#
# file:    skryabin.gap
#
# purpose: Defines the 15-dimensional simple Skryabin algebra
#
# created: Pasha Zusmanovich <pasha.zusmanovich@gmail.com> Oct 13 2014
#
# latest revision history: 
# Mar 12 2021  added 2-envelope; "names" for the standard basis elements;
#


# the 2-parametric family of 15-dimensional Skryabin algebras in p=2,
# as deformation of the corresponding semisimple algebra
# K - base field, should be extenions of GF(2)
SkryabinAlgebra := function (beta, delta, K)
	local T;

	T := EmptySCTable (15, Zero(K), "antisymmetric"); 

	#  1 e \otimes 1 
	#  2 e \otimes x
	#  3 e \otimes x^2
	#  4 e \otimes x^3
	#  5 h \otimes 1 
	#  6 h \otimes x
	#  7 h \otimes x^2
	#  8 h \otimes x^3
	#  9 f \otimes 1 
	# 10 f \otimes x
	# 11 f \otimes x^2
	# 12 f \otimes x^3
	# 13 g \otimes 1
	# 14 g \otimes x
	# 15 d

	# {e \otimes 1, e \otimes x} = beta g \otimes x
	SetEntrySCTable (T, 1, 2, [beta, 14]);
  
	# {e \otimes 1, e \otimes x^2} = delta g \otimes 1
	SetEntrySCTable (T, 1, 3, [delta, 13]);

	# {e \otimes 1, e \otimes x^3} = delta g \otimes x + d
	SetEntrySCTable (T, 1, 4, [delta, 14, 1, 15]);

	# {e \otimes x, e \otimes x^2} = delta g \otimes x + d
	SetEntrySCTable (T, 2, 3, [delta, 14, 1, 15]);

	# {e \otimes x, e \otimes x^3} = h \otimes 1
	SetEntrySCTable (T, 2, 4, [1, 5]);

	# {e \otimes x^2, e \otimes x^3} = h \otimes x
	SetEntrySCTable (T, 3, 4, [1, 6]);

	# {e \otimes 1, h \otimes 1} = e \otimes 1
	SetEntrySCTable (T, 1, 5, [1, 1]);

	# {e \otimes 1, h \otimes x} = e \otimes x
	SetEntrySCTable (T, 1, 6, [1, 2]);

	# {e \otimes 1, h \otimes x^2} = e \otimes x^2
	SetEntrySCTable (T, 1, 7, [1, 3]);

	# {e \otimes 1, h \otimes x^3} = e \otimes x^3
	SetEntrySCTable (T, 1, 8, [1, 4]);

	# {e \otimes x, h \otimes 1} = e \otimes x
	SetEntrySCTable (T, 2, 5, [1, 2]);

	# {e \otimes x, h \otimes x} = 0

	# {e \otimes x, h \otimes x^2} = e \otimes x^3
	SetEntrySCTable (T, 2, 7, [1, 4]);

	# {e \otimes x, h \otimes x^3} = f \otimes 1
	SetEntrySCTable (T, 2, 8, [1, 9]);

	# {e \otimes x^2, h \otimes 1} = e \otimes x^2
	SetEntrySCTable (T, 3, 5, [1, 3]);

	# {e \otimes x^2, h \otimes x} = e \otimes x^3
	SetEntrySCTable (T, 3, 6, [1, 4]);

	# {e \otimes x^2, h \otimes x^2} = 0

	# {e \otimes x^2, h \otimes x^3} = f \otimes x
	SetEntrySCTable (T, 3, 8, [1, 10]);

	# {e \otimes x^3, h \otimes 1} = e \otimes x^3
	SetEntrySCTable (T, 4, 5, [1, 4]);

	# {e \otimes x^3, h \otimes x} = 0
	# {e \otimes x^3, h \otimes x^2} = 0

	# {e \otimes x^3, h \otimes x^3} = f \otimes x^2
	SetEntrySCTable (T, 4, 8, [1, 11]);

	# {e \otimes 1, f \otimes 1} = h \otimes 1
	SetEntrySCTable (T, 1, 9, [1, 5]);

	# {e \otimes 1, f \otimes x} = h \otimes x
	SetEntrySCTable (T, 1, 10, [1, 6]);

	# {e \otimes 1, f \otimes x^2} = h \otimes x^2
	SetEntrySCTable (T, 1, 11, [1, 7]);

	# {e \otimes 1, f \otimes x^3} = h \otimes x^3
	SetEntrySCTable (T, 1, 12, [1, 8]);

	# {e \otimes x, f \otimes 1} = h \otimes x
	SetEntrySCTable (T, 2, 9, [1, 6]);

	# {e \otimes x, f \otimes x} = 0

	# {e \otimes x, f \otimes x^2} = h \otimes x^3
	SetEntrySCTable (T, 2, 11, [1, 8]);

	# {e \otimes x, f \otimes x^3} = g \otimes 1
	SetEntrySCTable (T, 2, 12, [1, 13]);

	# {e \otimes x^2, f \otimes 1} = h \otimes x^2
	SetEntrySCTable (T, 3, 9, [1, 7]);

	# {e \otimes x^2, f \otimes x} = h \otimes x^3
	SetEntrySCTable (T, 3, 10, [1, 8]);

	# {e \otimes x^2, f \otimes x^2} = 0

	# {e \otimes x^2, f \otimes x^3} = g \otimes x
	SetEntrySCTable (T, 3, 12, [1, 14]);

	# {e \otimes x^3, f \otimes 1} = h \otimes x^3
	SetEntrySCTable (T, 4, 9, [1, 8]);

	# {e \otimes x^3, f \otimes x} = g \otimes 1
	SetEntrySCTable (T, 4, 10, [1, 13]);

	# {e \otimes x^3, f \otimes x^2} = g \otimes x
	SetEntrySCTable (T, 4, 11, [1, 14]);

	# {e \otimes x^3, f \otimes x^3} = 0

	# {e \otimes 1, g \otimes 1} = f \otimes 1
	SetEntrySCTable (T, 1, 13, [1, 9]);

	# {e \otimes 1, g \otimes x} = f \otimes x
	SetEntrySCTable (T, 1, 14, [1, 10]);

	# {e \otimes x, g \otimes 1} = f \otimes x
	SetEntrySCTable (T, 2, 13, [1, 10]);

	# {e \otimes x, g \otimes x} = 0

	# {e \otimes x^2, g \otimes 1} = f \otimes x^2
	SetEntrySCTable (T, 3, 13, [1, 11]);

	# {e \otimes x^2, g \otimes x} = f \otimes x^3
	SetEntrySCTable (T, 3, 14, [1, 12]);

	# {e \otimes x^3, g \otimes 1} = f \otimes x^3
	SetEntrySCTable (T, 4, 13, [1, 12]);

	# {e \otimes x^3, g \otimes x} = 0
	# {h \otimes 1, h \otimes x} = 0
	# {h \otimes 1, h \otimes x^2} = 0
	# {h \otimes 1, h \otimes x^3} = 0
	# {h \otimes x, h \otimes x^2} = 0

	# {h \otimes x, h \otimes x^3} = g \otimes 1
	SetEntrySCTable (T, 6, 8, [1, 13]);

	# {h \otimes x^2, h \otimes x^3} = g \otimes x
	SetEntrySCTable (T, 7, 8, [1, 14]);

	# {h \otimes 1, f \otimes 1} = f \otimes 1
	SetEntrySCTable (T, 5, 9, [1, 9]);

	# {h \otimes 1, f \otimes x} = f \otimes x
	SetEntrySCTable (T, 5, 10, [1, 10]);

	# {h \otimes 1, f \otimes x^2} = f \otimes x^2
	SetEntrySCTable (T, 5, 11, [1, 11]);

	# {h \otimes 1, f \otimes x^3} = f \otimes x^3
	SetEntrySCTable (T, 5, 12, [1, 12]);

	# {h \otimes x, f \otimes 1} = f \otimes x
	SetEntrySCTable (T, 6, 9, [1, 10]);

	# {h \otimes x, f \otimes x} = 0

	# {h \otimes x, f \otimes x^2} = f \otimes x^3
	SetEntrySCTable (T, 6, 11, [1, 12]);

	# {h \otimes x, f \otimes x^3} = 0

	# {h \otimes x^2, f \otimes 1} = f \otimes x^2
	SetEntrySCTable (T, 7, 9, [1, 11]);

	# {h \otimes x^2, f \otimes x} = f \otimes x^3
	SetEntrySCTable (T, 7, 10, [1, 12]);

	# {h \otimes x^2, f \otimes x^2} = 0
	# {h \otimes x^2, f \otimes x^3} = 0

	# {h \otimes x^3, f \otimes 1} = f \otimes x^3
	SetEntrySCTable (T, 8, 9, [1, 12]);

	# {h \otimes x^3, f \otimes x} = 0
	# {h \otimes x^3, f \otimes x^2} = 0
	# {h \otimes x^3, f \otimes x^3} = 0
	# {h \otimes 1, g \otimes 1} = 0
	# {h \otimes 1, g \otimes x} = 0
	# {h \otimes x, g \otimes 1} = 0
	# {h \otimes x, g \otimes x} = 0
	# {h \otimes x^2, g \otimes 1} = 0
	# {h \otimes x^2, g \otimes x} = 0
	# {h \otimes x^3, g \otimes 1} = 0
	# {h \otimes x^3, g \otimes x} = 0
	# {f \otimes 1, f \otimes x} = 0
	# {f \otimes 1, f \otimes x^2} = 0
	# {f \otimes 1, f \otimes x^3} = 0
	# {f \otimes x, f \otimes x^2} = 0
	# {f \otimes x, f \otimes x^3} = 0
	# {f \otimes x^2, f \otimes x^3} = 0
	# {f \otimes 1, g \otimes 1} = 0
	# {f \otimes 1, g \otimes x} = 0
	# {f \otimes x, g \otimes 1} = 0
	# {f \otimes x, g \otimes x} = 0
	# {f \otimes x^2, g \otimes 1} = 0
	# {f \otimes x^2, g \otimes x} = 0
	# {f \otimes x^3, g \otimes 1} = 0
	# {f \otimes x^3, g \otimes x} = 0
	# {g \otimes 1, g \otimes x} = 0

	# {e \otimes 1, d} = beta f \otimes x^3
	SetEntrySCTable (T, 1, 15, [beta, 12]);

	# {e \otimes x, d} = e \otimes 1
	SetEntrySCTable (T, 2, 15, [1, 1]);

	# {e \otimes x^2, d} = e \otimes x
	SetEntrySCTable (T, 3, 15, [1, 2]);

	# {e \otimes x^3, d} = e \otimes x^2
	SetEntrySCTable (T, 4, 15, [1, 3]);

	# {h \otimes 1, d} = 0

	# {h \otimes x, d} = h \otimes 1
	SetEntrySCTable (T, 6, 15, [1, 5]);

	# {h \otimes x^2, d} = h \otimes x
	SetEntrySCTable (T, 7, 15, [1, 6]);

	# {h \otimes x^3, d} = h \otimes x^2
	SetEntrySCTable (T, 8, 15, [1, 7]);

	# {f \otimes 1, d} = 0

	# {f \otimes x, d} = f \otimes 1
	SetEntrySCTable (T, 10, 15, [1, 9]);

	# {f \otimes x^2, d} = f \otimes x
	SetEntrySCTable (T, 11, 15, [1, 10]);

	# {f \otimes x^3, d} = f \otimes x^2
	SetEntrySCTable (T, 12, 15, [1, 11]);

	# {g \otimes 1, d} = 0

	# {g \otimes x, d} = g \otimes 1
	SetEntrySCTable (T, 14, 15, [1, 13]);

	return (LieAlgebraByStructureConstants (K, T));
end;




# defines the 19-dimensional 2-envelope of the Skryabin algebra, 
# as given by multiplication in the standard basis
# K - base field of characteristic 2
2EnvelopeSkryabinAlgebra := function (K)
	local T;

	T := EmptySCTable (19, Zero(K), "antisymmetric"); 

	#  1 b1
	#  2 b2
	#  3 b3
	#  4 b4
	#  5 b5
	#  6 b6
	#  7 b7
	#  8 b8
	#  9 b9
	# 10 c1
	# 11 c2
	# 12 c3
	# 13 c4
	# 14 c5
	# 15 d
	# 16 b1^2
	# 17 b4^2
	# 18 b7^2
	# 19 c3^2

	# nonzero multiplications in the standard basis

	# [b1,b2] = b3
	SetEntrySCTable (T, 1, 2, [1,3]);

	# [b1,b3] = b1
	SetEntrySCTable (T, 1, 3, [1,1]);

	# [b1,b5] = b6
	SetEntrySCTable (T, 1, 5, [1,6]);

	# [b1,b6] = b4
	SetEntrySCTable (T, 1, 6, [1,4]);

	# [b1,b8] = b9
	SetEntrySCTable (T, 1, 8, [1,9]);

	# [b1,b9] = b7
	SetEntrySCTable (T, 1, 9, [1,7]);

	# [b1,c1] = d
	SetEntrySCTable (T, 1, 10, [1,15]);

	# [b1,c2] = c3
	SetEntrySCTable (T, 1, 11, [1,12]);

	# [b1,c3] = c1
	SetEntrySCTable (T, 1, 12, [1,10]);

	# [b1,c4] = b2
	SetEntrySCTable (T, 1, 13, [1,2]);

	# [b1,c5] = b5
	SetEntrySCTable (T, 1, 14, [1,5]);

	# [b2,b3] = b2
	SetEntrySCTable (T, 2, 3, [1,2]);

	# [b2,b4] = b6
	SetEntrySCTable (T, 2, 4, [1,6]);

	# [b2,b6] = b5
	SetEntrySCTable (T, 2, 6, [1,5]);

	# [b2,b7] = b9
	SetEntrySCTable (T, 2, 7, [1,9]);

	# [b2,b9] = b8
	SetEntrySCTable (T, 2, 9, [1,8]);

	# [b2,c1] = c3
	SetEntrySCTable (T, 2, 10, [1,12]);

	# [b2,c3] = c2
	SetEntrySCTable (T, 2, 12, [1,11]);

	# [b3,b4] = b4
	SetEntrySCTable (T, 3, 4, [1,4]);

	# [b3,b5] = b5
	SetEntrySCTable (T, 3, 5, [1,5]);

	# [b3,b7] = b7
	SetEntrySCTable (T, 3, 7, [1,7]);

	# [b3,b8] = b8
	SetEntrySCTable (T, 3, 8, [1,8]);

	# [b3,c1] = c1
	SetEntrySCTable (T, 3, 10, [1,10]);

	# [b3,c2] = c2
	SetEntrySCTable (T, 3, 11, [1,11]);

	# [b4,b7] = d
	SetEntrySCTable (T, 4, 7, [1,15]);

	# [b4,b8] = c3
	SetEntrySCTable (T, 4, 8, [1,12]);

	# [b4,b9] = c1
	SetEntrySCTable (T, 4, 9, [1,10]);

	# [b4,c1] = b3
	SetEntrySCTable (T, 4, 10, [1,3]);

	# [b4,c2] = c4
	SetEntrySCTable (T, 4, 11, [1,13]);

	# [b4,c3] = b2
	SetEntrySCTable (T, 4, 12, [1,2]);

	# [b4,c4] = b5
	SetEntrySCTable (T, 4, 13, [1,5]);

	# [b4,d] = b1
	SetEntrySCTable (T, 4, 15, [1,1]);

	# [b5,b7] = c3
	SetEntrySCTable (T, 5, 7, [1,12]);

	# [b5,b9] = c2
	SetEntrySCTable (T, 5, 9, [1,11]);

	# [b5,c1] = c4
	SetEntrySCTable (T, 5, 10, [1,13]);

	# [b5,d] = b2
	SetEntrySCTable (T, 5, 15, [1,2]);

	# [b6,b7] = c1
	SetEntrySCTable (T, 6, 7, [1,10]);

	# [b6,b8] = c2
	SetEntrySCTable (T, 6, 8, [1,11]);

	# [b6,c3] = c4
	SetEntrySCTable (T, 6, 12, [1,13]);

	# [b6,d] = b3
	SetEntrySCTable (T, 6, 15, [1,3]);

	# [b7,c1] = b6
	SetEntrySCTable (T, 7, 10, [1,6]);

	# [b7,c2] = c5
	SetEntrySCTable (T, 7, 11, [1,14]);

	# [b7,c3] = b5
	SetEntrySCTable (T, 7, 12, [1,5]);

	# [b7,c4] = b8
	SetEntrySCTable (T, 7, 13, [1,8]);

	# [b7,c5] = c2
	SetEntrySCTable (T, 7, 14, [1,11]);

	# [b7,d] = b4
	SetEntrySCTable (T, 7, 15, [1,4]);

	# [b8,c1] = c5
	SetEntrySCTable (T, 8, 10, [1,14]);

	# [b8,d] = b5
	SetEntrySCTable (T, 8, 15, [1,5]);

	# [b9,c3] = c5
	SetEntrySCTable (T, 9, 12, [1,14]);

	# [b9,d] = b6
	SetEntrySCTable (T, 9, 15, [1,6]);

	# [c1,c3] = b8
	SetEntrySCTable (T, 10, 12, [1,8]);

	# [c1,c4] = c2
	SetEntrySCTable (T, 10, 13, [1,11]);

	# [c1,d] = b7
	SetEntrySCTable (T, 10, 15, [1,7]);

	# [c2,d] = b8
	SetEntrySCTable (T, 11, 15, [1,8]);

	# [c3,d] = b9
	SetEntrySCTable (T, 12, 15, [1,9]);

	# [c5,d] = c4
	SetEntrySCTable (T, 14, 15, [1,13]);

	# [b1,c_3^2] = b8
	SetEntrySCTable (T, 1, 19, [1,8]);
	
	# [b2,b1^2] = b1
	SetEntrySCTable (T, 2, 16, [1,1]);

	# [b4,b7^2] = b4
	SetEntrySCTable (T, 4, 18, [1,4]);

	# [b4,c3^2] = c2
	SetEntrySCTable (T, 4, 19, [1,11]);

	# [b5,b1^2] = b4
	SetEntrySCTable (T, 5, 16, [1,4]);

	# [b5,b7^2] = b5
	SetEntrySCTable (T, 5, 18, [1,5]);

	# [b6,b7^2] = b6
	SetEntrySCTable (T, 6, 18, [1,6]);

	# [b7,b4^2] = b1
	SetEntrySCTable (T, 7, 17, [1,1]);

	# [b8,b1^2] = b7
	SetEntrySCTable (T, 8, 16, [1,7]);

	# [b8,b4^2] = b2
	SetEntrySCTable (T, 8, 17, [1,2]);

	# [b9,b4^2] = b3
	SetEntrySCTable (T, 9, 17, [1,3]);

	# [c1,b4^2] = b4
	SetEntrySCTable (T, 10, 17, [1,4]);

	# [c1,b7^2] = c1
	SetEntrySCTable (T, 10, 18, [1,10]);

	# [c2,b1^2] = c1
	SetEntrySCTable (T, 11, 16, [1,10]);

	# [c2,b4^2] = b5
	SetEntrySCTable (T, 11, 17, [1,5]);

	# [c2,b7^2] = c2
	SetEntrySCTable (T, 11, 18, [1,11]);

	# [c3,b1^2] = d
	SetEntrySCTable (T, 12, 16, [1,15]);

	# [c3,b4^2] = b6
	SetEntrySCTable (T, 12, 17, [1,6]);

	# [c3,b7^2] = c3
	SetEntrySCTable (T, 12, 18, [1,12]);

	# [c4,b1^2] = b3
	SetEntrySCTable (T, 13, 16, [1,3]);

	# [c5,b1^2] = b6
	SetEntrySCTable (T, 14, 16, [1,6]);

	# [c5,b7^2] = c5
	SetEntrySCTable (T, 14, 18, [1,14]);

	# [d,b7^2] = d
	SetEntrySCTable (T, 15, 18, [1,15]);

	# [d,c3^2] = c5
	SetEntrySCTable (T, 15, 19, [1,14]);

	# [b1^2,c3^2] = b9
	SetEntrySCTable (T, 16, 19, [1,9]);

	# [b4^2,c3^2] = c4
	SetEntrySCTable (T, 17, 19, [1,13]);

	return (LieAlgebraByStructureConstants (K, T));
end;




# now we define:
# L: the Skryabin algebra over the given field K of char. 2
# L2: the 2-envelope of the Skryabin algebra
# bas: the standard basis of L2
# b1,...,b9,c1,...c5,d, b12, b42, b72, c32: elements of the standard basis

K := GF(2);
#K := GF(2^2);
#K := GF(2^4);
L2 := 2EnvelopeSkryabinAlgebra (K);
bas := CanonicalBasis (L2);

b1  := bas[1]; 
b2  := bas[2]; 
b3  := bas[3];  
b4  := bas[4];  
b5  := bas[5]; 
b6  := bas[6]; 
b7  := bas[7];  
b8  := bas[8];  
b9  := bas[9]; 
c1  := bas[10];
c2  := bas[11]; 
c3  := bas[12]; 
c4  := bas[13];
c5  := bas[14];
d   := bas[15];
b12 := bas[16];  # b1^2
b42 := bas[17];  # b4^2
b72 := bas[18];  # b7^2
c32 := bas[19];  # c3^2

L := Subalgebra (L2, 
                 [b1, b2, b3, b4, b5, b6, b7, b8, b9, c1, c2, c3, c4, c5, d], 
                 "basis");

# names for the standard basis elements in L2
# used to display results in a human-friendly form
basis_names := ["b1", "b2", "b3", "b4", "b5", "b6", "b7", "b8", "b9", 
                "c1", "c2", "c3", "c4", "c5", "d", 
                "b1^2", "b4^2", "b7^2", "c3^2"];

# eof