The upper triangular algebra loop of degree 4

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1 The upper triangular algebra loop of degree 4 Michael Munywoki Joint With K.W. Johnson and J.D.H. Smith Iowa State University Department of Mathematics Third Mile High Conference on Nonassociative Mathematics August 13, 2013

2 Quasigroups Loops Loop conjugacy classes Introduction We define a natural loop structure on the set U 4 of unimodular upper-triangular matrices over a given field. It is shown that the loop is non-associative and nilpotent, of class 3. The conjugacy classes are computed and its seen that they lie between group conjugacy classes and superclasses.

3 Quasigroups Loops Loop conjugacy classes Introduction We define a natural loop structure on the set U 4 of unimodular upper-triangular matrices over a given field. It is shown that the loop is non-associative and nilpotent, of class 3. The conjugacy classes are computed and its seen that they lie between group conjugacy classes and superclasses.

4 Quasigroups Loops Loop conjugacy classes Quasigroups Definition A quasigroup, written as Q or (Q,, /, \) is a set Q equipped with three binary operations of multiplication, right division / and left division \, satisfying the identities: (SL) y (y\x) = x ; (SR) x = (x/y) y : (IL) y\(y x) = x ; (IR) x = (x y)/y.

5 Quasigroups Loops Loop conjugacy classes Loops Definition A Loop, is a quasigroup Q with an identity element 1 such that 1 x = x = x 1 for all x in Q.

6 Quasigroups Loops Loop conjugacy classes Loop conjugacy classes The inner multiplication group Inn Q is the stabilizer Mlt Q 1 in Mlt Q of the identity element 1. For example, if Q is a group, then Inn Q is the inner automorphism group of Q, although for a general loop Q, elements of Inn Q are not necessarily automorphisms of Q.

7 Quasigroups Loops Loop conjugacy classes Loop conjugacy classes The inner multiplication group Inn Q is the stabilizer Mlt Q 1 in Mlt Q of the identity element 1. For example, if Q is a group, then Inn Q is the inner automorphism group of Q, although for a general loop Q, elements of Inn Q are not necessarily automorphisms of Q.

8 Quasigroups Loops Loop conjugacy classes For elements q, r of Q, define the conjugation T (q) = R(q)L(q) 1, (1) the right inner mapping R(q, r) = R(q)R(r)R(qr) 1, (2) and the left inner mapping L(q, r) = L(q)L(r)L(rq) 1 (3) in Mlt Q 1.

9 Quasigroups Loops Loop conjugacy classes Collectively, (1)-(3) are known as inner mappings. The orbits of Inn Q on Q are defined as the (loop) conjugacy classes of Q. If Q is a group, the loop conjugacy classes of Q are just the usual group conjugacy classes.

10 The loop multiplication The loop multiplication The right division The left division The algebra loop Inner mappings Properties of the algebra loop Consider matrices 1 x 12 x 13 x 14 x = 0 1 x 23 x x and 1 y 12 y 13 y 14 y = 0 1 y 23 y y with entries x ij, y ij from a field F. We define the set of such matrices as U 4.

11 The loop multiplication The right division The left division The algebra loop Inner mappings Properties of the algebra loop Then the quasigroup product is 1 x 12 x 13 x 14 1 y 12 y 13 y x 23 x x y 23 y y 34 (4) x 12 + y 12 x 13 + y 13 + x 12 y 23 [x y] 14 = 0 1 x 23 + y 23 x 24 + y 24 + x 23 y x 34 + y with [x y] 14 = x 14 + y 14 + x 12 y 24 + x 13 y 34 + x 12 x 23 y 34 + x 12 y 23 y 34 (5) as the (1,4)-th entry of the product

12 The loop multiplication The right division The left division The algebra loop Inner mappings Properties of the algebra loop The summands in (5) correspond to paths of respective lengths 1, 2, 3 from 1 to 4 in the chain 1 < 2 < 3 < 4, with labels chosen from x over the former part of the path, and y over the latter part. The other entries in the product have a similar (but simpler) structure. Note that the above product has the matrix I 4 as its identity element.

13 The loop multiplication The right division The left division The algebra loop Inner mappings Properties of the algebra loop The summands in (5) correspond to paths of respective lengths 1, 2, 3 from 1 to 4 in the chain 1 < 2 < 3 < 4, with labels chosen from x over the former part of the path, and y over the latter part. The other entries in the product have a similar (but simpler) structure. Note that the above product has the matrix I 4 as its identity element.

14 The right division The loop multiplication The right division The left division The algebra loop Inner mappings Properties of the algebra loop With matrices x and y as above, consider the right division z = y/x, namely a solution to the equation z x = y. 1 z 12 z 13 z 14 z = 0 1 z 23 z z Lemma There is a unique solution z = yr(x) 1 to z x = y.

15 The right division The loop multiplication The right division The left division The algebra loop Inner mappings Properties of the algebra loop With matrices x and y as above, consider the right division z = y/x, namely a solution to the equation z x = y. 1 z 12 z 13 z 14 z = 0 1 z 23 z z Lemma There is a unique solution z = yr(x) 1 to z x = y.

16 The loop multiplication The right division The left division The algebra loop Inner mappings Properties of the algebra loop Proof. The entries z ij, for 1 i < j 4, are obtained by recursion on the length j i of the path from i to j in the chain 1 < 2 < 3 < 4. Paths of length 1: z 12 = (y 12 x 12 ), z 23 = (y 23 x 23 ), z 34 = (y 34 x 34 ). Paths of length 2: z 13 + x 13 + z 12 x 23 = y 13, so z 13 = y 13 x 13 z 12 x 23 = (y 13 x 13 ) (y 12 x 12 )x 23. Similarly, z 24 = (y 24 x 24 ) (y 23 x 23 )x 34.

17 The loop multiplication The right division The left division The algebra loop Inner mappings Properties of the algebra loop Proof (Cont.) The path of length 3: z 14 + x 14 + z 12 x 24 + z 13 x 34 + z 12 z 23 x 34 + z 12 x 23 x 34 = y 14, so z 14 = y 14 x 14 z 12 x 24 z 13 x 34 z 12 z 23 x 34 z 12 x 23 x 34 = (y 14 x 14 ) + (y 12 x 12 )( x 24 ) + (y 13 x 13 )( x 34 ) + (y 12 x 12 )(y 23 x 23 )( x 34 ). Note that in each case, the coefficient z ij is uniquely specified in terms of x and y.

18 The left division The loop multiplication The right division The left division The algebra loop Inner mappings Properties of the algebra loop With matrices x and y as above, consider the left division z = x\y, namely a solution to the equation x z = y. 1 z 12 z 13 z 14 z = 0 1 z 23 z z Lemma There is a unique solution z = yl(x) 1 to x z = y.

19 The left division The loop multiplication The right division The left division The algebra loop Inner mappings Properties of the algebra loop With matrices x and y as above, consider the left division z = x\y, namely a solution to the equation x z = y. 1 z 12 z 13 z 14 z = 0 1 z 23 z z Lemma There is a unique solution z = yl(x) 1 to x z = y.

20 The loop multiplication The right division The left division The algebra loop Inner mappings Properties of the algebra loop Proof. The proof is similar to the right division lemma.

21 The algebra loop The loop multiplication The right division The left division The algebra loop Inner mappings Properties of the algebra loop Proposition With the product (4), the algebra group U 4 over a field F forms a loop. Definition The loop U 4 is known as the upper triangular algebra loop of degree 4.

22 The algebra loop The loop multiplication The right division The left division The algebra loop Inner mappings Properties of the algebra loop Proposition With the product (4), the algebra group U 4 over a field F forms a loop. Definition The loop U 4 is known as the upper triangular algebra loop of degree 4.

23 Inner mappings The loop multiplication The right division The left division The algebra loop Inner mappings Properties of the algebra loop Within the loop U 4, the effects of the inner mappings (1)-(3) may be computed using the work of (4), the right division and the left division. Consider elements x = [x ij ], q = [q ij ], r = [r ij ] of U 4. Then xt (q) = xr(q)l(q) 1 = q\xq 1 x 12 x 13 + x 12 q 12 x 23 q 23 [xt (q)] x 23 x 24 + x 23 q 23 = x 34 q x

24 The loop multiplication The right division The left division The algebra loop Inner mappings Properties of the algebra loop with [xt (q)] 14 = x 14 + x 12 q 12 x 24 q 24 + x 13 q 13 x 34 q 34 + x 12 q 12 x 23 x 34 x 23 q 34 + x 12 q 12 x 23 q 34 q 23 q 34. Furthermore, one has [xr(q, r)] 14 = [(xq r)/qr] 14 = x 14 + q 12 x 23 r 34 x 12 r 23 q 34 and [xl(q, r)] 14 = [rq\(r qx)] 14 = x 14 + r 12 x 23 q 34 q 12 r 23 x 34, with [xr(q, r)] ij = [xl(q, r)] ij = x ij for 1 i < j 4 and j i < 3.

25 The loop multiplication The right division The left division The algebra loop Inner mappings Properties of the algebra loop with [xt (q)] 14 = x 14 + x 12 q 12 x 24 q 24 + x 13 q 13 x 34 q 34 + x 12 q 12 x 23 x 34 x 23 q 34 + x 12 q 12 x 23 q 34 q 23 q 34. Furthermore, one has [xr(q, r)] 14 = [(xq r)/qr] 14 = x 14 + q 12 x 23 r 34 x 12 r 23 q 34 and [xl(q, r)] 14 = [rq\(r qx)] 14 = x 14 + r 12 x 23 q 34 q 12 r 23 x 34, with [xr(q, r)] ij = [xl(q, r)] ij = x ij for 1 i < j 4 and j i < 3.

26 Properties of the algebra loop The loop multiplication The right division The left division The algebra loop Inner mappings Properties of the algebra loop Definition For given 1 l < m 4, the elementary element E lm of U 4 is defined by { [E lm 1 if i = l and j = m, or if i = j ; ] ij = 0 otherwise.

27 The loop multiplication The right division The left division The algebra loop Inner mappings Properties of the algebra loop Properties of the algebra loop (Cont.) Theorem Over a given field F, the upper triangular algebra loop U 4 of degree 4 is neither commutative nor associative.

28 The loop multiplication The right division The left division The algebra loop Inner mappings Properties of the algebra loop Proof. It was already observed that U 4 forms a loop. Consider elementary elements x = E 23, q = E 12, r = E 34 of U 4. The computations of the inner mappings show that [xt (q)] 13 = x 13 + x 12 q 12 x 23 q 23 = 1 0 = x 13, so the loop is not commutative, and [xr(q, r)] 14 = x 14 + q 12 x 23 r 34 x 12 r 23 q 34 = 1 0 = x 14, so the loop is not associative.

29 Nilpotence and center Zeroes on the superdiagonal The remaining cases Summary Consider the upper triangular algebra loop U 4 of degree 4 over a given field F. We demonstrate that U 4 is nilpotent, and determine the loop conjugacy class of each element x = [x ij ] of U 4.

30 Nilpotence and center Zeroes on the superdiagonal The remaining cases Summary Recall that the center Z or Z(Q) of a loop Q is the set {z Q q, r Q, zt (q) = zr(q, r) = zl(q, r) = z} In other words, the center Z(Q) consists precisely of the set of elements of Q which lie in singleton conjugacy classes.

31 Nilpotence and center Zeroes on the superdiagonal The remaining cases Summary Recall that the center Z or Z(Q) of a loop Q is the set {z Q q, r Q, zt (q) = zr(q, r) = zl(q, r) = z} In other words, the center Z(Q) consists precisely of the set of elements of Q which lie in singleton conjugacy classes.

32 Nilpotence and center Zeroes on the superdiagonal The remaining cases Summary Center of U 4 Proposition The set { x = [xij ] U 4 xij = 0 if 1 j i < 3 } (6) forms the center of U 4. Thus the center of U 4 is the set x 14 Z(Q) = , x 14 F

33 Nilpotence and center Zeroes on the superdiagonal The remaining cases Summary Center of U 4 Proposition The set { x = [xij ] U 4 xij = 0 if 1 j i < 3 } (6) forms the center of U 4. Thus the center of U 4 is the set x 14 Z(Q) = , x 14 F

34 Nilpotence and center Zeroes on the superdiagonal The remaining cases Summary Nilpotence of U 4 Theorem The loop U 4 is nilpotent, of class 3. Indeed, Z 4 k (U 4 ) = {x = [x ij ] U 4 x ij = 0 if 1 j i < k} (7) for 1 k 4.

35 Nilpotence and center Zeroes on the superdiagonal The remaining cases Summary Zeroes on the superdiagonal Proposition If the vector (x 13, x 24 ) is non-zero, the conjugacy class of 1 0 x 13 x 14 x = x is 1 0 x 13 a x a F. (8)

36 Nilpotence and center Zeroes on the superdiagonal The remaining cases Summary The remaining cases Proposition If x 23 is non-zero, the conjugacy class of x is 1 x 12 b a 0 1 x 23 c x 34 a, b, c F. (9)

37 Nilpotence and center Zeroes on the superdiagonal The remaining cases Summary The remaining cases(cont.) Proposition Suppose that x 23 = 0, while the vector (x 12, x 34 ) is non-zero. Then the conjugacy class of x is 1 x 12 x 13 + bx 12 a x 24 bx x 34 a, b F. (10)

38 Nilpotence and center Zeroes on the superdiagonal The remaining cases Summary The remaining cases(cont.) Corollary Suppose that x 23 = 0, while the vector (x 12, x 34 ) is non-zero. Then the conjugacy class of x has cardinality F 2. Remark The Corollary shows that if F is finite, the size of the loop conjugacy class of E 12 + E 34 1 is F 2. On the other hand, the computations (in the language of Diaconis/Isaacs) show that the superclass of E 12 + E 34 1 has size F 3. Thus the loop conjugacy classes in U 4 do not necessarily coincide with the superclasses.

39 Nilpotence and center Zeroes on the superdiagonal The remaining cases Summary The remaining cases(cont.) Corollary Suppose that x 23 = 0, while the vector (x 12, x 34 ) is non-zero. Then the conjugacy class of x has cardinality F 2. Remark The Corollary shows that if F is finite, the size of the loop conjugacy class of E 12 + E 34 1 is F 2. On the other hand, the computations (in the language of Diaconis/Isaacs) show that the superclass of E 12 + E 34 1 has size F 3. Thus the loop conjugacy classes in U 4 do not necessarily coincide with the superclasses.

40 Nilpotence and center Zeroes on the superdiagonal The remaining cases Summary Type of element Size of class Number of classes q 0 F 0 0 q q 2 1 F F 0 F q 3 q 2 (q 1) F F 0 F q 2 q(q 2 1)

41 Nilpotence and center Zeroes on the superdiagonal The remaining cases Summary Summary The Table lists the sizes and number of each kind of loop conjugacy class in U 4. The element types are identified by the pattern of matrix entries above the diagonal, in conjunction with the reference to the proposition giving the full description of the type. The symbol is used as a wild card to denote a (potentially) non-zero field element. As a pattern entry, the symbol F denotes arbitrary elements of F that appear in the class. The symbol q stands for the cardinality of the underlying field F.

42 I M. Aguiar et al., Supercharacters, symmetric functions in non-commuting variables, and related Hopf algebras, Adv. Math. 229 (2012), C.A.M. André, Basic characters of the unitriangular group, J. Algebra 175 (1995), Bruck, R.H., Contributions to the theory of loops, Trans. Amer. Math. Soc. 60 (1946), Bruck, R.H., A Survey of Binary Systems, Springer, Berlin, 1958.

43 II P. Diaconis and I.M. Isaacs, Supercharacters and superclasses for algebra groups, Trans. Amer. Math. Soc. 360 (2008), K.W. Johnson and J.D.H. Smith, Characters of finite quasigroups III: quotients and fusion, Eur. J. Comb. 10 (1989), J.D.H. Smith, An Introduction to Quasigroups and Their Representations, Chapman and Hall/CRC, Boca Raton, FL, J.D.H. Smith and A.B. Romanowska, Post-Modern Algebra, Wiley, New York, NY, 1999.

The upper triangular algebra loop of degree 4

Comment.Math.Univ.Carolin. 55,4(2014) 457 470 457 The upper triangular algebra loop of degree 4 K.W. Johnson, M. Munywoki, Jonathan D.H. Smith Abstract. A natural loop structure is defined on the set U