scholarly journals A combinatorial identity for the derivative of a theta series of a finite type root lattice

2003 ◽  
Vol 172 ◽  
pp. 1-30
Author(s):  
Satoshi Naito
Keyword(s):  
Representation Theory ◽  
Lie Algebra ◽  
Finite Type ◽  
Cartan Subalgebra ◽  
Theta Series ◽  
Combinatorial Identity ◽  
Root Lattice ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Affine Lie Algebra

AbstractLet be a (not necessarily simply laced) finite-dimensional complex simple Lie algebra with the Cartan subalgebra and Q ⊂ * the root lattice. Denote by ΘQ(q) the theta series of the root lattice Q of . We prove a curious “combinatorial” identity for the derivative of ΘQ(q), i.e. for by using the representation theory of an affine 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 BGG reciprocity for current algebras

2015 ◽  
Vol 151 (7) ◽  
pp. 1265-1287 ◽  
Author(s):  
Vyjayanthi Chari ◽  
Bogdan Ion
Keyword(s):  
Lie Algebra ◽  
Current Algebra ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Affine Lie Algebra ◽  
A Current ◽  
Current Algebras

In Bennett et al. [BGG reciprocity for current algebras, Adv. Math. 231 (2012), 276–305] it was conjectured that a BGG-type reciprocity holds for the category of graded representations with finite-dimensional graded components for the current algebra associated to a simple Lie algebra. We associate a current algebra to any indecomposable affine Lie algebra and show that, in this generality, the BGG reciprocity is true for the corresponding category of representations.

AUTOMORPHISMS OF CERTAIN NILPOTENT LIE ALGEBRAS OVER COMMUTATIVE RINGS

2007 ◽  
Vol 17 (03) ◽  
pp. 527-555 ◽  
Author(s):  
YOU'AN CAO ◽  
DEZHI JIANG ◽  
JUNYING WANG
Keyword(s):  
Automorphism Group ◽  
Commutative Ring ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Commutative Rings ◽  
Nilpotent Lie Algebras ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Positive Roots ◽  
Nilpotent Subalgebra

Let L be a finite-dimensional complex simple Lie algebra, Lℤ be the ℤ-span of a Chevalley basis of L and LR = R⊗ℤLℤ be a Chevalley algebra of type L over a commutative ring R. Let [Formula: see text] be the nilpotent subalgebra of LR spanned by the root vectors associated with positive roots. The aim of this paper is to determine the automorphism group of [Formula: see text].

scholarly journals Weyl modules and Weyl functors for hyper-map algebras

2020 ◽  
pp. 2140007
Author(s):  
Angelo Bianchi ◽  
Samuel Chamberlin
Keyword(s):  
Lie Algebra ◽  
Finitely Generated ◽  
Weyl Modules ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Universal Properties ◽  
Definition Of ◽  
Unitary Algebra

We investigate the representations of the hyperalgebras associated to the map algebras [Formula: see text], where [Formula: see text] is any finite-dimensional complex simple Lie algebra and [Formula: see text] is any associative commutative unitary algebra with a multiplicatively closed basis. We consider the natural definition of the local and global Weyl modules, and the Weyl functor for these algebras. Under certain conditions, we prove that these modules satisfy certain universal properties, and we also give conditions for the local or global Weyl modules to be finite-dimensional or finitely generated, respectively.

One-Dimensional Representations of the Cycle Subalgebra of a Semi-Simple Lie Algebra

1970 ◽  
Vol 13 (4) ◽  
pp. 463-467 ◽  
Author(s):  
F. W. Lemire
Keyword(s):  
Lie Algebra ◽  
Cartan Subalgebra ◽  
Mass Function ◽  
Algebra Homomorphism ◽  
Closed Field ◽  
Noncommutative Algebra ◽  
One Dimensional ◽  
Finite Dimensional ◽  
The Difference ◽  
Mass Functions

Let L denote a semi-simple, finite dimensional Lie algebra over an algebraically closed field K of characteristic zero. If denotes a Cartan subalgebra of L and denotes the centralizer of in the universal enveloping algebra U of L, then it has been shown that each algebra homomorphism (called a "mass-function" on ) uniquely determines a linear irreducible representation of L. The technique involved in this construction is analogous to the Harish-Chandra construction [2] of dominated irreducible representations of L starting from a linear functional . The difference between the two results lies in the fact that all linear functionals on are readily obtained, whereas since is in general a noncommutative algebra the construction of mass-functions is decidedly nontrivial.

scholarly journals Hom-structures on semi-simple Lie algebras

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Wenjuan Xie ◽  
Quanqin Jin ◽  
Wende Liu
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Jacobi Identity ◽  
Algebra Homomorphism ◽  
Closed Field ◽  
Simple Lie Algebras ◽  
3 Dimensional ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Reduction Methods

AbstractA Hom-structure on a Lie algebra (g,[,]) is a linear map σ W g σ g which satisfies the Hom-Jacobi identity: [σ(x), [y,z]] + [σ(y), [z,x]] + [σ(z),[x,y]] = 0 for all x; y; z ∈ g. A Hom-structure is referred to as multiplicative if it is also a Lie algebra homomorphism. This paper aims to determine explicitly all the Homstructures on the finite-dimensional semi-simple Lie algebras over an algebraically closed field of characteristic zero. As a Hom-structure on a Lie algebra is not necessarily a Lie algebra homomorphism, the method developed for multiplicative Hom-structures by Jin and Li in [J. Algebra 319 (2008): 1398–1408] does not work again in our case. The critical technique used in this paper, which is completely different from that in [J. Algebra 319 (2008): 1398– 1408], is that we characterize the Hom-structures on a semi-simple Lie algebra g by introducing certain reduction methods and using the software GAP. The results not only improve the earlier ones in [J. Algebra 319 (2008): 1398– 1408], but also correct an error in the conclusion for the 3-dimensional simple Lie algebra sl2. In particular, we find an interesting fact that all the Hom-structures on sl2 constitute a 6-dimensional Jordan algebra in the usual way.

scholarly journals Quantised affine algebras and parameter-dependent R-matrices

1995 ◽  
Vol 51 (2) ◽  
pp. 177-194 ◽  
Author(s):  
Anthony J. Bracken ◽  
Mark D. Gould ◽  
Yao-Zhong Zhang
Keyword(s):  
Explicit Formula ◽  
Lie Algebra ◽  
Simple Lie Algebra ◽  
Affine Lie Algebra ◽  
Affine Algebras ◽  
Parameter Dependent ◽  
Concrete Representations

Let Uq(G(1)) be a quantised non-twisted affine Lie algebra with Uq(G) the corresponding quantised simple Lie algebra. Using the previously obtained universal R-matrices for and , explicitly spectral-dependent universal R-matrices for Uq(A1) and Uq(A2) are determined. These spectral-dependent universal R-matrices are evaluated in some concrete representations; well-known results for the fundamental representations are reproduced, and an explicit formula for the spectral-dependent R-matrix associated with the V(3) ⊗ V(6) module is derived, where V(3) and V(6) carry the 3- and 6-dimensional representations of Uq(A2), respectively.

Normality in Elementary Subgroups of Chevalley Groups Over Rings

1976 ◽  
Vol 28 (2) ◽  
pp. 420-428 ◽  
Author(s):  
James F. Hurley
Keyword(s):  
Normal Subgroup ◽  
Commutative Ring ◽  
Lie Algebra ◽  
Chevalley Group ◽  
Complex Field ◽  
Normal Subgroups ◽  
Chevalley Groups ◽  
Simple Lie Algebra ◽  
Elementary Subgroup ◽  
Finite Dimensional

In [6] we have constructed certain normal subgroups G7 of the elementary subgroup GR of the Chevalley group G(L, R) over R corresponding to a finite dimensional simple Lie algebra L over the complex field, where R is a commutative ring with identity. The method employed was to augment somewhat the generators of the elementary subgroup EI of G corresponding to an ideal I of the underlying Chevalley algebra LR;EI is thus the group generated by all xr(t) in G having the property that ter ⊂ I. In [6, § 5] we noted that in general EI actually had to be enlarged for a normal subgroup of GR to be obtained.

scholarly journals Yangians versus minimal W-algebras: A surprising coincidence

2020 ◽  
pp. 2050036
Author(s):  
Victor G. Kac ◽  
Pierluigi Möseneder Frajria ◽  
Paolo Papi
Keyword(s):  
Lie Algebra ◽  
Monic Polynomial ◽  
Adjoint Representation ◽  
Conformal Weight ◽  
Quantum Fields ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Matrix Formula ◽  
Polynomial Formula

We prove that the singularities of the [Formula: see text]-matrix [Formula: see text] of the minimal quantization of the adjoint representation of the Yangian [Formula: see text] of a finite dimensional simple Lie algebra [Formula: see text] are the opposite of the roots of the monic polynomial [Formula: see text] entering in the OPE expansions of quantum fields of conformal weight [Formula: see text] of the universal minimal affine [Formula: see text]-algebra at level [Formula: see text] attached to [Formula: see text].

scholarly journals Abelian Ideals with Given Dimension in Borel Subalgebras

2012 ◽  
Vol 19 (04) ◽  
pp. 755-770
Author(s):  
Li Luo
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Borel Subalgebra ◽  
Simple Lie Algebras ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Dimensional Distribution

A well-known Peterson's theorem says that the number of abelian ideals in a Borel subalgebra of a rank-r finite-dimensional simple Lie algebra is exactly 2r. In this paper, we determine the dimensional distribution of abelian ideals in a Borel subalgebra of finite-dimensional simple Lie algebras, which is a refinement of Peterson's theorem capturing more Lie algebra invariants.

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