scholarly journals A denominator identity for affine Lie superalgebras with zero dual Coxeter number

2012 ◽  
Vol 6 (5) ◽  
pp. 1043-1059 ◽  
Author(s):  
Maria Gorelik ◽  
Shifra Reif
Keyword(s):  
Lie Superalgebras ◽  
Coxeter Number ◽  
Dual Coxeter Number ◽  
Denominator Identity

scholarly journals Coloured Generalised Young Diagrams for Affine Weyl-Coxeter Groups

10.37236/931 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
R. C. King ◽  
T. A. Welsh
Keyword(s):  
Lie Algebras ◽  
Coxeter Group ◽  
Coxeter Groups ◽  
Slight Variation ◽  
Affine Lie Algebras ◽  
Young Diagrams ◽  
Bijective Correspondence ◽  
Coxeter Number ◽  
Finite Dimensional ◽  
Dual Coxeter Number

Coloured generalised Young diagrams $T(w)$ are introduced that are in bijective correspondence with the elements $w$ of the Weyl-Coxeter group $W$ of $\mathfrak{g}$, where $\mathfrak{g}$ is any one of the classical affine Lie algebras $\mathfrak{g}=A^{(1)}_\ell$, $B^{(1)}_\ell$, $C^{(1)}_\ell$, $D^{(1)}_\ell$, $A^{(2)}_{2\ell}$, $A^{(2)}_{2\ell-1}$ or $D^{(2)}_{\ell+1}$. These diagrams are coloured by means of periodic coloured grids, one for each $\mathfrak{g}$, which enable $T(w)$ to be constructed from any expression $w=s_{i_1}s_{i_2}\cdots s_{i_t}$ in terms of generators $s_k$ of $W$, and any (reduced) expression for $w$ to be obtained from $T(w)$. The diagram $T(w)$ is especially useful because $w(\Lambda)-\Lambda$ may be readily obtained from $T(w)$ for all $\Lambda$ in the weight space of $\mathfrak{g}$. With $\overline{\mathfrak{g}}$ a certain maximal finite dimensional simple Lie subalgebra of $\mathfrak{g}$, we examine the set $W_s$ of minimal right coset representatives of $\overline{W}$ in $W$, where $\overline{W}$ is the Weyl-Coxeter group of $\overline{\mathfrak{g}}$. For $w\in W_s$, we show that $T(w)$ has the shape of a partition (or a slight variation thereof) whose $r$-core takes a particularly simple form, where $r$ or $r/2$ is the dual Coxeter number of $\mathfrak{g}$. Indeed, it is shown that $W_s$ is in bijection with such partitions.

THE CRITICAL REPRESENTATIONS OF AFFINE LIE ALGEBRAS

1986 ◽  
Vol 01 (10) ◽  
pp. 557-564 ◽  
Author(s):  
D. ALTSCHÜLER
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Central Charge ◽  
Affine Lie Algebras ◽  
Affine Algebra ◽  
Open Problems ◽  
Simple Lie Algebra ◽  
Coxeter Number ◽  
Dual Coxeter Number ◽  
Current Algebras

A critical representation of an affine algebra Ĝ is a representation with central charge k=−g, g being the dual Coxeter number of the underlying simple Lie algebra G. These representations arise naturally in the study of conformal current algebras and BRS cohomology. The author shows how to construct them explicitly in a number of cases, and some intriguing open problems are mentioned.

scholarly journals GROUP THEORY FACTORS FOR FEYNMAN DIAGRAMS

1999 ◽  
Vol 14 (01) ◽  
pp. 41-96 ◽  
Author(s):  
T. VAN RITBERGEN ◽  
A. N. SCHELLEKENS ◽  
J. A. M. VERMASEREN
Keyword(s):  
Group Theory ◽  
Feynman Diagrams ◽  
Computer Algorithms ◽  
Coxeter Number ◽  
Exceptional Algebras ◽  
Dual Coxeter Number ◽  
Invariant Tensors

We present algorithms for the group independent reduction of group theory factors of Feynman diagrams. We also give formulas and values for a large number of group invariants in which the group theory factors are expressed. This includes formulas for various contractions of symmetric invariant tensors, formulas and algorithms for the computation of characters and generalized Dynkin indices and trace identities. Tables of all Dynkin indices for all exceptional algebras are presented, as well as all trace identities to order equal to the dual Coxeter number. Further results are available through efficient computer algorithms.

scholarly journals Peterson-type dimension formulas for graded Lie superalgebras

2001 ◽  
Vol 163 ◽  
pp. 107-144 ◽  
Author(s):  
Seok-Jin Kang ◽  
Jae-Hoon Kwon ◽  
Young-Tak Oh
Keyword(s):  
Abelian Group ◽  
Differential Operators ◽  
Free Abelian Group ◽  
Recursive Formula ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Finiteness Conditions ◽  
Highest Weight ◽  
Multiplicity Formula ◽  
Denominator Identity

Let be a free abelian group of finite rank, let Γ be a sub-semigroup of satisfying certain finiteness conditions, and let be a (Γ × Z2)-graded Lie superalgebra. In this paper, by applying formal differential operators and the Laplacian to the denominator identity of , we derive a new recursive formula for the dimensions of homogeneous subspaces of . When applied to generalized Kac-Moody superalgebras, our formula yields a generalization of Peterson’s root multiplicity formula. We also obtain a Freudenthal-type weight multiplicity formula for highest weight modules over generalized Kac-Moody superalgebras.

scholarly journals Irreducible modules with highest weight vectors over modular Witt and special Lie superalgebras

2019 ◽  
Vol 17 (1) ◽  
pp. 1381-1391
Author(s):  
Keli Zheng ◽  
Yongzheng Zhang
Keyword(s):  
Lie Superalgebras ◽  
Arbitrary Field ◽  
Special Linear ◽  
Highest Weight ◽  
Irreducible Modules

Abstract Let 𝔽 be an arbitrary field of characteristic p > 2. In this paper we study irreducible modules with highest weight vectors over Witt and special Lie superalgebras of 𝔽. The same irreducible modules of general and special linear Lie superalgebras, which are the 0-th part of Witt and special Lie superalgebras in certain ℤ-grading, are also considered. Then we establish a certain connection called a P-expansion between these modules.

scholarly journals Domain walls in 4d $$ \mathcal{N} $$ = 1 SYM

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Diego Delmastro ◽  
Jaume Gomis
Keyword(s):  
Domain Wall ◽  
Domain Walls ◽  
Simons Theory ◽  
Partition Functions ◽  
Quantum Field Theories ◽  
Yang Mills ◽  
Stable Domain ◽  
Coxeter Number ◽  
Topological Quantum ◽  
Simply Connected

Abstract 4d$$ \mathcal{N} $$ N = 1 super Yang-Mills (SYM) with simply connected gauge group G has h gapped vacua arising from the spontaneously broken discrete R-symmetry, where h is the dual Coxeter number of G. Therefore, the theory admits stable domain walls interpolating between any two vacua, but it is a nonperturbative problem to determine the low energy theory on the domain wall. We put forward an explicit answer to this question for all the domain walls for G = SU(N), Sp(N), Spin(N) and G2, and for the minimal domain wall connecting neighboring vacua for arbitrary G. We propose that the domain wall theories support specific nontrivial topological quantum field theories (TQFTs), which include the Chern-Simons theory proposed long ago by Acharya-Vafa for SU(N). We provide nontrivial evidence for our proposals by exactly matching renormalization group invariant partition functions twisted by global symmetries of SYM computed in the ultraviolet with those computed in our proposed infrared TQFTs. A crucial element in this matching is constructing the Hilbert space of spin TQFTs, that is, theories that depend on the spin structure of spacetime and admit fermionic states — a subject we delve into in some detail.

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