Diagrammatics for real supergroups (Q6580066)

Many recent developments in representation theory involve one or more of the following interrelated concepts: dual pairs, invariant theory and interpolating categories. In the case of the general linear group, the connection between the above concepts is as follows. The oriented Brauer category \(\mathcal{OB}(d)\) is the free rigid symmetric \(\mathbb C\)-linear monoidal category on a generating object of categorical dimension d. Since the category of modules over the general linear group \(GL(m, C), m \in N\), is rigid symmetric monoidal, there exists a functor\N\[\NG : \mathcal{OB}(m) \to GL(m, C)\text{-mod}\N\]\NThe fullness of \(G\) is sometimes referred to as the first fundamental theorem of invariant theory. (Describing the kernel is the second fundamental theorem.)\N\NAn analogous picture exists for the orthogonal and symplectic groups. In these cases, the natural module is self-dual. Thus, the oriented Brauer category is replaced by the unoriented Brauer category \(\mathcal B(d)\), which is the free rigid symmetric \(\Bbbk\)-linear monoidal category on a symmetrically self-dual object of categorical dimension \(d\). The move to the super world also leads to additional free categories. First, one observes that an isomorphism of a module with its dual can be even or odd. The even case corresponds to the Brauer category. The odd case leads to the periplectic Brauer supercategory \(\mathcal B^1\), which is the free rigid symmetric \( \Bbbk\)-linear monoidal supercategory on an odd-self-dual object (which necessarily has categorical dimension zero). Another free supercategory arises from the super version of Schur's lemma. Since we work over the complex numbers, Schur's lemma implies that the endomorphism algebra of a simple module is a complex division superalgebra. In the non-super setting, the only possibility is \(\mathbb C\). However, in the super setting, there is one additional possibility, which is the two-dimensional complex Clifford superalgebra \(Cl(\mathbb C)\). This observation leads to the definition of the oriented Brauer-Clifford category \(\mathcal{OBC}\) of [\textit{J. Brundan} et al., Can. J. Math. 71, No. 5, 1061--1101 (2019; Zbl 1470.17005)], which is the free rigid symmetric monoidal supercategory on a generating object whose endomorphism algebra is \(Cl(\mathbb C)\).\N\NThe goal of this paper is to extend this line of research to real supergroups instead of complex ones. To do so, first to any associative superalgebra A over a field \(\Bbbk\), a diagrammatic supercategory \(\mathcal{OB}_{\Bbbk}(A)\) is introduced, which is the free rigid symmetric monoidal supercategory on an object with endomorphism superalgebra A. For \(A = \Bbbk\) this is the oriented Brauer category. Its universal property implies that for any Lie superalgebra \(g\), and any \((g, A)\)-superbimodule \(V\), there is an oriented incarnation superfunctor\N\[\N\mathcal{OB}_{\Bbbk}(A^{\text{op}}) \to g\text{-smod}\N\]\NOne main result of this paper (Theorem 6.10) is that, when \(A\) is a central real division superalgebra and \(V = A^{m|n}\), the functor \(\mathcal{OB}_{\mathbb R}(A^{\text{op}}; m\text{-}n) \to gl(m|n, A)\text{-smod}\) is full.\N\NThen the unoriented (i.e. self-dual) cases are studied. There, the situation is a bit more involved, since one must carefully analyze which types of self-duality can arise. It turns out that only four of the ten real division superalgebras admit anti-involutions: the real numbers, the complex numbers, the quaternions, and the two-dimensional complex Clifford superalgebra. The second main result of this paper (Theorem 10.5) is that, in these cases, the unoriented incarnation superfunctor is full.\N\NTaking the oriented and unoriented cases together, these results handle real supergroups corresponding to all real forms of the general linear, orthosymplectic, periplectic, and isomeric Lie superalgebras. As another application, equivalences between supercategories of tensor supermodules over the different real forms of a complex supergroup are deduced.