R. Fioresi
We show a correspondence between tensor representations of the super general linear group GL(m|n) and tensor representations of the general linear superalgebra gl(m|n) constructed by Berele and Regev.
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We give conditions for unitarizability of Harish-Chandra super modules for Lie supergroups and superalgebras.
We examine the cohomology and representation theory of a family of finite supergroup schemes of the form $(\mathbb G_a^-\times \mathbb G_a^-)\rtimes (\mathbb G_{a(r)}\times (\mathbb Z/p)^s)$. In particular, we show that a certain relation holds in the cohomology ring, and deduce that for finite supergroup schemes having this as a quotient, both cohomology mod nilpotents and projectivity of modules is detected on proper sub-super\-group schemes. This special case feeds into the proof of a more general detection theorem for unipotent finite supergroup schemes, in a separate work of the authors j…
By using certain quantum differential operators, we construct a super representation for the quantum queer supergroup U_v(q_n). The underlying space of this representation is a deformed polynomial superalgebra in 2n^2 variables whose homogeneous components can be used as the underlying spaces of queer q-Schur superalgebras. We then extend the representation to its formal power series algebra which contains a (super) submodule isomorphic to the regular representation of U_v(q_n). A monomial basis M for U_v(q_n) plays a key role in proving the isomorphism. In this way, we may present the quantum…
We give a combinatorial construction for the canonical bases of the $\pm$-parts of the quantum enveloping superalgebra $\bfU(\mathfrak{gl}_{m|n})$ and discuss their relationship with the Kazhdan-Lusztig bases for the quantum Schur superalgebras $\bsS(m|n,r)$ introduced in \cite{DR}. We will also extend this relationship to the induced bases for simple polynomial representations of $\bfU(\mathfrak{gl}_{m|n})$.
Let k be any field. J-P. Serre proved that the spectrum of the Grothendieck ring of the k-representation category of a group is connected, and that the same holds in characteristic zero for the representation category of a Lie algebra over k. We say that a tensor category C over k is virtually indecomposable if its Grothendieck ring contains no nontrivial central idempotents. We prove that the following tensor categories are virtually indecomposable: Tensor categories with the Chevalley property; representation categories of affine group schemes; representation categories of formal groups; rep…
The purpose of this paper is to extend the theory of Super Harish-Chandra pairs, originally developed by Koszul for Lie supergroups, to analytic and algebraic supergroups, in order to obtain information also about their representations. We also define the distribution superalgebra for algebraic and analytic supergroups and study its relation with the universal enveloping superalgebra
Invariant Einstein metrics on basic classical Lie supergroups
本文系统研究基本经典李超群上的不变爱因斯坦度量，通过将爱因斯坦条件化为可解代数系统，证明除少数例外情形外，每个实基本经典李超代数至少容许两个不同的爱因斯坦度量，并在特定系列中发现了非超情形未见的Ricci平坦与非Ricci平坦度量共存现象。
The real compact supergroup $S^{1|1}$ is analized from different perspectives and its representation theory is studied. We prove it is the only (up to isomorphism) supergroup, which is a real form of $({\mathbf C}^{1|1})^\times$ with reduced Lie group $S^1$, and a link with SUSY structures on ${\mathbf C}^{1|1}$ is established. We describe a large family of complex semisimple representations of $S^{1|1}$ and we show that any $S^{1|1}$-representation whose weights are all nonzero is a direct sum of members of our family. We also compute the matrix elements of the members of this family and we g…
Infinitesimal supersymmetries over classical Lie groups that do not necessarily integrate to Lie supergroups are described. They yield a notion of supersymmetry that is less rigid than the assumption of a Lie supergroup action but still implies an underlying action of a Lie group. In contrast to Lie supergroups, the arising representation-theoretical Lie supergroups (RTLSG) occur as families associated to Harish-Chandra superpairs. However morphisms of RTLSGs directly correspond to morphisms of Harish-Chandra superpairs. Particular RTLSGs can be derived from the explicit constructions of Lie s…
We introduce categories of homogeneous strict polynomial functors, $\Pol^\I_{d,\k}$ and $\Pol^\II_{d,\k}$, defined on vector superspaces over a field $\k$ of characteristic not equal 2. These categories are related to polynomial representations of the supergroups $GL(m|n)$ and Q(n), respectively. In particular, we prove an equivalence between $\Pol^\I_{d,\k}$, $\Pol^\II_{d,\k}$ and the category of finite dimensional supermodules over the Schur superalgebra $\Sc(m|n,d)$, $\Qc(n,d)$ respectively provided $m,n \ge d$. We also discuss some aspects of Sergeev duality from the viewpoint of the categ…
We propose to use interpolation categories to study PBW deformations, and demonstrate this idea for the orthosymplectic supergroups. Employing a combinatorial calculus based on pseudographs and partitions which we derive from a suitable Jacobi identity, we classify PBW deformations in (quotients of) Deligne's interpolation categories for the orthosymplectic groups. As special cases, our classification recovers families of infinitesimal Hecke algebras found by Etingof-Gan-Ginzburg (2005) for the symplectic groups and by Tsymbaliuk (2015) for the orthogonal groups together with their respective …
As a homomorphic image of the hyperalgebra $U_{q,R}(m|n)$ associated with the quantum linear supergroup $U_\upsilon(\mathfrak{gl}_{m|n})$, we first give a presentation for the $q$-Schur superalgebra $S_{q,R}(m|n,r)$ over a commutative ring $R$. We then develop a criterion for polynomial supermodules of $U_{q,F}(m|n)$ over a filed $F$ and use this to determine a classification of polynomial irreducible supermodules at roots of unity. This also gives classifications of irreducible $S_{q,F}(m|n,r)$-supermodules for all $r$. As an application when $m=n\geq r$ and motivated by the beautiful work \c…
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