arXiv subjects
Publications and source records attributed to Jonathan R. Kujawa.
We introduce and study a one-parameter family of algebras that naturally generalize the Schur algebras. We show the Schur algebra is canonically a quotient when the parameter is a nonnegative integer, characterize when they are semisimple, show they are based quasi-hereditary, and that their category of representations is a highest weight category that can be identified as a subcategory of parabolic category $\mathcal{O}$ for the general linear Lie algebra.
We consider the finite Weyl groups of classical type -- $W(A_{r})$ for $r \geq 1$, $W(B_{r}) = W(C_{r})$ for $r \geq 2$, and $W(D_{r})$ for $r \geq 4$ -- as supergroups in which the reflections are of odd superdegree. Viewing the corresponding complex group algebras as Lie superalgebras via the graded commutator bracket, we determine the structure of the Lie sub-superalgebras generated by the sets of reflections. In each case, this Lie superalgebra is equal to the full derived subalgebra of the group algebra plus the span of the class sums of the reflections.
We consider the group algebra over the field of complex numbers of the Weyl group of type B (the hyperoctahedral group, or the group of signed permutations) and of the Weyl group of type D (the demihyperoctahedral group, or the group of even-signed permutations), viewed as Lie algebras via the commutator bracket, and determine the structure of the Lie subalgebras generated by the sets of reflections.
Let $A$ be an associative superalgebra over a field of characteristic zero. Let $n \geq d+1$. The main result of the paper establishes an equivalence of categories between supermodules for the wreath product $ S_{d} \wr A$ and an explicitly defined category of supermodules for the general linear Lie algebra $\mathfrak{gl}_{n}(A)$. We also give an example showing the bound $n \geq d+1$ cannot be improved.
Let $\mathbb{k}$ be a characteristic zero domain. We define and study a diagrammatic monoidal $\mathbb{k}$-linear supercategory $\mathbf{Web}^{aff}_{A}$ associated to any locally unital Frobenius $\mathbb{k}$-superalgebra $A$. This category can be viewed variously as an affinization of the finite web category $\mathbf{Web}_{A}$ previously defined by the authors and Zhu, as a thickening of the degenerate affine wreath product algebras defined by Savage, or as a Frobenius deformation of affine web categories defined by Song and Wang. We show that there is an asymptotically faithful family of functors from $\mathbf{Web}^{aff}_{A}$ to the monoidal supercategory of endofunctors of $\mathfrak{gl}_n(A)$-modules for every $n \geq 1$, and use this to establish a basis of `decorated double coset diagrams' for morphism spaces in $\mathbf{Web}^{aff}_{A}$. We also define and establish basis results for the cyclotomic quotient category $\mathbf{Web}^Λ_{A}$ associated with a cyclotomic datum $Λ$.
We consider the group algebra of the symmetric group as a superalgebra, and describe its Lie subsuperalgebra generated by the transpositions. The updated version corrects some of the arguments made in Sections 4.5 - 4.7. The statements of the main results are unaffected.
This paper introduces type P web supercategories. They are defined as diagrammatic monoidal $k$-linear supercategories via generators and relations. We study the structure of these categories and provide diagrammatic bases for their morphism spaces. We also prove these supercategories provide combinatorial models for the monoidal supercategory generated by the symmetric powers of the natural module and their duals for the Lie superalgebra of type P.
We show that the transition matrix from the standard basis to the web basis for a Specht module of the Hecke algebra is unitriangular and satisfies a strong positivity property whenever the Specht module is labeled by a partition with at most two parts. This generalizes results of Russell--Tymoczko and Rhoades.
Let $\mathbb{k}$ be a characteristic zero domain. For a locally unital $\mathbb{k}$-superalgebra $A$ with distinguished idempotents $I$and even subalgebra $a \subseteq A_{\bar 0}$, we define and study an associated diagrammatic monoidal $\mathbb{k}$-linear supercategory $\mathbf{Web}^{A,a}_I$. This supercategory yields a diagrammatic description of the generalized Schur algebras $T^A_a(n,d)$. We also show there is an asymptotically faithful functor from $\mathbf{Web}^{A,a}_I$ to the monoidal supercategory of $\mathfrak{gl}_n(A)$-modules generated by symmetric powers of the natural module. When this functor is full, the single diagrammatic supercategory $\mathbf{Web}^{A,a}_I$ provides a combinatorial description of this module category for all $n \geq 1$. We also use these results to establish Howe dualities between $\mathfrak{gl}_{m}(A)$ and $\mathfrak{gl}_{n}(A)$ when $A$ is semisimple.
We survey the current state of various support variety theories for Lie superalgebras and finite supergroup schemes. We pay particular attention to the theory in characteristic zero developed by Boe, Kujawa, and Nakano using relative Lie superalgebra cohomology, and to the theory developed in positive characteristic in our previous work.
This paper investigates cohomology and support varieties for Lie superalgebras and restricted Lie superalgebras over a field of characteristic 2. The existence of an underlying ordinary Lie algebra allows us to obtain results that are still open in odd characteristic, and also to establish results that have no non-super analogues in characteristic 2.
We adapt a construction due to Troesch to the category of strict polynomial superfunctors in order to construct complexes of injective objects whose cohomology is isomorphic to Frobenius twists of the (super)symmetric power functors. We apply these complexes to construct injective resolutions of the even and odd Frobenius twist functors, to investigate the structure of the Yoneda algebra of the Frobenius twist functor, and to compute other extension groups between strict polynomial superfunctors.
We establish classical and categorical Howe dualities between the Lie superalgebras $\mathfrak{p}(m)$ and $\mathfrak{p}(n)$, for $m,n \geq 1$. We also describe a presentation via generators and relations as well as a Kostant $\mathbb{Z}$-form for the universal enveloping superalgebra $U(\mathfrak{p}(m))$.
We compute the complexity, z-complexity, and support varieties of the (thick) Kac modules for the Lie superalgebras of type P. We also show the complexity and the z-complexity have geometric interpretations in terms of support and associated varieties; these results are in agreement with formulas previously discovered for other classes of Lie superalgebras. Our main technical tool is a recursive algorithm for constructing projective resolutions for the Kac modules. The indecomposable projective summands which appear in a given degree of the resolution are explicitly described using the combinatorics of weight diagrams. Surprisingly, the number of indecomposable summands in each degree can be computed exactly: we give an explicit formula for the corresponding generating function.
Webs are combinatorial diagrams used to encode homomorphisms between representations of Lie (super)algebras and related objects. This paper extends the theory of webs to the quantum group of type Q. We define a monoidal supercategory of quantum type Q webs and show it admits a full, essentially surjective functor onto the monoidal supercategory of $U_q(\mathfrak{q}_n)$-modules generated by the quantum symmetric powers of the natural representation and their duals. We also show that a certain subcategory of the web category is a ribbon category and discuss applications to the representation theory of $U_q(\mathfrak{q}_n)$ and to invariants of oriented, framed links.
We investigate cohomological support varieties for finite-dimensional Lie superalgebras defined over fields of odd characteristic. Verifying a conjecture from our previous work, we show the support variety of a finite-dimensional supermodule can be realized as an explicit subset of the odd nullcone of the underlying Lie superalgebra. We also show the support variety of a finite-dimensional supermodule is zero if and only if the supermodule is of finite projective dimension. As a consequence, we obtain a positive characteristic version of a theorem of Bøgvad, showing that if a finite-dimensional Lie superalgebra over a field of odd characteristic is absolutely torsion free, then its enveloping algebra is of finite global dimension.
Let $\mathfrak g$ be a complex simple Lie algebra and let $U_ζ({\mathfrak g})$ be the corresponding Lusztig ${\mathbb Z}[q,q^{-1}]$-form of the quantized enveloping algebra specialized to an $\ell$th root of unity. Moreover, let $\mod(U_ζ({\mathfrak g}))$ be the braided monoidal category of finite-dimensional modules for $U_ζ({\mathfrak g})$. In this paper we classify the thick tensor ideals of $\mod(U_ζ({\mathfrak g}))$ and compute the prime spectrum of the stable module category associated to $\text{mod}(U_ζ({\mathfrak g}))$ as defined by Balmer.
We investigate support schemes for infinitesimal unipotent supergroups and their representations. Our main results provide a non-cohomological description of these schemes which generalizes the classical work of Suslin, Friedlander, and Bendel. As a consequence, support schemes in this setting have the desired features of such a theory, including naturality with respect to group homomorphisms, the tensor product property, and realizability. As an application of the theory developed here, we investigate support varieties for certain finite-dimensional Hopf subalgebras of the Steenrod algebra.