Academia.eduAcademia.edu

Outline

Title
Abstract
FAQs
Figures
Introduction
References

An Easy Construction of G2

Profile image of Norman WildbergerNorman Wildberger

2008

Abstract

Abstract. We show how to construct the simple exceptional Lie algebra of type G2 by explicitly constructing its 7 dimensional representation. No knowledge of Lie theory is required, and all relevant coefficients can be remembered by use of the simple mnemonic βαββαβ. 1.

FAQs

sparkles

AI

What unique methods facilitate the construction of the G2 Lie algebra?add

The paper demonstrates a novel construction using the directed G2 Hexagon graph combined with mutations, yielding an easier approach compared to traditional methods. Specifically, it requires no prior knowledge of Lie theory, significantly simplifying the process.

How do mutations and graphs relate to G2's structure constants?add

The study shows that mutations in directed graphs help derive weights and structure constants for G2. The explicit commutation relations can be traced back to observed relationships during mutations across the G2 Hexagon.

What is the dimensionality of the Lie algebra g2 constructed in this paper?add

The construction reveals that the Lie algebra g2 has a dimension of 14, originating from the operators defined on the graph's vertices. Each operator corresponds to vertices in the G2 Hexagon, facilitating easier computations of relations.

What role do dual mutations play in the Numbers Game for G2?add

Dual mutations in the Numbers Game adaptively change vertex functions, leading to a symmetric structure of roots. This process generates positive roots reflecting key properties of the algebra and aids in the construction of the G2 framework.

How does the construction differ from traditional Lie theory approaches?add

Unlike traditional constructions reliant on complex Lie theory, this paper uses graph theory and explicit operator definitions. This fresh perspective not only simplifies understanding but also broadens the possible applications within representation theory.

Figures (1)

Related papers

Explicit constructions of some infinite families of finite-dimensional irreducible representations of the type _6 and _7 simple Lie algebras
Molly Dunkum

2021

We construct every finite-dimensional irreducible representation of the simple Lie algebra of type _7 whose highest weight is a nonnegative integer multiple of the dominant minuscule weight associated with the type _7 root system. As a consequence, we obtain constructions of each finite-dimensional irreducible representation of the simple Lie algebra of type _6 whose highest weight is a nonnegative integer linear combination of the two dominant minuscule _6-weights. Our constructions are explicit in the sense that, if the representing space is d-dimensional, then a weight basis is provided such that all entries of the d × d representing matrices of the Chevalley generators are obtained via explicit, non-recursive formulas. To effect this work, we introduce what we call _6- and _7-polyminuscule lattices that analogize certain lattices associated with the famous special linear Lie algebra representation constructions obtained by Gelfand and Tsetlin.

View PDFchevron_right
Exceptional quantum subgroups for the rank two Lie algebras B2 and G2
Robert Coquereaux

Journal of Mathematical Physics, 2010

Exceptional modular invariants for the Lie algebras B 2 (at levels 2, 3, 7, 12) and G 2 (at levels 3, 4) can be obtained from conformal embeddings. We determine the associated algebras of quantum symmetries and discover or recover, as a by-product, the graphs describing exceptional quantum subgroups of type B 2 or G 2 which encode their module structure over the associated fusion category. Global dimensions are given.

View PDFchevron_right
Basic Twist Quantization of the Exceptional Lie Algebra G_2
Andrzej Borowiec

2004

We present the formulae for twist quantization of g2, corresponding to the solution of classical YB equation with support in the 8-dimensional Borel subalgebra of g2. The considered chain of twists consists of the four factors describing the four steps of quantization: Jordanian twist, the two twist factors extending Jordanian twist and the deformed Jordanian or in second variant additional Abelian twist. The first two steps describe as well the sl(3) quantization. The coproducts are calculated for each step in explicite form, and for that purpose we present new formulas for the calculation of similarity transformations on tensor product. We introduce new basic generators in universal enveloping algebra U (g2) which provide nonlinearities in algebraic sector maximally simplifying the deformed coproducts.

View PDFchevron_right
Basic twist quantization of the exceptional Lie algebra g[sub 2]
Andrzej Borowiec

Journal of Mathematical Physics, 2005

We present the formulae for twist quantization of g2, corresponding to the solution of classical YB equation with support in the 8-dimensional Borel subalgebra of g2. The considered chain of twists consists of the four factors describing the four steps of quantization: Jordanian twist, the two twist factors extending Jordanian twist and the deformed Jordanian or in second variant additional Abelian twist. The first two steps describe as well the sl(3) quantization. The coproducts are calculated for each step in explicite form, and for that purpose we present new formulas for the calculation of similarity transformations on tensor product. We introduce new basic generators in universal enveloping algebra U (g2) which provide nonlinearities in algebraic sector maximally simplifying the deformed coproducts. * Supported by KBN grant1PO3BO1828 † on leave

View PDFchevron_right
Lie–Yamaguti algebras related to g2
Pilar Benito

Journal of Pure and Applied Algebra, 2005

Lie-Yamaguti algebras (or generalized Lie triple systems) are intimately related to reductive homogeneous spaces. Simple Lie-Yamaguti algebras whose standard enveloping Lie algebra is the simple Lie algebra of type G2 are described, making use of the octonions. These examples reveal the much greater complexity of these systems, compared to Lie triple systems.

View PDFchevron_right
A family of simple Lie algebras in characteristic two
Giuseppe Jurman

Journal of Algebra, 2004

In this paper we describe a family of simple Lie algebras defined over a field of characteristic two depending on two integer parameters. We also describe their loop algebras.

View PDFchevron_right
On 1-dimensional representations of finite W-algebras associated to simple Lie algebras of exceptional type
Gerhard Roehrle

2010

We consider the finite W -algebra U (g, e) associated to a nilpotent element e ∈ g in a simple complex Lie algebra g of exceptional type. Using presentations obtained through an algorithm based on the PBW-theorem, we verify a conjecture of Premet, that U (g, e) always has a 1-dimensional representation, when g is of type G 2 , F 4 , E 6 or E 7 . Thanks to a theorem of Premet, this allows one to deduce the existence of minimal dimension representations of reduced enveloping algebras of modular Lie algebras of the above types. In addition, a theorem of Losev allows us to deduce that there exists a completely prime primitive ideal in U (g) whose associated variety is the coadjoint orbit corresponding to e.

View PDFchevron_right
Representations of finite $W$-algebras associated to simple Lie algebras of exceptional type
Gerhard Roehrle

arXiv (Cornell University), 2009

We consider the finite $W$-algebra $U(\g,e)$ associated to a nilpotent element $e \in \g$ in a simple complex Lie algebra $\g$ of exceptional type. Using presentations obtained through an algorithm based on the PBW-theorem, we verify a conjecture of Premet, that $U(\g,e)$ always has a 1-dimensional representation, when $\g$ is of type $G_2$, $F_4$, $E_6$ or $E_7$. Thanks to a theorem of Premet, this allows one to deduce the existence of minimal dimension representations of reduced enveloping algebras of modular Lie algebras of the above types. In addition, we deduce that there exists a completely prime primitive ideal in $U(\g)$ whose associated variety is the coadjoint orbit corresponding to $e$.

View PDFchevron_right
A Combinatorial Construction of G 2
Norman Wildberger

2003

We show how to construct the simple exceptional Lie algebra of type G 2 by explicitly constructing its 7 dimensional representation. Technically no knowledge of Lie theory is required. The structure constants have a combinatorial meaning involving convex subsets of a partially ordered multiset of six elements. These arise from playing the Numbers and Mutation games on a certain directed multigraph.

View PDFchevron_right
Exceptional Lie algebras, SU(3) and Jordan pairs: part 2. Zorn-type representations
Alessio Marrani

Journal of Physics A: Mathematical and Theoretical, 2014

A representation of the exceptional Lie algebras reflecting a simple unifying view, based on realizations in terms of Zorn-type matrices, is presented. The role of the underlying Jordan pair and Jordan algebra content is crucial in the development of the structure. Each algebra contains three Jordan pairs sharing the same Lie algebra of automorphisms and the same external su(3) symmetry. Applications in physics are outlined.

View PDFchevron_right

References (18)

  1. X β , X α ] = X α+β [X α+β , X β ] = 2X α+2β [X α+2β , X β ] = 3X α+3β [X α+3β , X α ] = X 2α+3β [X α+2β , X α+β ] = 3X 2α+3β [Y β , Y α ] = Y α+β [Y α+β , Y β ] = 2Y α+2β [Y α+2β , Y β ] = 3Y α+3β [Y α+3β , Y α ] = Y 2α+3β [Y α+2β , Y α+β ] = 3Y 2α+3β [X α , Y α+β ] = -Y β [X α , Y 2α+3β ] = -Y α+3β [X β , Y α+3β ] = -Y α+2β [X β , Y α+2β ] = -2Y α+β [X β , Y α+β ] = 3Y α [X α+β , Y α = X β [X α+β , Y β ] = -3X α [X α+β , Y α+2β ] = -2Y β [X α+2β , Y β ] = 2X α+β [X α+2β , Y α+β ] = 2X β [X α+2β , Y α+3β = Y α [X α+2β , Y 2α+3β = Y α+β [X α+3β , Y 2α+3β ] = -Y α [X 2α+3β , Y α ] = X α+3β [X 2α+3β , Y α+β ] = -X α+2β
  2. X 2α+3β , Y α+2β ] = -X α+β [X 2α+3β , Y α+3β ] = X α [Y β , X β ] = H β [Y α , X α ] = H α [Y α+β , X α+β ] = H α+β = 3H α + H β [Y α+2β , X α+2β ] = H α+2β = 2H α + 3H β [Y α+3β , X α+3β ] = H α+3β = H α + H β [Y 2α+3β , X 2α+3β ] = H 2α+3β = 2H α + H β [X γ , H β ] = H β (γ)X γ [Y γ , H β ] = -H β (γ)Y γ [X γ , H α ] = H α (γ)X γ [Y γ , H α ] = -H α (γ)Y γ . example? If so, what is the References
  3. F. Adams, Lectures on Exceptional Lie Groups, Chicago Lectures in Mathematics, University of Chicago Press, 1996.
  4. E. Cartan, Thèse, Paris, 1894. 2nd ed., Vuibert, Paris, 1933.
  5. K. Eriksson, Strong convergence and a game of numbers, European J. Combin. 17 (1996), no.4 379-390.
  6. W. Fulton and J. Harris, Representation Theory, Graduate Texts in Mathematics, Springer-Verlag, New York 1991.
  7. Harish-Chandra, On some applications of the universal enveloping algebra of a semisimple Lie algebra, Trans. Amer. Math. Soc. 70, (1951), 28-96.
  8. J. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, Springer-Verlag, New York 1972.
  9. N. Jacobson, Lie Algebras, J. Wiley & Sons, New York 1962.
  10. N. Jacobson, Exceptional Lie Algebras, Lecture Notes in Pure and Applied Mathematics I, Marcel Dekker, New York, 1971.
  11. W. Killing, Die Zusammensetzung der stetigen endlichen Transformationsgruppen I,II,III and IV, Math. Ann. 31 (1888), 33 (1889), 34 (1889) 36 (1890).
  12. S. Mozes, Reflection processes on graphs and Weyl groups, J. Combinatorial Theory A 53 (1990), 128-142.
  13. R. Proctor, Minuscule elements of Weyl groups, the numbers game. and d-complete posets, J. Alg. 213 (1999), no.1, 272-303.
  14. H. Samelson, Notes on Lie algebras, Universitext, Springer-Verlag, New York 1990.
  15. J. Tits, Sur les constants de structure et le théoème d'existence des algèbres de Lie semi-simples, Publ. Math. I.H.E.S. 31, (1966), 21-58.
  16. N. J. Wildberger, A combinatorial construction for simply-laced Lie algebras, submitted, 2000.
  17. N. J. Wildberger, Two new models for the representations of sl(3), preprint, 2001.
  18. N. J. Wildberger, The Mutation Game, Coxeter Graphs, and Partially Ordered Multisets, preprint, 2001. E-mail address: n.wildberger@unsw.edu.au School of Mathematics UNSW Sydney NSW 2052 Australia

Related papers

A Geometric Construction of the Exceptional Lie Algebras F 4 and E 8
José Figueroa-O'Farrill

Communications in Mathematical Physics, 2008

We present a geometric construction of the exceptional Lie algebras F 4 and E 8 starting from the round spheres S 8 and S 15 , respectively, inspired by the construction of the Killing superalgebra of a supersymmetric supergravity background.

View PDFchevron_right
The exceptional Lie algebra E 7(−25) : multiplets and invariant differential operators
Vladimir Dobrev

Journal of Physics A: Mathematical and Theoretical, 2009

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact exceptional algebra E 7(−25). Our choice of this particular algebra is motivated by the fact that it belongs to a narrow class of algebras, which we call 'conformal Lie algebras', which have very similar properties to the conformal algebras of n-dimensional Minkowski space-time. This class of algebras is identified and summarized in a table. Another motivation is related to the AdS/CFT correspondence. We give the multiplets of indecomposable elementary representations, including the necessary data for all relevant invariant differential operators. 1 Actually, A ∼ = SO(1, 1) × • • • × SO(1, 1), r = dim A copies. 2 The group decomposition is global which means that each element g of G can be represented by the group multiplication of three elements from the respective subgroups g = kan, k ∈ K, a ∈ A, n ∈ N. Similarly, each element W ∈ G, can be represented as the sum W = X ⊕ Y ⊕ Z, X ∈ K, Y ∈ A, Z ∈ N. 3 This group decomposition is almost global, which means that the decomposition g = manñ, (m ∈ M ,ñ ∈Ñ) is valid except for a subset of G of lower dimensionality. But the algebra decomposition W = U ⊕ Y ⊕ Z ⊕Z, (U ∈ M,Z ∈Ñ), is valid as above for each element W ∈ G.

View PDFchevron_right
Tits Construction of The Exceptional Simple Lie Algebras
Alberto Elduque

Pure and Applied Mathematics Quarterly, 2011

The classical Tits construction of the exceptional simple Lie algebras has been extended in a couple of directions by using either Jordan superalgebras or composition superalgebras. These extensions are reviewed here. The outcome has been the discovery of some new simple modular Lie superalgebras.

View PDFchevron_right
Explicit constructions of some infinite families of finite-dimensional irreducible representations of the type $${\textsf {E}}_{6}$$ and $${\textsf {E}}_{7}$$ simple Lie algebras
Molly Dunkum

Applicable Algebra in Engineering, Communication and Computing

We construct every finite-dimensional irreducible representation of the simple Lie algebra of type E 7 whose highest weight is a nonnegative integer multiple of the dominant minuscule weight associated with the type E 7 root system. As a consequence, we obtain constructions of each finite-dimensional irreducible representation of the simple Lie algebra of type E 6 whose highest weight is a nonnegative integer linear combination of the two dominant minuscule E 6-weights. Our constructions are explicit in the sense that, if the representing space is d-dimensional, then a weight basis is provided such that all entries of the d×d representing matrices of the Chevalley generators are obtained via explicit, non-recursive formulas. To effect this work, we introduce what we call E 6-and E 7-polyminuscule lattices that analogize certain lattices associated with the famous special linear Lie algebra representation constructions obtained by Gelfand and Tsetlin.

View PDFchevron_right
Explicit constructions of some infinite families of finite-dimensional irreducible representations of the type $\mbox{E}_{6}$ and $\mbox{E}_{7}$ simple Lie algebras
Molly Dunkum

2021

We construct every finite-dimensional irreducible representation of the simple Lie algebra of type E7 whose highest weight is a nonnegative integer multiple of the dominant minuscule weight associated with the type E7 root system. As a consequence, we obtain constructions of each finite-dimensional irreducible representation of the simple Lie algebra of type E6 whose highest weight is a nonnegative integer linear combination of the two dominant minuscule E6-weights. Our constructions are explicit in the sense that, if the representing space is d-dimensional, then a weight basis is provided such that all entries of the d×d representing matrices of the Chevalley generators are obtained via explicit, non-recursive formulas. To effect this work, we introduce what we call E6and E7-polyminuscule lattices that analogize certain lattices associated with the famous special linear Lie algebra representation constructions obtained by Gelfand and Tsetlin. Mathematics Subject Classification: 17B...

View PDFchevron_right
580 California St., Suite 400
San Francisco, CA, 94104
© 2025 Academia. All rights reserved