Lie Superalgebras of Matrix Differential Operators David A. RichterThis file was last edited on July 23, 2007. It runs best on Maple, version 9.51.Introduction This package is for mining and exploring Lie superalgebras of matrix differential operators (MDO's). The aim is to obtain a catalogue of examples rich enough for a study of the theory of Lie superalgebras and their actions in the contexts of spectrum-generating algebras and differential geometry. Currently this package is an archive of realizations of Lie superalgebras by MDO's. If one wants to see one of these in action, so to speak, then one should execute the section "Essential Ingredients" and, if desired, choose the most appropriate version of "weylmult". (The three-variable version for MDO's is the default.) If desired or required, one may also execute one or more of the groups in "Clifford algebras". The Clifford algebras are defined inductively, so one cannot use the 16-dimensional Clifford algebra, for example, without using the 4-dimensional Clifford algebra.For example, if one wants to see a realization of $\mathrm{osp}(3,2)$ using $8\times 8$ MDO's in two variables, then one must execute the section "Essential Ingredients" and the first three groups in the section "Clifford algebras" (and the two-variable "weylmult" for MDO's if one desires speedier computation).

The most complicated part of this section is the function "weylmult". This function yields the multiplication between elements of a Weyl algebra or a Weyl-Clifford algebra. (A Weyl-Clifford algebra is a tensor product of a Weyl algebra and a Clifford algebra.) Elements of a Weyl algebra are represented as polynomials in $x$, $p=d/dx$, $y$, $q=d/dy$, $z$, and $r=d/dz$.One finds here a grand total of six versions of the function "weylmult". Two choices determine which function the user may care to use, (a) the number of variables used and (b) whether or not to use the Matrix version. (Presumably, since this package is nominally designed for working with matrix differential operators, one would probably prefer the Matrix version.) The functions "Lie", "Sym", and "Act" should be self-explanatory. Given any Lie superalgebra, one aways needs a commutator ("Lie") and an anti-commutator ("Sym"). If one wants also to consider representations, then one may use the left action ("Act"). The definitions of these do not depend on the version of "weylmult" used.Currently the number of variables is restricted to 3 or fewer. More variables would lead to heavier computations and the costs would start to outweigh the benefits. Recall that the purpose here is merely to find/study specimens. The parameter "mx" tells the maximum order of the differential operators used. As above, using a small value for "mx" saves some time.with(LinearAlgebra): interface(rtablesize=32): The Kronecker delta symbol del:=(m,n)->1-abs(signum(m-n)): The elemenary zero-one matrix em:=(m,n)->Matrix(dm,dm,(i,j)->del(m,i)*del(n,j)): The elementary zero-one vector ev:=m->Vector(dm,i->del(m,i)): The Kroecker tensor product of two matrices, as tailored to suit the needs of this package tens:=proc(m1,m2) local r1,c1,r2,c2,i,j,prd: r1:=RowDimension(m1): c1:=ColumnDimension(m1): r2:=RowDimension(m2): c2:=ColumnDimension(m2): prd:=Matrix(r1*r2,c1*c2,0): for i from 1 to r1*r2 do for j from 1 to c1*c2 do prd[i,j]:=m1[1+floor((i-1)/r2),1+floor((j-1)/c2)]*m2[1+((i-1) mod r2),1+((j-1)mod c2)]: end do: end do: return prd: end proc: These are the standard commutator and anticommutator. Lie:=(f,g)->expand(weylmult(f,g)-weylmult(g,f)): Sym:=(f,g)->expand(weylmult(f,g)+weylmult(g,f)): This is the standard left action of a Weyl(-Clifford) algebra on polynomials/vector polynomials. Act:=(f,g)->subs(p=0,q=0,r=0,weylmult(f,g)): The version of "weylmult" for MDO's in three variables (default): mx:=3: weylmult:= proc(f,g) local i,j,k,s: s:=f.g: for i from 0 to mx do for j from 0 to mx do for k from 0 to mx do if i+j+k>0 then s:=s+map(h->diff(h,p$i,q$j,r$k),f).map(h->diff(h,x$i,y$j,z$k),g)/(i!*j!*k!): end if: end do: end do: end do: return map(expand,s): end proc:

weylmult:= proc(f,g) local i,s: s:=f*g: for i from 1 to mx do s:=s+diff(f,p$i)*diff(g,x$i)/(i!): end do: return expand(s): end proc:

weylmult:= proc(f,g) local i,j,s: s:=f*g: for i from 0 to mx do for j from 0 to mx do if i+j>0 then s:=s+diff(f,p$i,q$j)*diff(g,x$i,y$j)/(i!*j!): end if: end do: end do: return expand(s): end proc:

weylmult:= proc(f,g) local i,j,k,s: s:=f*g: for i from 0 to mx do for j from 0 to mx do for k from 0 to mx do if i+j+k>0 then s:=s+diff(f,p$i,q$j,r$k)*diff(g,x$i,y$j,z$k)/(i!*j!*k!): end if: end do: end do: end do: return expand(s): end proc:

weylmult:= proc(f,g) local i,s: s:=f.g: for i from 1 to mx do s:=s+map(h->diff(h,p$i),f).map(h->diff(h,x$i),g)/i!: end do: return map(expand,s): end proc:

weylmult:= proc(f,g) local i,j,s: s:=f.g: for i from 0 to mx do for j from 0 to mx do if i+j>0 then s:=s+map(h->diff(h,p$i,q$j),f).map(h->diff(h,x$i,y$j),g)/(i!*j!): end if: end do: end do: return map(expand,s): end proc:

weylmult:= proc(f,g) local i,j,k,s: s:=f.g: for i from 0 to mx do for j from 0 to mx do for k from 0 to mx do if i+j+k>0 then s:=s+map(h->diff(h,p$i,q$j,r$k),f).map(h->diff(h,x$i,y$j,z$k),g)/(i!*j!*k!): end if: end do: end do: end do: return map(expand,s): end proc:

If one desires to work with $2^n\times 2^n$ matrices, then one must use the $4^n$-dimensional Clifford algebra. The 4-dimensional Clifford algebraId:=IdentityMatrix(2): H:=Matrix([[1,0],[0,-1]]): X:=Matrix([[0,1],[0,0]]): Y:=Matrix([[0,0],[1,0]]):The 16-dimensional Clifford algebraII:=tens(Id,Id): IH:=tens(Id,H): HI:=tens(H,Id): HH:=tens(H,H): IX:=tens(Id,X): IY:=tens(Id,Y): XI:=tens(X,Id): YI:=tens(Y,Id): HX:=tens(H,X): HY:=tens(H,Y): XH:=tens(X,H): YH:=tens(Y,H): XX:=tens(X,X): XY:=tens(X,Y): YX:=tens(Y,X): YY:=tens(Y,Y):Some elements of the 64-dimensional Clifford algebraIII:=tens(Id,II): IIH:=tens(Id,IH): IHI:=tens(Id,HI): IHH:=tens(Id,HH): HII:=tens(H,II): HIH:=tens(H,IH): HHI:=tens(H,HI): HHH:=tens(H,HH): IIX:=tens(Id,IX): IHX:=tens(Id,HX): HIX:=tens(H,IX): HHX:=tens(H,HX): IIY:=tens(Id,IY): IHY:=tens(Id,HY): HIY:=tens(H,IY): HHY:=tens(H,HY): IXI:=tens(Id,XI): IXH:=tens(Id,XH): HXI:=tens(H,XI): HXH:=tens(H,XH): IYI:=tens(Id,YI): IYH:=tens(Id,YH): HYI:=tens(H,YI): HYH:=tens(H,YH): XII:=tens(X,II): XIH:=tens(X,IH): XHI:=tens(X,HI): XHH:=tens(X,HH): YII:=tens(Y,II): YIH:=tens(Y,IH): YHI:=tens(Y,HI): YHH:=tens(Y,HH): IYX:=tens(Id,YX): YIX:=tens(Y,IX): YHX:=tens(Y,HX): XHX:=tens(X,HX): XHY:=tens(X,HY): YHY:=tens(Y,HY): YXI:=tens(Y,XI): XXI:=tens(X,XI): YYI:=tens(Y,YI): IXX:=tens(IX,X): IYY:=tens(Id,YY): XYI:=tens(X,YI): IXY:=tens(Id,XY): Some elements of the 256-dimensional Clifford algebraIIII:=tens(Id,III): IIIH:=tens(Id,IIH): IIHI:=tens(Id,IHI): IHII:=tens(Id,HII): HIII:=tens(H,III): IIIX:=tens(Id,IIX): IXII:=tens(Id,XII): IXHH:=tens(Id,XHH): IIYX:=tens(Id,IYX): YXII:=tens(Y,XII): YHXH:=tens(Y,HXH): YHXI:=tens(Y,HXI): IYIX:=tens(Id,YIX): IYHX:=tens(Id,YHX):

These are included so that one may hope to build MDO realizations of Lie superalgebras which include these in their even components.

This is the infinitesimal version of the usual action of $SL(2)$ on holomorphic line bundles over $\mathbb{P}^1$.mx:=1: X0:=-x^2*p+mu*x: H0:=2*x*p-mu: Y0:=p:

mx:=1: X1:=-x^2*p-x*y*q-(y-x*z)*r+lambda*x: X2:=-mu*x*z-z*r*y+z^2*r*x-y^2*q+mu*y-x*y*p+y*lambda: X3:=y*p-z^2*r+mu*z: Y1:=p: Y2:=q: Y3:=x*q+r: H1:=2*x*p+y*q-z*r-lambda: H3:=y*q-x*p+2*z*r-mu:

mx:=1: X1:=y*r+x*(mu-x*p-y*q-z*r): X2:=y*(mu-x*p-y*q-z*r): X3:=-y*p-z*(-mu+x*p+y*q+z*r): X4:=z*p: Y1:=p+z*q: Y2:=q: Y3:=r-x*q: Y4:=x*r: H1:=2*x*p+2*y*q-mu: H4:=-x*p+z*r:

mx:=1: euleroperator:=x*p+y*q+z*r-mu: quadraticform:=x*z-y^2: X1:=r*quadraticform-x*euleroperator: X2:=-q*quadraticform-2*y*euleroperator: X3:=p*quadraticform-z*euleroperator: X4:=z*q+2*y*p: Y1:=p: Y2:=q: Y3:=r: Y4:=x*q+2*y*r: H2:=2*euleroperator: H4:=-2*x*p+2*z*r:

This is a canonical example of a Lie superalgebra of matrix differential operators. This uses the 4-dimensional Clifford algebra. dm:=2: The even subalgebra. The operator K is central in the even subalgebra. X0:=(-x^2*p+mu*x)*Id-(x/2)*(Id-H): H0:=(2*x*p-mu)*Id+(1/2)*(Id-H): Y0:=p*Id: K:=(Id-H): The odd operators. B1:=Y: B2:=p*X: A1:=x*Y: A2:=(x*p-mu)*X:

dm:=4:mx:=1: The even subalgebra (three dimensions of it). X0:=(mu*x-x^2*p)*II+x*(em(1,1)+2*em(2,2)+em(4,4)): H0:=(-mu+2*x*p)*II-(em(1,1)+2*em(2,2)+em(4,4)): Y0:=p*II: The odd operators: unassign('a1','a2','a3','a4'): unassign('b1','b2','b3','b4'): A1:=a1*em(1,3)+a2*em(2,4)+p*(a3*em(1,2)+a4*em(3,4)): B1:=p*(b1*em(3,1)+b2*em(4,2))+b3*em(2,1)+b4*em(4,3): A2:=Lie(X0,A1): B2:=Lie(X0,B1): a4:=-a2*a3/a1: b4:=-b2*b3/b1: b2:=a1*b1/a2:

dm:=4: mx:=2: The even subalgebra X0:=(mu*x-x^2*p)*II+2*x*(em(3,3)+em(4,4)): H0:=(2*x*p-mu)*II-2*(em(3,3)+em(4,4)): Y0:=p*II: The odd operators B:=-a1*em(1,2)+em(3,4): A0:=em(3,1)+a1*em(4,2)+a2*p*(em(2,1)-a1*em(4,3)): A1:=Lie(X0,A0): A2:=Lie(X0,A1):

dm:=4:mx:=2: The even subalgebra: X1:=IX: Y1:=IY: H1:=IH: X2:=(mu*x-x^2*p)*II-(x/2)*(II-HI): H2:=(-mu+2*x*p)*II+(1/2)*(II-HI): Y2:=p*II: The odd operators: B0:=XY+p*YY: B1:=XH+p*YH: B2:=XX+p*YX: A0:=x*XY+(x*p-mu)*YY: A1:=x*XH+(x*p-mu)*YH: A2:=x*XX+(x*p-mu)*YX:

dm:=4:mx:=2: The even subalgebra: X1:=(mu*x-x^2*p+2*x)*II+x*(HI-IH): H1:=(2*x*p-mu-2)*II-(HI-IH): Y1:=p*II: X2:=XX: H2:=(1/2)*(IH+HI): Y2:=YY: The odd operators. unassign('b1','b2','b3','b4'): B0:=b1*em(2,1)+b2*em(4,3)+p^2*(b3*em(4,2)+b4*em(3,1)): B1:=Lie(X1,B0): B2:=Lie(X1,B1)/2: A0:=Lie(X2,B0): A1:=Lie(X2,B1): A2:=Lie(X2,B2): A constraint in order to ensure Lie superalgebra structure. b3:=-b2*b4/b1:An alternate presentation of the odd operators, using the 16-dimensional Clifford algebra: B0:= IY +p^2*YH: B1:= x*IY +(-mu+x*p-2)*p*YH +p*YI: B2:=x^2*IY+(7+5*mu+mu^2-4*x*p-2*mu*x*p+x^2*p^2)*YH+(2*x*p-2*mu-5)*YI: A0:= XH -p^2*IX: A1:= x*XH -(-mu+x*p-2)*p*IX +p*HX: A2:=x^2*XH-(7+5*mu+mu^2-4*x*p-2*mu*x*p+x^2*p^2)*IX+(2*x*p-2*mu-5)*HX:

dm:=8:mx:=2: The even subalgebra: X1:=YXI+IYX: H1:=(1/2)*(IIH-HII): Y1:=XYI+IXY: X2:=(mu*x-x^2*p)*III+(1/2)*x*(3*III+HII+IHI+IIH): H2:=(2*x*p-mu)*III-(1/2)*(3*III+HII+IHI+IIH): Y2:=p*III: The odd operators: unassign('a1','a2','a3','a4','b1','b2','b3','b4'): B0:= a1*em(1,5)+a2*em(2,6)+a3*em(3,7)+a4*em(4,8) +p*(b1*em(2,1)+b2*em(4,3)+b3*em(6,5)+b4*em(8,7)): B1:=Lie(X1,B0): B2:=Lie(X1,B1)/2: A0:=Lie(X2,B0): A1:=Lie(X2,B1): A2:=Lie(X2,B2): Some superalgebra conditions on the undetermined coefficients: b1:=-a2*b3/a1: b4:=-b2*a3/a4: a3:=a2:b3:=b2:

This is a long example. Much to explain here....dm:=8:mx:=3: The even subalgebra: X1:=(mu*x-x^2*p)*III+2*x*(em(1,1)+2*em(2,2)+em(3,3)+2*em(4,4)+em(6,6)+em(8,8)): H1:=(2*x*p-mu)*III-2*(em(1,1)+2*em(2,2)+em(3,3)+2*em(4,4)+em(6,6)+em(8,8)): Y1:=p*III: X2:=(lambda*y-y^2*q)*III+y*(3*em(1,1)+2*em(2,2)+2*em(3,3)+em(4,4)+2*em(5,5)+em(6,6)+em(7,7)): H2:=(2*y*q-lambda)*III-(3*em(1,1)+2*em(2,2)+2*em(3,3)+em(4,4)+2*em(5,5)+em(6,6)+em(7,7)): Y2:=q*III: The odd operators: unassign('a1','a2','a3','a4','b1','b2','b3','b4','b5','c1','c2','c3','c4', 'd1','d2','d3','d4','e1','e2','e3','e4','e5','f1','f2','f3','f4'): B0:= a1*em(1,5)+a2*em(2,6)+a3*em(3,7)+a4*em(4,8)+ p*(b1*em(1,3)+b2*em(2,4)+b3*em(5,7)+b4*em(6,8)+b5*em(3,6))+ p^2*(c1*em(1,2)+c2*em(3,4)+c3*em(5,6)+c4*em(7,8))+ q*(d1*em(2,1)+d2*em(4,3)+d3*em(6,5)+d4*em(8,7))+ p*q*(e1*em(3,1)+e2*em(4,2)+e3*em(7,5)+e4*em(8,6)+e5*em(6,3))+ p^2*q*(f1*em(5,1)+f2*em(6,2)+f3*em(7,3)+f4*em(8,4)): B1:=Lie(X1,B0): B2:=Lie(X1,B1)/2: A0:=Lie(X2,B0): A1:=Lie(X2,B1): A2:=Lie(X2,B2): Constraints ensuring that B0^2=0. a1:=1:a2:=1:a3:=1:a4:=1: b3:=-b1:b4:=-b2: c3:=-c1:c4:=-c2: d3:=-d1:d4:=-d2: e3:=-e1:e4:=-e2: f3:=f1:f2:=f1:f4:=f1: c2:=-c1*b2/b1: d1:=-b2*d2/b1: e1:=b2*e2/b1: f1:=-b2*(e2*b1-c1*d2)/b1: A quantization phenomenon. If b5 and e5 are both zero, then these are also required in order to ensure that one has a Lie superalgebra: mu:=-3: lambda:=-5/2: An alternate presentation of the odd operators, explicitly using a Weyl-Clifford algebra. In this example, b5 and e5 are both zero. B0:=XII+p*HXI+(1/2)*p^2*HHX-q*HHY+p*q*HYI-(1/2)*p^2*q*YII: B1:=2*x*XII-IXI+(1+2*x*p)*HXI+HXH-p*IHX+(p+x*p^2)*HHX -2*x*q*HHY-q*IYI+(q+2*x*p*q)*HYI+q*HYH +(-p*q-x*p^2*q)*YII-p*q*YIH: B2:=x^2*XII-x*IXI+(x+x^2*p)*HXI+x*HXH +(-x*p-1/2)*IHX+((1/2)*x^2*p^2+x*p+1/2)*HHX -x^2*q*HHY-x*q*IYI+(x*q+x^2*p*q)*HYI+x*q*HYH +(-x*p*q-(1/2)*x^2*p^2*q-(1/2)*q)*YII+(-x*p*q-(1/2)*q)*YIH: A0:=y*XII+y*p*HXI+(1/2)*y*p^2*HHX +(1/2)*IHY+(1/2)*(HIY)+(-1/2-y*q)*HHY -(1/2)*p*IYI+((1/2)*p+y*p*q)*HYI-(1/2)*p*HYH +(-(1/2)*y*p^2*q-(1/4)*p^2)*YII+(1/4)*p^2*YIH+(1/4)*p^2*YHI: A1:=2*x*y*XII+y*(-IXI+(1+2*x*p)*HXI+HXH) +y*p*(-IHX+(1+x*p)*HHX) +x*(IHY+HIY-(1+2*y*q)*HHY) +(-1-x*p-y*q)*IYI+(1/2)*(1+2*x*p)*(1+2*y*q)*HYI+(-x*p+y*q)*HYH +(1/2)*p*(-(x*p+2*y*q+2*x*y*p*q)*YII+(x*p-2*y*q)*YIH+(1+x*p)*YHI+YHH): A2:=x^2*y*XII +x*y*(-IXI+(1+x*p)*HXI+HXH) +(1/2)*y*(-(1+2*x*p)*IHX+(1+2*x*p+x^2*p^2)*HHX) +(1/2)*x^2*(IHY+HIY-(1+2*y*q)*HHY) +(1/2)*x*(-(2+x*p+2*y*q)*IYI+(1+x*p)*(1+2*y*q)*HYI+(-x*p+2*y*q)*HYH) +(1/4)*((-2*y*q-x^2*p^2-4*x*y*p*q-2*x^2*y*p^2*q)*YII +(-2*y*q+x^2*p^2-4*x*y*p*q)*YIH+(1+2*x*p+x^2*p^2)*YHI+(1+2*x*p)*YHH)):A good choice of coefficients. Varying $\lambda$ and $\mu$ yields a ``dense'' subset of the finite-dimensional modules. a1:=1:a2:=1:a3:=1:a4:=1: b3:=-b1:b4:=-b2: c1:=1/4*b1*b5*(mu+2)^2/(mu+3):c2:=-1/4*b5*(mu+2)^2*b2/(mu+3): c3:=-1/4*b1*b5*(mu+4)^2/(mu+3):c4:=1/4*b2*b5*(mu+4)^2/(mu+3): d1:=-2*(mu+8+2*lambda)/((mu+4)*(mu+2)^2*(3+lambda)*b1*b5): d2:=2*(mu+8+2*lambda)/((mu+4)^2*(lambda+2)*b2*b5*(mu+2)): d3:=2*(mu+8+2*lambda)/((mu+4)*(mu+2)^2*(3+lambda)*b1*b5): d4:=-2*(mu+8+2*lambda)/((mu+4)^2*(lambda+2)*b2*b5*(mu+2)): e1:=2*(lambda+2)/((mu+2)*(mu+4)*b1*(3+lambda)): e2:=2/((mu+4)^2*b2): e3:=-2/((mu+2)^2*b1): e4:=-2*(3+lambda)/((mu+4)*(lambda+2)*b2*(mu+2)): e5:=2*(mu+8+2*lambda)*(2+2*lambda-mu)/((mu+4)^2*(mu+2)^2*(3+lambda)*b5*(lambda+2)): f1:=-1/2*(2+2*lambda-mu)/((mu+2)*(mu+3)*(3+lambda)): f2:=-1/2*(mu+2)*(2+2*lambda-mu)/((mu+4)^2*(mu+3)*(3+lambda)): f3:=-1/2*(mu+4)*(2+2*lambda-mu)/((lambda+2)*(mu+2)^2*(mu+3)): f4:=-1/2*(2+2*lambda-mu)/((mu+4)*(lambda+2)*(mu+3)):

The even subalgebra is the infinitesimal version of an action by $SO(5)$ on a line bundle over a 3-dimensional homogeneous space, the quotient by a maximal parabolic subgroup.dm:=2: mx:=2: The even subalgebra: X1:=(y*r+x*(mu-x*p-y*q-z*r))*Id-x*em(2,2): X2:=y*(mu-x*p-y*q-z*r)*Id-y*em(2,2): X3:=(-y*p-z*(-mu+x*p+y*q+z*r))*Id-z*em(2,2): X4:=z*p*Id: Y1:=(p+z*q)*Id: Y2:=q*Id: Y3:=(r-x*q)*Id: Y4:=x*r*Id: The odd operators: A0:=em(1,2)+q*em(2,1): A1:=Lie(X1,A0): A2:=Lie(X2,A0): A3:=Lie(X3,A0):

After seeing this, it is probably not hard to guess a general formula for $\mathrm{osp}(2r+1,2)$; one uses the Clifford algebra $\mathrm{Cl}(2r+2)$....dm:=8: mx:=1: The even subalgebra: X0:=(mu*x-x^2*p)*III-(1/2)*x*(III-HII): H0:=(2*x*p-mu)*III+(1/2)*(III-HII): Y0:=p*III: X1:=IXX: X2:=IXH: X3:=IXY: X4:=IIY: Y1:=IYY: Y2:=IYH: Y3:=IYX: Y4:=IIX: The odd operators: B0:=XYI+c0*p*YYI: B2:=XHH+c0*p*YHH: B1:=XHX+c0*p*YHX: B3:=XHY+c0*p*YHY: B4:=XXI+c0*p*YXI: A0:=x*XYI+c0*(x*p-mu)*YYI: A1:=x*XHX+c0*(x*p-mu)*YHX: A2:=x*XHH+c0*(x*p-mu)*YHH: A3:=x*XHY+c0*(x*p-mu)*YHY: A4:=x*XXI+c0*(x*p-mu)*YXI:

dm:=4:mx:=2: The even subalgebra: X1:=(y*r+x*(mu-x*p-y*q-z*r))*II+x*(em(1,1)+2*em(2,2)+em(4,4)): X2:=y*(mu-x*p-y*q-z*r)*II+y*(em(1,1)+2*em(2,2)+em(4,4)): X3:=(-y*p-z*(-mu+x*p+y*q+z*r))*II+z*(em(1,1)+2*em(2,2)+em(4,4)): X4:=z*p*II: Y1:=(p+z*q)*II: Y2:=q*II: Y3:=(r-x*q)*II: Y4:=x*r*II: The odd operators: unassign('a1','a2','a3','a4','b1','b2','b3','b4'): A0:=a1*em(2,1)+a2*em(4,3)+q*(a3*em(3,1)+a4*em(4,2)): A1:=Lie(X3,A0): A2:=Lie(X1,A0): A3:=Lie(X3,A2): B0:=q*(b1*em(1,2)+b2*em(3,4))+b3*em(1,3)+b4*em(2,4): B1:=Lie(X3,B0): B2:=Lie(X1,B0): B3:=Lie(X3,B2): Constraints which ensure Lie superalgebra structure: a4:=-a2*a3/a1: b1:=1/a1: b2:=1/a2: b3:=1/a3: b4:=-a1/(a2*a3):

This realization has a quantization phenomenon which is relatively easy to see. The value of $\mu$ determines the isomorphism class of the Lie superalgebra, but it also determines the highest weight of the representation. The dimension of the representation is $4\mu+2$ when $\mu$ is a positive integer. There are three types of singular values for $\mu$. First, if $\mu\in\{0,-1\}$, then the Lie superalgebra degenerates into a copy of $\mathrm{spl}(2,2)$ with an extraneous summand of $\mathfrak{sl}_2$. Second, if $\mu\in\{1,-2,-1/2\}$, then the Lie superalgebra is isomorphic to $\mathrm{osp}(4,2)$. Third, if $\omega$ is a primitive 3rd root of unity, then the structural constants are ``balanced'' in that the outer automorphism group of the Lie superalgebra is the full symmetric group on 3 letters.dm:=4:mx:=1: The even subalgebra: X0:=(mu*x-x^2*p)*II-(1/2)*x*(II+HH): H0:=(2*x*p-mu)*II+(1/2)*(II+HH): Y0:=p*II: X1:=XX: H1:=(HI+IH)/2: Y1:=YY: X2:=XY: H2:=(HI-IH)/2: Y2:=YX: The odd operators: B0:=em(3,1)+p*em(4,2): B1:=em(3,4)-p*em(1,2): B2:=em(2,1)-p*em(4,3): B3:=em(2,4)+p*em(1,3): A0:=x*em(3,1)+(x*p-mu)*em(4,2): A1:=x*em(3,4)-(x*p-mu)*em(1,2): A2:=x*em(2,1)-(x*p-mu)*em(4,3): A3:=x*em(2,4)+(x*p-mu)*em(1,3):

This is a one-parameter family of mutually inequivalent Lie superalgebras. Perhaps due to the existence of this parameter, there seems to be a lot of freedom for experimentation. The even subalgebra is defined concisely, as compared to the definition of the 8 odd operators. Here is an experiment: Leave all the even vectors, X0, H0, ..., H2 as they are, and then vary the 20 coefficients a0, a1, ..., d1 used to define B0. The operator B0 is assumed to be a lowest-weight vector for the even subalgebra; in other words, it is annihilated by Y0, Y1, and Y2. Since B0 is therefore a generator for this 8-dimensional representation, the remaining 7 operators are obtained by applying the raising operators X0, X1, and X2 naturally enough. As it turns out, 8 constraints on these coefficients are requied in order to ensure that B0 is a lowest-weight vector, and these are specified below. Hence, only 12 parameters remain.The even subalgebra. dm:=16: mx:=2: X0:=(mu*x-x^2*p)*IIII+(1/2)*x*(IIIH+IIHI+IHII+HIII): H0:=(2*x*p-mu)*IIII-(1/2)*(IIIH+IIHI+IHII+HIII): Y0:=p*IIII: X1:=IIYX+YXII: Y1:=Transpose(X1): X2:=IYHX+YHXI: Y2:=Transpose(X2): H1:=Lie(X1,Y1): H2:=Lie(X2,Y2): The odd operators. There are 20 undetermined coefficients, as described above. unassign('a0','a1','a2','a3','a4','a5','a6','a7','c0','c1', 'b0','b1','b2','b3','b4','b5','b6','b7','d0','d1'): B0:=a0*em(1,9)+a1*em(2,10)+a2*em(3,11)+a3*em(4,12) +a4*em(5,13)+a5*em(6,14)+a6*em(7,15)+a7*em(8,16) +c0*em(2,7)+c1*em(10,15) +p*(b0*em(2,1)+b1*em(4,3)+b2*em(6,5)+b3*em(8,7) +b4*em(10,9)+b5*em(12,11)+b6*em(14,13)+b7*em(16,15) +d0*em(7,9)+d1*em(8,10)):B1:=Lie(X1,B0): B1:=Lie(X1,B0): B2:=Lie(X2,B0): B3:=Lie(X2,B1): A0:=Lie(X0,B0): A1:=Lie(X0,B1): A2:=Lie(X0,B2): A3:=Lie(X0,B3): The constraints required to ensure that B0 is annihilated by Y1 and Y2. (It is automatically annihilated by Y0.) a2:=a1+c0:b2:=d0+b4:a5:=a6+c1:b5:=b3+d1: a1:=a4+c0:b4:=b1+d0:a6:=c1+a3:b3:=b6+d1:These constraints, in addition to those above, ensure that Sym(B0,B0) is zero. b7:=-a0*b0/a7: b0:=-1/2*(a4*b1-b6*a3)/a0: b6:=-(a4*b1+4*d0*c0+2*d0*a4+2*b1*c0)/a3: d0:=-d1*c1/c0: a3:=-c1*(2*c0+a4)/c0:A particularly good choice of coefficients. Varying c0 yields all elements of the family. The singular values of c0 are $\{0,1,1/2,\omega\}$, where $\omega$ is a primitive 6th root of unity. If c0 is $\omega$, then the Lie superalgebra degenerates into $\mathrm{spl}(2,2)$, where the Lie superalgebra generated by the odd operators does not contain the MDO's X0, H0, and Y0. It's also interesting to note how varying the paramter $\mu$ can affect the structure. a2:=a1+c0:b2:=d0+b4:a5:=a6+c1:b5:=b3+d1: a1:=a4+c0:b4:=b1+d0:a6:=c1+a3:b3:=b6+d1: b7:=-a0*b0/a7: b0:=-1/2*(a4*b1-b6*a3)/a0: b6:=-(a4*b1+4*d0*c0+2*d0*a4+2*b1*c0)/a3: d0:=-d1*c1/c0: a3:=-c1*(2*c0+a4)/c0: c1:=1: d1:=1: a4:=1-2*c0: b1:=1: a7:=1: a0:=1:

mx:=2: dm:=8: X1:=(-x^2*p-x*y*q-(y-x*z)*r +lambda*x)*III +x*(2*em(1,1)+em(2,2)+2*em(3,3)+em(4,4)+em(5,5)+em(7,7)): X2:=(-x*y*p-y^2*q-z*(y-x*z)*r +lambda*y+mu*(y-x*z))*III +y*(2*em(1,1)+em(2,2)+2*em(3,3)+em(4,4)+em(5,5)+em(7,7)) +(y-x*z)*(em(1,1)+2*em(2,2)+em(4,4)+em(5,5)+2*em(6,6)+em(8,8)): X3:=(y*p-z^2*r +mu*z)*III +z*(em(1,1)+2*em(2,2)+em(4,4)+em(5,5)+2*em(6,6)+em(8,8)): Y1:=(p)*III: Y2:=(q)*III: Y3:=(x*q+r)*III: H1:=(2*x*p+y*q-z*r-lambda)*III -(2*em(1,1)+em(2,2)+2*em(3,3)+em(4,4)+em(5,5)+em(7,7)): H3:=(y*q-x*p+2*z*r-mu)*III -(em(1,1)+2*em(2,2)+em(4,4)+em(5,5)+2*em(6,6)+em(8,8)):unassign('a1','a2','a3','a4','a5','a6','b1','b2','b3','b4','b5','b6'): A1:=a1*em(1,5)+a2*em(2,6)+a3*em(3,7)+a4*em(4,8)+a5*em(1,4)+a6*em(5,8) +(p+z*q)*(b1*em(2,1)+b2*em(4,3)+b3*em(6,5)+b4*em(8,7)+b6*em(5,3)+b7*em(6,4)): # +q*(c1*em(3,1)+c2*em(4,2)+c3*em(7,5)+c4*em(8,6)+c5*em(5,2)+c6*em(7,4)): A2:=Lie(X1,A1): A3:=Lie(X2,A1):