Typical and atypical modules for Lie superalgebras

Question

There are two types highest weight representations for a Basic classical simple Lie superalgebra $\mathfrak{g}$ which are defined as typical (representation for which highest weight vector is the only vector killed by generators corresponding to positive roots) and atypical (not typical) but there is no such distinction in case of Complex semisimple Lie algebras.

Can anybody please explain to me what is happening in the super case? It has to do something with odd simple roots. But I am interested in understanding it clearly.

Also, how the definition of typicality of $V(\Lambda)$ is equivalent to the condition that $(\Lambda+\rho)(\alpha) \ne 0 \,\forall\, \alpha \in \bar{\Delta}_1^{+}$ ? where $\bar{\Delta}_1^{+}$ is the set of all odd roots whose twise is not an even root.

Thank you.

PS: Kindly tell me what is known about the denominator identity for $\mathfrak g$ and the character formula for atypical modules. Thanks again.

Answer 1

Regarding the "what is happening in the super case"; yes i agree that in some sense, it has to do with the odd simple roots but i think it is deeper than that:
In the case of semisimple, complex, Lie algebras, every reducible representation is completely reducible. However, this result is not true for the basic, classical, simple, complex, Lie superalgebras (BCSCLS for short). One might hope that the reducible reps which are not completely reducible constitute somewhat exceptional cases; however this is not true either. This situation, has led to distinguishing the graded, irreducible reprepesentations of BCSCLS into two classes: the typical and the atypical ones, by means of the following definition:

An irreducible, graded rep $V(\Lambda)$ of a BCSCLS, $L_s$, with highest weight $\Lambda$, is defined to be typical, if any reducible, graded, rep of $L_s$ with highest weight $\Lambda$, can be written as a direct sum of $V(\Lambda)$ with some other graded rep of $L_s$.
An irreducible, graded rep which is not typical, is defined to be atypical.

The above definition implies that if $V(\Lambda)$ is an atypical graded representation of $L_s$ with highest weight $\Lambda$, then there exists at least one graded, reducible representation of $L_s$ with highest weight $\Lambda$, that is not completely reducible.
V. G. Kac, has shown the following theorem:

An irreducible, graded representation of a BCSCLS, with highest weight $\Lambda$, is atypical if and only if $$ (\Lambda+\rho,\alpha)=0 $$ for some positive root $\alpha \in \bar{\Delta}_1^{+}$.

For the proof see:

• Representations of classical lie superalgebras (see Theorem 1, p. 615-616),
• Characters of typical representations of classical lie superalgebras

The condition $(\Lambda+\rho,\alpha)=(\Lambda+\rho)(\alpha) \ne 0$, $\forall\, \alpha \in \bar{\Delta}_1^{+}$ you are refering to, is essentially the "translation" of the verbal description of typicality provided in the first paragraph of the OP.

For some more references, you can find a sketch of these results on Dictionary on Lie superalgebras, by Frappat, Sorba and Sciarrino, section 39, p. 58-59.

Furthermore, the character formula for the singly atypical representations of the LS $A(m,n)$ and $C(n+1)$, is provided in section 7, p. 11–12 of the same reference. For more details see:

• A character formula for singly atypical modules of the lie superalgebra sl(m/n),

• Character formulas for irreducible modules of the Lie superalgebras sl(m/n),

• Dimension formulas for the Lie superalgebra sl(m/n)

• A conjectured character formula for atypical irreducible modules of the Lie superalgebra $sl(m/n)$ (where multiply atypical representations are also discussed),

  Finally, for some more classes of Lie superalgebras, including $gl(m/n)$, $osp(m/2n)$, $q(n)$ and some affine Lie superalgebras, the character formulae for their atypical modules and denominator identities, see the discussion and the references at:

• Mathematical Foundations of Supersymmetry, appendix A, sect. A.6, p. 254-256