Braid symmetries on bosonic extensions

We introduce a family of automorphisms on the bosonic extension of arbitrary type and show that they satisfy the braid relations. They preserve the global basis and the crystal basis. Using this braid group action, we define a subalgebra for each positive braid word, which possesses the PBW type basis. As an application, we show that the tensor product decomposition of the positive bosonic extionsion
Comment: 38 pages