جبر هوپف

جبر هوپف Hopf algebra هوأحد فروع الجبر التجريدي، مسمى على اسم هاينز هوپف. وله استخدامات عدة ضمن نظريات ميكانيكا الكم.

أمثلة

	تعتمد على	Comultiplication	Counit	Antipode	Commutative	Cocommutative	ملاحظات
group algebra KG	group G	Δ(g) = g ⊗ g for all g in G	ε(g) = 1 for all g in G	S(g) = g⁻¹ for all g in G	if and only if G is abelian	yes
functions f from a finite group to K, K^G (with pointwise addition and multiplication)	finite group G	Δ(f)(x,y) = f(xy)	ε(f) = f(1_G)	S(f)(x) = f(x⁻¹)	yes	if and only if G is commutative
Representative functions on a compact group	compact group G	Δ(f)(x,y) = f(xy)	ε(f) = f(1_G)	S(f)(x) = f(x⁻¹)	yes	if and only if G is commutative	Conversely, every commutative involutive reduced Hopf algebra over C with a finite Haar integral arises in this way, giving one formulation of Tannaka–Krein duality.
Regular functions on an algebraic group		Δ(f)(x,y) = f(xy)	ε(f) = f(1_G)	S(f)(x) = f(x⁻¹)	yes	if and only if G is commutative	Conversely, every commutative Hopf algebra over a field arises from a group scheme in this way, giving an antiequivalence of categories.
Tensor algebra T(V)	vector space V	Δ(x) = x ⊗ 1 + 1 ⊗ x, x in V, Δ(1) = 1 ⊗ 1	ε(x) = 0	S(x) = −x for all x in 'T¹(V) (and extended to higher tensor powers)	If and only if dim(V)=0,1	yes	symmetric algebra and exterior algebra (which are quotients of the tensor algebra) are also Hopf algebras with this definition of the comultiplication, counit and antipode
Universal enveloping algebra U(g)	Lie algebra g	Δ(x) = x ⊗ 1 + 1 ⊗ x for every x in g (this rule is compatible with commutators and can therefore be uniquely extended to all of U)	ε(x) = 0 for all x in g (again, extended to U)	S(x) = −x	if and only if g is abelian	yes
Sweedler's Hopf algebra H=K[c, x]/c² = 1, x² = 0 and xc = −cx.	K is a field with characteristic different from 2	Δ(c) = c ⊗ c, Δ(x) = c ⊗ x + x ⊗ 1, Δ(1) = 1 ⊗ 1	ε(c) = 1 and ε(x) = 0	S(c) = c⁻¹ = c and S(x) = −cx	no	no	The underlying vector space is generated by {1, c, x, cx and thus has dimension 4. This is the smallest example of a Hopf algebra that is both non-commutative and non-cocommutative.
ring of symmetric functions		in terms of complete homogeneous symmetric functions h_k (k ≥ 1): Δ(h_k) = 1 ⊗ h_k + h₁ ⊗ h_k−1 + ... + h_k−1 ⊗ h₁ + h_k ⊗ 1.	ε(h_k) = 0	S(h_k) = (−1)^ke_k	yes	yes

Note that functions on a finite group can be identified with the group ring, though these are more naturally thought of as dual – the group ring consists of finite sums of elements, and thus pairs with functions on the group by evaluating the function on the summed elements.

انظر أيضاً

Quasitriangular Hopf algebra
Algebra/set analogy
Representation theory of Hopf algebras
Ribbon Hopf algebra
Superalgebra
Supergroup
Anyonic Lie algebra
Sweedler's Hopf algebra
Hopf algebra of permutations
Milnor–Moore theorem

الهامش

^ The finiteness of G implies that K^G ⊗ K^G is naturally isomorphic to K^GxG. This is used in the above formula for the comultiplication. For infinite groups G, K^G ⊗ K^G is a proper subset of K^GxG. In this case the space of functions with finite support can be endowed with a Hopf algebra structure.
^ Hochschild, G (1965), Structure of Lie groups, Holden-Day, pp. 14–32
^ Jantzen, Jens Carsten (2003), Representations of algebraic groups, Mathematical Surveys and Monographs, 107 (2nd ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3527-2 , section 2.3
^ See Michiel Hazewinkel, Symmetric Functions, Noncommutative Symmetric Functions, and Quasisymmetric Functions, Acta Applicandae Mathematica, January 2003, Volume 75, Issue 1-3, pp 55–83