highest weight λ. We show that the canonical basis for ⋯(λ) (see Kashiwara [6, §0] and Lusztig [18, 14.4.12]) and the standard monomial basis (see [11, §§2.4 and 2.5]) for ⋯(λ) coincide.

scholarly journals Abelianizations of derivation Lie algebras of the free associative algebra and the free Lie algebra

2013 ◽  
Vol 162 (5) ◽  
pp. 965-1002 ◽  
Author(s):  
Shigeyuki Morita ◽  
Takuya Sakasai ◽  
Masaaki Suzuki
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Free Associative Algebra ◽  
Free Lie Algebra

On basic Cycles of An, Bn, Cn and Dn

1985 ◽  
Vol 37 (1) ◽  
pp. 122-140 ◽  
Author(s):  
D. J. Britten ◽  
F. W. Lemire
Keyword(s):  
Lie Algebra ◽  
Dual Space ◽  
Cartan Subalgebra ◽  
Closed Field ◽  
Generating Set ◽  
Enveloping Algebra ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Irreducible Modules ◽  
Positive Roots

In this paper, we investigate a conjecture of Dixmier [2] on the structure of basic cycles. Our interest in basic cycles arises primarily from the fact that the irreducible modules of a simple Lie algebra L having a weight space decomposition are completely determined by the irreducible modules of the cycle subalgebra of L. The basic cycles form a generating set for the cycle subalgebra.First some notation: F denotes an algebraically closed field of characteristic 0, L a finite dimensional simple Lie algebra of rank n over F, H a fixed Cartan subalgebra, U(L) the universal enveloping algebra of L, C(L) the centralizer of H in U(L), Φ the set of nonzero roots in H*, the dual space of H, Δ = {α1, …, αn} a base of Φ, and Φ+ = {β1, …, βm} the positive roots corresponding to Δ.

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.

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