# Tensor Representations of 𝔮 ( ∞ ) ${\mathfrak {q}} (\infty )$

Tensor Representations of 𝔮 ( ∞ ) ${\mathfrak {q}} (\infty )$ Algebr Represent Theor https://doi.org/10.1007/s10468-018-9803-7 Tensor Representations of q(∞) 1 2 D. Grantcharov · V. Serganova Received: 27 August 2017 / Accepted: 17 May 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018 Abstract We introduce a symmetric monoidal category of modules over the direct limit queer superalgebra q(∞). This category can be defined in two equivalent ways with the aid of the large annihilator condition. Tensor products of copies of the natural and the conatural representations are injective objects in this category. We obtain the socle filtrations and formulas for the tensor products of the indecomposable injectives. In addition, it is proven that the category is Koszul self-dual. Keywords Queer superalgebras · Tensor representations · Koszul duality Mathematics Subject Classification (2010) 17B65 · 17B10 · 16G10 1 Introduction Recently new symmetric monoidal categories have attracted considerable attention. Among them are the categories Trep g of modules over direct limit g of classical Lie algebras gen- erated as abelian tensor categories by the natural and conatural representations. Namely, g is one of the following: gl(∞) = lim gl(n), o(∞) = lim o(n) and sp(∞) = lim sp(n). −→ −→ −→ In [2] it is proven that these categories http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Algebras and Representation Theory Springer Journals

# Tensor Representations of 𝔮 ( ∞ ) ${\mathfrak {q}} (\infty )$

, Volume OnlineFirst – May 30, 2018
27 pages

/lp/springer_journal/tensor-representations-of-mathfrak-q-infty-k9nbqvhll4
Publisher
Springer Netherlands
Subject
Mathematics; Commutative Rings and Algebras; Associative Rings and Algebras; Non-associative Rings and Algebras
ISSN
1386-923X
eISSN
1572-9079
D.O.I.
10.1007/s10468-018-9803-7
Publisher site
See Article on Publisher Site

### Abstract

Algebr Represent Theor https://doi.org/10.1007/s10468-018-9803-7 Tensor Representations of q(∞) 1 2 D. Grantcharov · V. Serganova Received: 27 August 2017 / Accepted: 17 May 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018 Abstract We introduce a symmetric monoidal category of modules over the direct limit queer superalgebra q(∞). This category can be defined in two equivalent ways with the aid of the large annihilator condition. Tensor products of copies of the natural and the conatural representations are injective objects in this category. We obtain the socle filtrations and formulas for the tensor products of the indecomposable injectives. In addition, it is proven that the category is Koszul self-dual. Keywords Queer superalgebras · Tensor representations · Koszul duality Mathematics Subject Classification (2010) 17B65 · 17B10 · 16G10 1 Introduction Recently new symmetric monoidal categories have attracted considerable attention. Among them are the categories Trep g of modules over direct limit g of classical Lie algebras gen- erated as abelian tensor categories by the natural and conatural representations. Namely, g is one of the following: gl(∞) = lim gl(n), o(∞) = lim o(n) and sp(∞) = lim sp(n). −→ −→ −→ In [2] it is proven that these categories

### Journal

Algebras and Representation TheorySpringer Journals

Published: May 30, 2018

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