Representation Theory of Finite Groups and Associative by Charles W. Curtis
By Charles W. Curtis
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Extra resources for Representation Theory of Finite Groups and Associative Algebras
3) The proof above can be easily modified to construct a C-algebra isomorphism EH,C : D ,C (n) → UC (gln ), where the algebras are defined over C with respect to a non-root-of-unity z ∈ C∗ .
1) is defined over A. Hence, we can form a double Ringel–Hall algebra D (n)A . Then D ,C (n) ∼ = D (n)A ⊗ C and D (n) = D (n)A ⊗ Q(v). 2. Schiffmann–Hubery generators In this and the following sections, we will investigate the structure of D (n) by relating it with the quantum enveloping algebra of a generalized Kac–Moody algebra based on [67, 39]; see also [38, 14]. 1). Recall that an element of a Hopf algebra with comultiplication is called primitive if 38 2. Double Ringel–Hall algebras of cyclic quivers (x) = x ⊗ 1 + 1 ⊗ x.
Also, let U(gln )0 be i,0 , εn , θ−s ), for 1 the subalgebra of U(gln ) generated by the ki±1 . 1) induces that of U(gln ). 4. The multiplication map U(gln )+ ⊗ U(gln )0 ⊗ U(gln )− −→ U(gln ) is a Q(v)-space isomorphism. 5. 12)]. Indeed, under the isomorphism EH , C is identified with the central subalgebra Z (n) of D (n). (2) In  a Hopf algebra isomorphism H (n) 0 → U(gln )+ ⊗ U(gln )0 −1 was established and, moreover, the elements EH−1 (x+j,−1 k−1 j ) and EH (gi,±s ) in D (n) were explicitly described.