# The cohomological Hall algebra of a preprojective algebra

Y. Yang, G. Zhao, Proceedings of the London Mathematical Society 116 (2018) 1029–1074.

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Author

Yang, Yaping;
Zhao, Gufang

^{IST Austria}Department

Abstract

We introduce for each quiver Q and each algebraic oriented cohomology theory A, the cohomological Hall algebra (CoHA) of Q, as the A-homology of the moduli of representations of the preprojective algebra of Q. This generalizes the K-theoretic Hall algebra of commuting varieties defined by Schiffmann-Vasserot. When A is the Morava K-theory, we show evidence that this algebra is a candidate for Lusztig's reformulated conjecture on modular representations of algebraic groups.
We construct an action of the preprojective CoHA on the A-homology of Nakajima quiver varieties. We compare this with the action of the Borel subalgebra of Yangian when A is the intersection theory. We also give a shuffle algebra description of this CoHA in terms of the underlying formal group law of A. As applications, we obtain a shuffle description of the Yangian.

Publishing Year

Date Published

2018-05-01

Journal Title

Proceedings of the London Mathematical Society

Volume

116

Issue

5

Page

1029-1074

ISSN

IST-REx-ID

### Cite this

Yang Y, Zhao G. The cohomological Hall algebra of a preprojective algebra.

*Proceedings of the London Mathematical Society*. 2018;116(5):1029-1074. doi:10.1112/plms.12111Yang, Y., & Zhao, G. (2018). The cohomological Hall algebra of a preprojective algebra.

*Proceedings of the London Mathematical Society*,*116*(5), 1029–1074. https://doi.org/10.1112/plms.12111Yang, Yaping, and Gufang Zhao. “The Cohomological Hall Algebra of a Preprojective Algebra.”

*Proceedings of the London Mathematical Society*116, no. 5 (2018): 1029–74. https://doi.org/10.1112/plms.12111.Y. Yang and G. Zhao, “The cohomological Hall algebra of a preprojective algebra,”

*Proceedings of the London Mathematical Society*, vol. 116, no. 5, pp. 1029–1074, 2018.Yang Y, Zhao G. 2018. The cohomological Hall algebra of a preprojective algebra. Proceedings of the London Mathematical Society. 116(5), 1029–1074.

Yang, Yaping, and Gufang Zhao. “The Cohomological Hall Algebra of a Preprojective Algebra.”

*Proceedings of the London Mathematical Society*, vol. 116, no. 5, Oxford University Press (OUP), 2018, pp. 1029–74, doi:10.1112/plms.12111.**All files available under the following license(s):**

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arXiv 1407.7994