Formal sums over a vector space
Let π vector space over π π { π§ } formal sums over π π§ degree operator 1 fcalc
π· = π· π§ = π§ π π π§ , where π π π§ Formal derivative .
This is not strictly a degree operator , as π { π§ } graded vector space ,
however its subspaces π [ π§ ] π [ π§ , π§ β 1 ]
Properties
Over a graded vector space
Now taking π π graded with degree operator π β E n d β‘ π
π£ ( π§ ) = β π β π π£ π π§ π β π { π§ } π ( π§ ) = β π β π π₯ ( π ) π§ β π β ( E n d β‘ π ) { π§ } we have
π· π£ ( π§ ) = π π£ ( π§ ) βΊ ( β π β π ) [ d e g β‘ π£ π = π ] β π· π ( π§ ) = [ π , π ( π§ ) ] βΊ ( β π β π ) [ d e g β‘ π₯ ( π ) = π ] Let πΏ ( π§ ) β π [ [ π§ , π§ β 1 ] ] Formal delta . Then it follows from Properties that for any π£ ( π§ ) β π£ [ π§ , π§ β 1 ] π β π Γ
π£ ( π§ ) π· πΏ ( π π§ ) = π£ ( π β 1 ) π· πΏ ( π π§ ) β ( π· π£ ) ( π β 1 ) πΏ ( π π§ ) and that for any π ( π§ 1 , π§ 2 ) β ( E n d β‘ π ) [ [ π§ 1 , π§ β 1 1 , π§ 2 , π§ β 1 2 ] ] l i m π§ 1 β π§ 2 π ( π§ 1 , π§ 2 ) π β π Γ
π ( π§ 1 , π§ 2 ) π· 1 πΏ ( π π§ 1 / π§ 2 ) = π ( π β 1 π§ 2 , π§ 2 ) π· 1 πΏ ( π π§ 1 / π§ 2 ) β ( π· 1 π ) ( π β 1 π§ 2 , π§ 2 ) πΏ ( π π§ 1 / π§ 2 ) π ( π§ 1 , π§ 2 ) π· 2 πΏ ( π π§ 1 / π§ 2 ) = π ( π§ 1 , π π§ 1 ) π· 2 πΏ ( π π§ 1 / π§ 2 ) β ( π· 2 π ) ( π§ 1 , π π§ 1 ) πΏ ( π π§ 1 / π§ 2 ) where π· 1 = π· π§ 1 π· 2 = π· π§ 2
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