Linear aspects of quasigroup triality Alex W. Nowak Iowa State University Dissertation Defense April 13, 2020 A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Summary 1 Quasigroups and triality 2 Linear quasigroup theory 3 Modules over Mendelsohn triple systems 4 Abelian groups in MTS 5 Beyond set-theoretic triality A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Quasigroups and triality Quasigroups and triality A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Quasigroups and triality Triality: a combinatorial perspective (Q; ·) is a quasigroup when T = f(x; y; x · y) j (x; y) 2 Q2g has the Latin square property: 8(x1; x2; x3); (y1; y2; y3) 2 T; jf1 ≤ i ≤ 3 j xi = yigj 6= 2. g For all g 2 S3, T = f(x1g; x2g; x3g) j (x1; x2; x3) 2 T g also has the Latin square property. If H is a subgroup of the kernel of this permutation action, then T is H-symmetric. A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Quasigroups and triality Triality: a combinatorial perspective (Q; ·) is a quasigroup when T = f(x; y; x · y) j (x; y) 2 Q2g has the Latin square property: 8(x1; x2; x3); (y1; y2; y3) 2 T; jf1 ≤ i ≤ 3 j xi = yigj 6= 2. g For all g 2 S3, T = f(x1g; x2g; x3g) j (x1; x2; x3) 2 T g also has the Latin square property. If H is a subgroup of the kernel of this permutation action, then T is H-symmetric. A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Quasigroups and triality Triality: a combinatorial perspective (Q; ·) is a quasigroup when T = f(x; y; x · y) j (x; y) 2 Q2g has the Latin square property: 8(x1; x2; x3); (y1; y2; y3) 2 T; jf1 ≤ i ≤ 3 j xi = yigj 6= 2. g For all g 2 S3, T = f(x1g; x2g; x3g) j (x1; x2; x3) 2 T g also has the Latin square property. If H is a subgroup of the kernel of this permutation action, then T is H-symmetric. A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Quasigroups and triality Triality: an algebraic perspective (Q; ·; =; n) is a quasigroup when all the following hold: (IL) yn(y · x) = x; (IR) x = (x · y)=y; (SL) y · (ynx) = x; (SR) x = (x=y) · y: (12) (Q; ·; =; n) o / (Q; ◦; nn; ==) (23) 3; ck (23) s{ #+ (Q; n; ==; ·) (Q; nn; ◦; =) g 7 (12) (12) ' w (Q; ==; n; ◦) ks +3 (Q; =; ·; nn) (23) A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Quasigroups and triality Triality: an algebraic perspective (Q; ·; =; n) is a quasigroup when all the following hold: (IL) yn(y · x) = x; (IR) x = (x · y)=y; (SL) y · (ynx) = x; (SR) x = (x=y) · y: (12) (Q; ·; =; n) o / (Q; ◦; nn; ==) (23) 3; ck (23) s{ #+ (Q; n; ==; ·) (Q; nn; ◦; =) g 7 (12) (12) ' w (Q; ==; n; ◦) ks +3 (Q; =; ·; nn) (23) A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Quasigroups and triality Multiplication groups (Q; ·) is a quasigroup iff R(q): x 7! x · q and L(q): x 7! q · x are bijections. The group Mlt(Q) = hR(q);L(q)iSQ acts transitively on Q. ` Q is a subquasigroup of Q[X] = Q VhXiV and the subgroup of Mlt(Q[X]) generated by R(Q) [ L(Q) is the universal multiplication group of Q in V, U(Q; V). A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Quasigroups and triality Multiplication groups (Q; ·) is a quasigroup iff R(q): x 7! x · q and L(q): x 7! q · x are bijections. The group Mlt(Q) = hR(q);L(q)iSQ acts transitively on Q. ` Q is a subquasigroup of Q[X] = Q VhXiV and the subgroup of Mlt(Q[X]) generated by R(Q) [ L(Q) is the universal multiplication group of Q in V, U(Q; V). A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Quasigroups and triality Multiplication groups (Q; ·) is a quasigroup iff R(q): x 7! x · q and L(q): x 7! q · x are bijections. The group Mlt(Q) = hR(q);L(q)iSQ acts transitively on Q. ` Q is a subquasigroup of Q[X] = Q VhXiV and the subgroup of Mlt(Q[X]) generated by R(Q) [ L(Q) is the universal multiplication group of Q in V, U(Q; V). A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Quasigroups and triality Universal stabilizers U(Q; V) also acts transitively on Q, so for any e 2 Q, define U(Q; V)e to be the universal stabilizer of Q in V. rq L(rq=e) 7 g L(r) L(q=e) e 3+ q R(enq) R(enqr) R(r) ' qr w −1 −1 Define Te(q) = R(enq)L(q=e) , Re(q; r) = R(enq)R(r)R(enqr) , −1 and Le(q; r) = L(q=e)L(r)L(rq=e) −1 U(Q; V)e will act on the fiber p feg in a split extension p : E ! Q. A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Quasigroups and triality Universal stabilizers U(Q; V) also acts transitively on Q, so for any e 2 Q, define U(Q; V)e to be the universal stabilizer of Q in V. rq L(rq=e) 7 g L(r) L(q=e) e 3+ q R(enq) R(enqr) R(r) ' qr w −1 −1 Define Te(q) = R(enq)L(q=e) , Re(q; r) = R(enq)R(r)R(enqr) , −1 and Le(q; r) = L(q=e)L(r)L(rq=e) −1 U(Q; V)e will act on the fiber p feg in a split extension p : E ! Q. A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Quasigroups and triality Universal stabilizers U(Q; V) also acts transitively on Q, so for any e 2 Q, define U(Q; V)e to be the universal stabilizer of Q in V. rq L(rq=e) 7 g L(r) L(q=e) e 3+ q R(enq) R(enqr) R(r) ' qr w −1 −1 Define Te(q) = R(enq)L(q=e) , Re(q; r) = R(enq)R(r)R(enqr) , −1 and Le(q; r) = L(q=e)L(r)L(rq=e) −1 U(Q; V)e will act on the fiber p feg in a split extension p : E ! Q. A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Quasigroups and triality Universal stabilizers U(Q; V) also acts transitively on Q, so for any e 2 Q, define U(Q; V)e to be the universal stabilizer of Q in V. rq L(rq=e) 7 g L(r) L(q=e) e 3+ q R(enq) R(enqr) R(r) ' qr w −1 −1 Define Te(q) = R(enq)L(q=e) , Re(q; r) = R(enq)R(r)R(enqr) , −1 and Le(q; r) = L(q=e)L(r)L(rq=e) −1 U(Q; V)e will act on the fiber p feg in a split extension p : E ! Q. A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Quasigroups and triality H-symmetry classes Q LS RS C P TS A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Linear quasigroup theory Linear quasigroup theory A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Linear quasigroup theory H-symmetry: modules as models Q: ZhR; Li-modules −1 −1 −1 −1 x · y = xR + yL, x=y = xR − yLR , xny = yL − xRL ±1 C(xy = yx): Z[R ]-modules −1 −1 x · y = (x + y)R, x=y = xR − y, xny = yR − x ±1 LS(y · yx = x): Z[R ]-modules −1 x · y = xR − y = xny, x=y = (x + y)R , 3 P(y · xy = x): Z[R]=(R + 1)-modules −1 −1 x · y = xR + yR , x=y = xR + yR = xny TS: Z-modules x · y = x=y = xny = −(x + y) A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Linear quasigroup theory H-symmetry: modules as models Q: ZhR; Li-modules −1 −1 −1 −1 x · y = xR + yL, x=y = xR − yLR , xny = yL − xRL ±1 C(xy = yx): Z[R ]-modules −1 −1 x · y = (x + y)R, x=y = xR − y, xny = yR − x ±1 LS(y · yx = x): Z[R ]-modules −1 x · y = xR − y = xny, x=y = (x + y)R , 3 P(y · xy = x): Z[R]=(R + 1)-modules −1 −1 x · y = xR + yR , x=y = xR + yR = xny TS: Z-modules x · y = x=y = xny = −(x + y) A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Linear quasigroup theory H-symmetry: modules as models Q: ZhR; Li-modules −1 −1 −1 −1 x · y = xR + yL, x=y = xR − yLR , xny = yL − xRL ±1 C(xy = yx): Z[R ]-modules −1 −1 x · y = (x + y)R, x=y = xR − y, xny = yR − x ±1 LS(y · yx = x): Z[R ]-modules −1 x · y = xR − y = xny, x=y = (x + y)R , 3 P(y · xy = x): Z[R]=(R + 1)-modules −1 −1 x · y = xR + yR , x=y = xR + yR = xny TS: Z-modules x · y = x=y = xny = −(x + y) A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Linear quasigroup theory H-symmetry: modules as models Q: ZhR; Li-modules −1 −1 −1 −1 x · y = xR + yL, x=y = xR − yLR , xny = yL − xRL ±1 C(xy = yx): Z[R ]-modules −1 −1 x · y = (x + y)R, x=y = xR − y, xny = yR − x ±1 LS(y · yx = x): Z[R ]-modules −1 x · y = xR − y = xny, x=y = (x + y)R , 3 P(y · xy = x): Z[R]=(R + 1)-modules −1 −1 x · y = xR + yR , x=y = xR + yR = xny TS: Z-modules x · y = x=y = xny = −(x + y) A.