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1. G-quadratic, LG-quadratic, and Koszul quotients of exterior algebras

   https://doi.org/10.1080/00927872.2022.2029875  

   McCullough, Jason; Mere, Zachary (January 2022, Communications in Algebra)

   This paper introduces the study of LG-quadratic quotients of exterior algebras, showing that they are Koszul, as in the commutative case. We construct an example of an LG-quadratic algebra that is not G-quadratic and another example that is Koszul but not LG-quadratic. This is only the second known Koszul algebra that is not LG-quadratic and the first that is noncommutative. 

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2. A category $$\mathcal{O}$$ for oriented matroids

   https://doi.org/10.4171/JCA/71  

   Kowalenko, Ethan; Mautner, Carl (June 2023, Journal of Combinatorial Algebra)

   We associate to a sufficiently generic oriented matroid program and choice of linear system of parameters a finite-dimensional algebra, whose representation theory is analogous to blocks of Bernstein—Gelfand—Gelfand category\mathcal{O}. When the data above comes from a generic linear program for a hyperplane arrangement, we recover the algebra defined by Braden—Licata—Proudfoot—Webster. Applying our construction to non-linear oriented matroid programs provides a large new class of algebras. For Euclidean oriented matroid programs, the resulting algebras are quasi-hereditary and Koszul, as in the linear setting. In the non-Euclidean case, we obtain algebras that are not quasi-hereditary and not known to be Koszul, but still have a natural class of standard modules and satisfy numerical analogues of quasi-heredity and Koszulity on the level of graded Grothendieck groups. 

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3. Cyclic pairings and derived Poisson structures.

   Ramadoss, Ajay C; Zhang, Yining (April 2019, New York journal of mathematics)

   null (Ed.)

   By a fundamental theorem of D. Quillen, there is a natural duality - an instance of general Koszul duality - between differential graded (DG) Lie algebras and DG cocommutative coalgebras defined over a field k of characteristic 0. A cyclic pairing (i.e., an inner product satisfying a natural cyclicity condition) on the cocommutative coalgebra gives rise to an interesting structure on the universal enveloping algebra Ua of the Koszul dual Lie algebra a called the derived Poisson bracket. Interesting special cases of the derived Poisson bracket include the Chas-Sullivan bracket on string topology. We study the derived Poisson brackets on universal enveloping algebras Ua, and their relation to the classical Poisson brackets on the derived moduli spaces DRep_g(a) of representations of a in a finite dimensional reductive Lie algebra g. More specifically, we show that certain derived character maps of a intertwine the derived Poisson bracket with the classical Poisson structure on the representation homology HR(a, g) related to DRep_g(a). 

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4. Canonical resolutions over Koszul algebras

   https://doi.org/10.1007/978-3-030-91986-3_11  

   Faber, Eleonore; Juhnke-Kubitzke, Martina; Lindo, Haydee; Miller, Claudia; R. G., Rebecca; Seceleanu, Alexandra (January 2021, Association for Women in Mathematics series)

   Miller, Claudia; Striuli, Janet; Witt, Emily (Ed.)

   We generalize Buchsbaum and Eisenbud’s resolutions for the powers of the maximal ideal of a polynomial ring to resolve powers of the homogeneous maximal ideal over graded Koszul algebras. 

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5. The minimal free resolution of a general principal symmetric ideal

   https://doi.org/10.1090/tran/9314  

   Harada, Megumi; Seceleanu, Alexandra; Şega, Liana (December 2024, Transactions of the American Mathematical Society)

   We introduce the class of principal symmetric ideals, which are ideals generated by the orbit of a single polynomial under the action of the symmetric group. Fixing the degree of the generating polynomial, this class of ideals is parametrized by points in a suitable projective space. We show that the minimal free resolution of a principal symmetric ideal is constant on a non-empty Zariski open subset of this projective space and we determine this resolution explicitly. Along the way, we study two classes of graded algebras which we term narrow and extremely narrow; both of which are instances of compressed artinian algebras. 

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